Integrability, multi-soliton and rational solutions, and dynamical analysis for a relativistic Toda
本站小编 Free考研考试/2022-01-02
Meng-Li Qin, Xiao-Yong Wen,∗, Cui-Lian YuanSchool of Applied Science, Beijing Information Science and Technology University, Beijing 100192, China
First author contact:Author to whom any correspondence should be addressed Received:2021-02-8Revised:2021-03-21Accepted:2021-04-6Online:2021-05-06
Abstract Under investigation in this paper is a relativistic Toda lattice system with one perturbation parameter $\alpha$ abbreviated as RTL_($\alpha$) system by Suris, which may describe the motions of particles in lattices interacting through an exponential interaction force. First of all, an integrable lattice hierarchy associated with an RTL_($\alpha$) system is constructed, from which some relevant integrable properties such as Hamiltonian structures, Liouville integrability and conservation laws are investigated. Secondly, the discrete generalized (m, 2N − m)-fold Darboux transformation is constructed to derive multi-soliton solutions, higher-order rational and semi-rational solutions, and their mixed solutions of an RTL_($\alpha$) system. The soliton elastic interactions and details of rational solutions are analyzed via the graphics and asymptotic analysis. Finally, soliton dynamical evolutions are investigated via numerical simulations, showing that a small noise has very little effect on the soliton propagation. These results may provide new insight into nonlinear lattice dynamics described by RTL_($\alpha$) system. Keywords：RTL_($\alpha$) system;Hamiltonian structures;discrete generalized (m, 2N−m)-fold Darboux transformation;soliton and rational solutions;asymptotic analysis
PDF (0KB)MetadataMetricsRelated articlesExportEndNote|Ris|BibtexFavorite Cite this article Meng-Li Qin, Xiao-Yong Wen, Cui-Lian Yuan. Integrability, multi-soliton and rational solutions, and dynamical analysis for a relativistic Toda lattice system with one perturbation parameter. Communications in Theoretical Physics, 2021, 73(6): 065003- doi:10.1088/1572-9494/abf4f7
1. Introduction
Solitons are a class of nonlinear localized waves with particle-like properties [1]. In the field of mathematical physics, the studies of soliton are mainly divided into continuous cases and discrete cases. Nonlinear partial differential equations (NPDEs) and nonlinear differential-difference equations (NDDEs) are usually used to describe the dynamical behaviors of solitons in continuous and discrete cases, respectively [1]. A soliton in NDDEs is sometimes also referred to as a lattice-soliton [1]. There have been many studies on continuous solitons in NPDEs which are used to describe some physical phenomena, such as nonlinear surface gravity waves propagating over a horizontal sea bed, propagation of a short wave in ferromagnets, optical logic switches and ultrashort pulse lasers, the nonlinear dynamics of optical solitons in nonlinear optics, shallow water wave, etc [2-7]. Nevertheless, the discrete soliton in discrete NDDEs is still inadequate compared to their continuous part. In recent years, NDDEs, which are thought as spatially discrete counterparts of NPDEs, have received widespread attention because they may be used to describe various physical phenomena in different fields such as nonlinear lattice dynamics, pulses in biological chains, ladder type electric circuits, population dynamics, nonlinear optics, plasma physics and so on [1, 8-14]. Seeking discrete exact solutions, in particular discrete soliton solutions, plays a crucial role in understanding the physical phenomena described by NDDEs [8-14]. Many effective methods for solving NDDEs have been proposed in the literature like the discrete inverse scattering transformation [12], discrete Hirota transformation [13, 14], and discrete Darboux transformation (DT) [15-19] and discrete N-fold DT [20-26], etc. Among them, the discrete N-fold DT based upon Lax pair of integrable NDDEs is a powerful technique to get multi-soliton solutions without complicated iterative procedure [20-26]. And recently, a generalized technique on the basis of the usual discrete N-fold DT has also been proposed [27-29]. In comparison with the usual N-fold DT only providing the soliton solutions, the discrete generalized DT technique not only expresses the ordinary multi-soliton solutions but also produces new higher-order rational solutions, semi-rational solutions and their mixed solutions.
One of the most famous and deeply researched integrable NDDEs is undoubtedly the Toda lattice which was originally discovered by Toda [1, 8, 9]. Toda lattice may describe a lattice of particles interacting with nearest neighbors via forces exponentially depending on distances (see figure 1), and it is the first example of integrable NDDEs which also has important guiding significance in the study of nonlinear waves and ergode theory [1]. Regarding Toda lattice and its relativistic version, some integrable properties have been studied such as Lax pairs [30-33], Hamiltonian structures [33-36], conservation laws [37], the exp-function method [38], the rotational expansion method [39], the tanh-method [40], Bäcklund transformation [41], DT [16, 19, 24, 42] and so on. Toda lattice has an integrable generalization of relativistic version nowadays called the relativistic Toda lattice (RTL) with one perturbation parameter $\alpha$ proposed by Ruijsenaars [8, 30, 31] in the form of the following Newtonian equations of motion: $\begin{eqnarray}\begin{array}{rcl}{x}_{n,{tt}} & = & (1-\alpha {x}_{n+1,t})(1-\alpha {x}_{n,t})\displaystyle \frac{{{\rm{e}}}^{{x}_{n+1}-{x}_{n}}}{1+{\alpha }^{2}{{\rm{e}}}^{{x}_{n+1}-{x}_{n}}}\\ & & -\,(1-\alpha {x}_{n,t})(1+\alpha {x}_{n-1,t})\displaystyle \frac{{{\rm{e}}}^{{x}_{n}-{x}_{n-1}}}{1+{\alpha }^{2}{{\rm{e}}}^{{x}_{n}-{x}_{n-1}}},\end{array}\end{eqnarray}$ which may be viewed as a one-parameter perturbation of Toda lattice, where xn = x(n, t) is the function of the discrete spatial variable n and time variable t, in which n ∈ Z, t ∈ R, ${x}_{n,t}=\tfrac{{\rm{d}}x}{{\rm{d}}t},{x}_{n,{tt}}=\tfrac{{{\rm{d}}}^{2}x}{{\rm{d}}{t}^{2}}$, and $\alpha$ is a small parameter whose physical meaning is the inverse speed of light. When $\alpha$ → 0 or $\alpha$ = 0, equation (1) is just the usual Toda lattice. Introducing the following transformations $\begin{eqnarray}\begin{array}{rcl}1+\alpha {b}_{n} & = & \displaystyle \frac{1+{\alpha }^{2}{{\rm{e}}}^{{x}_{n}-{x}_{n-1}}}{1-\alpha {x}_{n,t}},\\ {a}_{n} & = & (1+\alpha {b}_{n}){{\rm{e}}}^{{x}_{n+1}-{x}_{n}},\end{array}\end{eqnarray}$ the time evolutions of an and bn are governed by the following equations of motion [8]: $\begin{eqnarray}\left\{\begin{array}{l}{a}_{n,t}=\tfrac{{a}_{n}({b}_{n+1}-{b}_{n})}{(\alpha {b}_{n}+1)(\alpha {b}_{n+1}+1)},\\ {b}_{n,t}=\tfrac{{a}_{n}(\alpha {b}_{n-1}+1)-{a}_{n-1}(\alpha {b}_{n+1}+1)}{(\alpha {b}_{n-1}+1)(\alpha {b}_{n+1}+1)},\end{array}\right.\end{eqnarray}$ where an = a(n, t) and bn = b(n, t) are the functions of discrete and time variables n and t respectively, and ${a}_{n,t}=\tfrac{{\rm{d}}{a}_{n}}{{\rm{d}}t},{b}_{n,t}=\tfrac{{\rm{d}}{b}_{n}}{{\rm{d}}t}$. Equation (3) turns out to have much richer structure than the original Newtonian motion equation (1). Equation (3) is the 'minus first'flow of the RTL ($\alpha$) hierarchy which is assigned an abbreviation as RTL_($\alpha$) system by Suris [8]. Hereafter, we will use this abbreviated name for equation (1) or (3) which will be mainly discussed in this paper. Here we need to point out that RTL_($\alpha$) system (3) with the denominator studied in this paper is different from RTL systems without the denominator in [16, 19, 30-32, 34-36, 38-42]. The 2 × 2 Lax representations for equation (3) and its discretization form have been given in [8]. For the convenience of later discussions, we here list the 2 × 2 Lax pair for equation (3) as follows: $\begin{eqnarray}\begin{array}{l}E{\phi }_{n}={U}_{n}{\phi }_{n}={U}_{n}(u,\lambda ){\phi }_{n}\\ =\left(\begin{array}{cc}{\lambda }^{2}(\alpha {b}_{n+1}+1)-1 & \alpha {a}_{n}\lambda \\ -\alpha \lambda & 0\end{array}\right){\phi }_{n},\end{array}\end{eqnarray}$
$\begin{eqnarray}{\phi }_{n,t}={V}_{n}{\phi }_{n}=\left(\begin{array}{cc}0 & -\tfrac{\lambda {a}_{n}}{\alpha {b}_{n+1}+1}\\ \tfrac{\lambda }{\alpha {b}_{n}+\lambda } & \tfrac{{\lambda }^{2}}{\alpha }-\tfrac{1}{\alpha (\alpha {b}_{n}+1)}\end{array}\right){\phi }_{n},\end{eqnarray}$ which is somewhat different from ones in [8], where λ is the spectral parameter independent of time t, E is the shift operator defined by Ef(n, t) = f(n + 1, t), E−1f(n, t) = f(n − 1, t), $u={({a}_{n},{b}_{n+1})}^{{\rm{T}}}$ are the potential functions of variables n, t, and ${\phi }_{n}={({\varphi }_{n},{\psi }_{n})}^{{\rm{T}}}$ (T means transpose) is an eigenfunction vector. The compatibility condition Un,t = (EVn)Un − UnVn between the spatial part (4) and time evolution part (5) of Lax pair yields equation (3). To the best of the authors'knowledge, some integrable properties via the Tu scheme [33], various exact solutions via the discrete generalized (m, 2N − m)-fold DT, and soliton dynamical behaviors via numerical simulations of equation (3) have not been reported in literature before. Different from the discrete generalized (m, N − m)-fold DT choosing the number m of different spectral parameter from 1 to N in [27-29], we will extend the discrete generalized (m, N − m)-fold DT to the discrete generalized (m, 2N − m)-fold DT by selecting the spectral parameter number m between 1 and 2N so that we may give some new exact solutions.
Therefore, in this paper, we will extend the Tu scheme [33] and discrete generalized (m, 2N − m)-DT to further study the integrability and exact solutions of equation (3). The rest of this article is organized as follows. In section 2, a discrete integrable lattice hierarchy associated with equation (3) is presented and some integrable properties including Hamiltonian structures and Liouville integrability will be discussed by using trace identity on the basis of the Tu scheme [33]. Section 3 deals with infinitely many conservation laws of equation (3). Section 4 is devoted to the discrete generalized (m, 2N − m)-fold DT of equation (3) based on its known Lax pair. In section 5, the exact solutions such as multi-soliton solutions, rational and semi-rational solutions and their mixed solutions for equation (3) are obtained by applying the resulting generalized DT, and their asymptotic states are analyzed. In section 6, the dynamical behaviors of multi-soliton solutions will be discussed via numerical simulations. Some conclusions and discussions are given in section 7.
2. A RTL_($\alpha$) hierarchy and its Hamiltonian structures
For the discrete 2 × 2 matrix spectral problem (4), we will use the Tu scheme to construct the lattice hierarchy of (3). Solving the following stationary discrete zero-curvature equation $\begin{eqnarray}{P}_{n+1}{U}_{n}-{U}_{n}{P}_{n}=0,\end{eqnarray}$ in which $\begin{eqnarray*}\begin{array}{rcl}{P}_{n} & = & \left(\begin{array}{cc}{A}_{n} & {B}_{n}\\ {C}_{n} & -{A}_{n}\end{array}\right)\\ & = & \left(\begin{array}{cc}{\sum }_{j=0}^{\infty }{A}_{n}^{\left(j\right)}{\lambda }^{-2j} & {\sum }_{j=0}^{\infty }{B}_{n}^{\left(j\right)}{\lambda }^{-2j-1}\\ {\sum }_{j=0}^{\infty }{C}_{n}^{\left(j\right)}{\lambda }^{-2j-1} & -{\sum }_{j=0}^{\infty }{A}_{n}^{\left(j\right)}{\lambda }^{-2j}\end{array}\right)\end{array}\end{eqnarray*}$ results in the following recursion relations $\begin{eqnarray}\left\{\begin{array}{l}(\alpha {b}_{n+1}+1)({A}_{n+1}^{\left(j+1\right)}-{A}_{n}^{\left(j+1\right)})-{A}_{n+1}^{\left(j\right)}+{A}_{n}^{\left(j\right)}-\alpha {B}_{n+1}^{\left(j\right)}-\alpha {a}_{n}{C}_{n}^{\left(j\right)}=0,\\ \alpha {a}_{n}{A}_{n+1}^{\left(j+1\right)}-(\alpha {b}_{n+1}+1){B}_{n}^{\left(j+1\right)}+{B}_{n}^{\left(j\right)}+\alpha {a}_{n}{A}_{n}^{\left(j+1\right)}=0,\\ (\alpha {b}_{n+1}+1){C}_{n+1}^{\left(j+1\right)}-{C}_{n+1}^{\left(j\right)}+\alpha {A}_{n+1}^{\left(j+1\right)}+\alpha {A}_{n}^{\left(j+1\right)}=0,\\ \alpha {a}_{n}{C}_{n+1}^{\left(j+1\right)}+\alpha {B}_{n}^{\left(j+1\right)}=0,\end{array}\right.\end{eqnarray}$ where ${A}_{n}^{\left(j\right)},$ ${B}_{n}^{\left(j\right)}$ and ${C}_{n}^{\left(j\right)}$ are the functions of an, bn. Now we choose the initial condition ${A}_{n}^{\left(0\right)}=-\tfrac{1}{2\alpha }$, the recursion relations (7) will determine the rest ${A}_{n}^{\left(j\right)},{B}_{n}^{\left(j\right)},{C}_{n}^{\left(j\right)}$ uniquely, and the first few coefficients are listed as below $\begin{eqnarray}\begin{array}{l}{B}_{n}^{\left(0\right)}=-\displaystyle \frac{{a}_{n}}{\alpha {b}_{n+1}+1},{C}_{n}^{\left(0\right)}=\displaystyle \frac{1}{\alpha {b}_{n}+1},\\ \ \ {A}_{n}^{\left(1\right)}=-\displaystyle \frac{\alpha {a}_{n}}{(\alpha {b}_{n}+1)(\alpha {b}_{n+1}+1)},\\ {B}_{n}^{\left(1\right)}=-\displaystyle \frac{{a}_{n}}{{\left(\alpha {b}_{n+1}+1\right)}^{2}}(1+\displaystyle \frac{{\alpha }^{2}{a}_{n}}{\alpha {b}_{n}+1}+\displaystyle \frac{{\alpha }^{2}{a}_{n+1}}{\alpha {b}_{n+2}+1}),\\ \ \ {C}_{n}^{\left(1\right)}=\displaystyle \frac{1}{{\left(\alpha {b}_{n}+1\right)}^{2}}(1+\displaystyle \frac{{\alpha }^{2}{a}_{n-1}}{\alpha {b}_{n-1}+1}+\displaystyle \frac{{\alpha }^{2}{a}_{n}}{\alpha {b}_{n+1}+1}),\\ {A}_{n}^{\left(2\right)}=-\left[\displaystyle \frac{\alpha {a}_{n}}{(\alpha {b}_{n}+1)(\alpha {b}_{n+1}+1)}\right]\left[\displaystyle \frac{1}{\alpha {b}_{n}+1}+\displaystyle \frac{1}{\alpha {b}_{n+1}+1}\right.\\ \ \ +\displaystyle \frac{{\alpha }^{2}{a}_{n-1}}{(\alpha {b}_{n-1}+1)(\alpha {b}_{n}+1)}+\displaystyle \frac{{\alpha }^{2}{a}_{n}}{(\alpha {b}_{n}+1)(\alpha {b}_{n+1}+1)}\\ \ \ \left.+\displaystyle \frac{{\alpha }^{2}{a}_{n+1}}{(\alpha {b}_{n+1}+1)(\alpha {b}_{n+2}+1)}\right],\\ {B}_{n}^{\left(2\right)}=-\left[\displaystyle \frac{{a}_{n}}{{\left(\alpha {b}_{n+1}+1\right)}^{3}}\right]\left[1+\displaystyle \frac{{\alpha }^{2}{a}_{n}}{\alpha {b}_{n}+1}+\displaystyle \frac{{\alpha }^{2}{a}_{n+1}}{\alpha {b}_{n+2}+1}\right]\\ \ \ -\left[\displaystyle \frac{{\alpha }^{2}{a}_{n}^{2}}{(\alpha {b}_{n}+1){\left(\alpha {b}_{n+1}+1\right)}^{2}}\right]\left[\displaystyle \frac{1}{\alpha {b}_{n}+1}+\displaystyle \frac{1}{\alpha {b}_{n+1}+1}\right.\\ +\displaystyle \frac{{\alpha }^{2}{a}_{n-1}}{(\alpha {b}_{n-1}+1)(\alpha {b}_{n}+1)}+\displaystyle \frac{{\alpha }^{2}{a}_{n}}{(\alpha {b}_{n}+1)(\alpha {b}_{n+1}+1)}\\ \ \ +\,\displaystyle \frac{{\alpha }^{2}{a}_{n+1}}{(\alpha {b}_{n+1}+1)(\alpha {b}_{n+2}+1)}]-\left[\displaystyle \frac{{\alpha }^{2}{a}_{n}{a}_{n+1}}{{\left(\alpha {b}_{n+1}+1\right)}^{2}(\alpha {b}_{n+2}+1)}\right]\\ \left[\displaystyle \frac{1}{\alpha {b}_{n+1}+1}+\displaystyle \frac{1}{\alpha {b}_{n+2}+1}+\displaystyle \frac{{\alpha }^{2}{a}_{n}}{(\alpha {b}_{n}+1)(\alpha {b}_{n+1}+1)}\right.\\ \ \ \left.+\,\displaystyle \frac{{\alpha }^{2}{a}_{n+1}}{(\alpha {b}_{n+1}+1)(\alpha {b}_{n+2}+1)}+\displaystyle \frac{{\alpha }^{2}{a}_{n+2}}{(\alpha {b}_{n+2}+1)(\alpha {b}_{n+3}+1)}\right],\\ {C}_{n}^{\left(2\right)}=\left[\displaystyle \frac{1}{{\left(\alpha {b}_{n}+1\right)}^{3}}\right]\left[1+\displaystyle \frac{{\alpha }^{2}{a}_{n-1}}{\alpha {b}_{n-1}+1}+\displaystyle \frac{{\alpha }^{2}{a}_{n}}{\alpha {b}_{n+1}+1}\right]\\ \ \ +\,\left[\displaystyle \frac{{\alpha }^{2}{a}_{n-1}}{(\alpha {b}_{n-1}+1){\left(\alpha {b}_{n}+1\right)}^{2}}\right]\left[\displaystyle \frac{1}{\alpha {b}_{n-1}+1}+\displaystyle \frac{1}{\alpha {b}_{n}+1}\right.\\ \ \ +\,\displaystyle \frac{{\alpha }^{2}{a}_{n-2}}{(\alpha {b}_{n-2}+1)(\alpha {b}_{n-1}+1)}+\displaystyle \frac{{\alpha }^{2}{a}_{n-1}}{(\alpha {b}_{n-1}+1)(\alpha {b}_{n}+1)}\\ \ \ +\,\displaystyle \frac{{\alpha }^{2}{a}_{n}}{(\alpha {b}_{n}+1)(\alpha {b}_{n+1}+1)}]+\left[\displaystyle \frac{{\alpha }^{2}{a}_{n}}{{\left(\alpha {b}_{n}+1\right)}^{2}(\alpha {b}_{n+1}+1)}\right]\\ \ \ \left[\displaystyle \frac{1}{\alpha {b}_{n}+1}+\displaystyle \frac{1}{\alpha {b}_{n+1}+1}+\displaystyle \frac{{\alpha }^{2}{a}_{n-1}}{(\alpha {b}_{n-1}+1)(\alpha {b}_{n}+1)}\right.\\ \ \ \left.+\,\displaystyle \frac{{\alpha }^{2}{a}_{n}}{(\alpha {b}_{n}+1)(\alpha {b}_{n+1}+1)}+\displaystyle \frac{{\alpha }^{2}{a}_{n+1}}{(\alpha {b}_{n+1}+1)(\alpha {b}_{n+2}+1)}\right],\ldots .\end{array}\end{eqnarray}$ If ${P}_{n}^{\left(m\right)}$ is defined by $\begin{eqnarray*}\begin{array}{l}{P}_{n}^{\left(m\right)}\\ \ ={\lambda }^{2m+2}{P}_{n}=\left(\begin{array}{cc}{\sum }_{j=0}^{m}{A}_{n}^{\left(j\right)}{\lambda }^{2m-2j+2} & {\sum }_{j=0}^{m}{B}_{n}^{\left(j\right)}{\lambda }^{2m-2j+1}\\ {\sum }_{j=0}^{m}{C}_{n}^{\left(j\right)}{\lambda }^{2m-2j+1} & -{\sum }_{j=0}^{m}{A}_{n}^{\left(j\right)}{\lambda }^{2m-2j+2}\end{array}\right),m\geqslant 0,\end{array}\end{eqnarray*}$ from equation (6) together with (7), we arrive at $\begin{eqnarray}\begin{array}{l}{{EP}}_{n}^{\left(m\right)}{U}_{n}-{U}_{n}{P}_{n}^{\left(m\right)}\,=\,\left(\begin{array}{cc}-{\lambda }^{2}(\alpha {b}_{n+1}+1)({A}_{n+1}^{\left(m+1\right)}-{A}_{n}^{\left(m+1\right)}) & \lambda {B}_{n}^{\left(m\right)}\\ -\lambda {C}_{n+1}^{\left(m\right)} & 0\end{array}\right).\end{array}\end{eqnarray}$ Since $\begin{eqnarray}{U}_{n,t}=\left(\begin{array}{cc}\alpha {\lambda }^{2}{b}_{n+1,t} & \alpha \lambda {a}_{n,t}\\ 0 & 0\end{array}\right),\end{eqnarray}$ we can see that (9) and (10) have different forms, in order to derive the desired lattice hierarchy, we need to seek an appropriate modification matrix δn to modify ${P}_{n}^{\left(m\right)}$. It is easy to verify if we take $\begin{eqnarray}{\delta }_{n}=\left(\begin{array}{cc}0 & 0\\ 0 & -\displaystyle \frac{1}{\alpha }{C}_{n}^{\left(m\right)}\end{array}\right),\end{eqnarray}$ then $\begin{eqnarray}{\delta }_{n+1}{U}_{n}-{U}_{n}{\delta }_{n}=\left(\begin{array}{cc}(0 & \lambda {a}_{n}{C}_{n}^{\left(m\right)}\\ \lambda {C}_{n+1}^{\left(m\right)} & 0\end{array}\right).\end{eqnarray}$ Let us set ${V}_{n}^{\left(m\right)}={P}_{n}^{\left(m\right)}+{\delta }_{n}$ such that $\begin{eqnarray}\begin{array}{l}E({P}_{n}^{\left(m\right)}+{\delta }_{n}){U}_{n}-{U}_{n}({P}_{n}^{\left(m\right)}+{\delta }_{n})\,=\,\left(\begin{array}{cc}-{\lambda }^{2}(\alpha {b}_{n+1}+1)({A}_{n+1}^{\left(m+1\right)}-{A}_{n}^{\left(m+1\right)}) & \lambda {B}_{n}^{\left(m\right)}+\lambda {a}_{n}{C}_{n}^{m}\\ 0 & 0\end{array}\right).\end{array}\end{eqnarray}$ Assuming that the time evolution of φn meets the equations ${\phi }_{n,{tm}}={V}_{n}^{\left(m\right)}{\phi }_{n}$, then the compatibility condition $E{\phi }_{n,{tm}}={(E{\phi }_{n})}_{{tm}}$ implies $\begin{eqnarray}{U}_{n,{tm}}=({{EV}}_{n}^{\left(m\right)}){U}_{n}-{U}_{n}{V}_{n}^{\left(m\right)},m\geqslant 0,\end{eqnarray}$ which yields the following integrable lattice hierarchy: $\begin{eqnarray}\left\{\begin{array}{l}{a}_{n,{t}_{m}}=\tfrac{{B}_{n}^{\left(m\right)}+{a}_{n}{C}_{n}^{\left(m\right)}}{\alpha },\\ {b}_{n,{t}_{m}}=-\tfrac{(\alpha {b}_{n}+1)({A}_{n}^{\left(m+1\right)}-{A}_{n-1}^{\left(m+1\right)})}{\alpha }.\end{array}\right.\end{eqnarray}$ (1)When m = 0, the lattice hierarchy in equation (15) reduces to RTL_($\alpha$) system (3) with t0 = t as $\begin{eqnarray}\left\{\begin{array}{l}{a}_{n,{t}_{0}}=\tfrac{{B}_{n}^{\left(0\right)}+{a}_{n}{C}_{n}^{\left(0\right)}}{\alpha }=\tfrac{{a}_{n}({b}_{n+1}-{b}_{n})}{(\alpha {b}_{n}+1)(\alpha {b}_{n+1}+1)},\\ {b}_{n,{t}_{0}}=-\tfrac{(\alpha {b}_{n}+1)({A}_{n}^{\left(1\right)}-{A}_{n-1}^{\left(1\right)})}{\alpha }=\tfrac{{a}_{n}(\alpha {b}_{n-1}+1)-{a}_{n-1}(\alpha {b}_{n+1}+1)}{(\alpha {b}_{n-1}+1)(\alpha {b}_{n+1}+1)},\end{array}\right.\end{eqnarray}$ whose time part of Lax pair is $\begin{eqnarray}{\phi }_{n,{t}_{0}}={V}_{n}^{\left(0\right)}{\phi }_{n}=\left(\begin{array}{cc}-\tfrac{{\lambda }^{2}}{2\alpha } & -\tfrac{\lambda {a}_{n}}{\alpha {b}_{n+1}+1}\\ \tfrac{\lambda }{\alpha {b}_{n}+1} & \tfrac{{\lambda }^{2}}{2\alpha }-\tfrac{1}{\alpha (\alpha {b}_{n}+1)}\end{array}\right){\phi }_{n},\end{eqnarray}$ which is a little different from equation (5). In fact, they are equivalent under gauge transformation.
(2)When m = 1, from the lattice hierarchy in equation (15), we arrive at $\begin{eqnarray}\left\{\begin{array}{l}{a}_{n,{t}_{1}}=-\tfrac{{a}_{n}}{\alpha {\left(\alpha {b}_{n+1}+1\right)}^{2}}(1+\tfrac{{\alpha }^{2}{a}_{n}}{\alpha {b}_{n}+1}+\tfrac{{\alpha }^{2}{a}_{n+1}}{\alpha {b}_{n+2}+1})+\tfrac{{a}_{n}}{\alpha {\left(\alpha {b}_{n}+1\right)}^{2}}(1+\tfrac{{\alpha }^{2}{a}_{n-1}}{\alpha {b}_{n-1}+1}+\tfrac{{\alpha }^{2}{a}_{n}}{\alpha {b}_{n+1}+1}),\\ {b}_{n,{t}_{1}}=\tfrac{{a}_{n-1}}{\alpha {b}_{n-1}+1}\left[\tfrac{1}{\alpha {b}_{n-1}+1}+\tfrac{1}{\alpha {b}_{n}+1}+\tfrac{{\alpha }^{2}{a}_{n-2}}{(\alpha {b}_{n-2}+1)(\alpha {b}_{n-1}+1)}+\tfrac{{\alpha }^{2}{a}_{n-1}}{(\alpha {b}_{n-1}+1)(\alpha {b}_{n}+1)}\right]\\ \,-\tfrac{{a}_{n}}{\alpha {b}_{n+1}+1}\left[\tfrac{1}{\alpha {b}_{n}+1}+\tfrac{1}{\alpha {b}_{n+1}+1}+\tfrac{{\alpha }^{2}{a}_{n}}{(\alpha {b}_{n}+1)(\alpha {b}_{n+1}+1)}+\tfrac{{\alpha }^{2}{a}_{n+1}}{(\alpha {b}_{n+1}+1)(\alpha {b}_{n+2}+1)}\right],\end{array}\right.\end{eqnarray}$ whose time part of Lax pair is $\begin{eqnarray}\begin{array}{rcl}{\phi }_{n,{t}_{1}} & = & {V}_{n}^{\left(1\right)}{\phi }_{n}=\left(\begin{array}{cc}{\lambda }^{4}{A}_{n}^{\left(0\right)}+{\lambda }^{2}{A}_{n}^{\left(1\right)} & {\lambda }^{3}{B}_{n}^{\left(0\right)}+\lambda {B}_{n}^{\left(1\right)}\\ {\lambda }^{3}{C}_{n}^{\left(0\right)}+\lambda {C}_{n}^{\left(1\right)} & -{\lambda }^{4}{A}_{n}^{\left(0\right)}-{\lambda }^{2}{A}_{n}^{\left(1\right)}-\displaystyle \frac{1}{\alpha }{C}_{n}^{\left(1\right)}\end{array}\right){\phi }_{n}\,=\,\left(\begin{array}{cc}{V}_{11}(n) & {V}_{12}(n)\\ {V}_{21}(n) & {V}_{22}(n)\end{array}\right){\phi }_{n},\end{array}\end{eqnarray}$ in which $\begin{eqnarray*}\begin{array}{rcl}{V}_{11}(n) & = & -\displaystyle \frac{{\lambda }^{4}}{2\alpha }-\displaystyle \frac{\alpha {\lambda }^{2}{a}_{n}}{\left(\alpha {b}_{n}+1\right)(\alpha {b}_{n+1}+1)},\\ {V}_{12}(n) & = & -\displaystyle \frac{{\lambda }^{3}{a}_{n}}{\alpha {b}_{n+1}+1}-\displaystyle \frac{\lambda {a}_{n}}{{\left(\alpha {b}_{n+1}+1\right)}^{2}}\\ & & \times \,\left(1+\displaystyle \frac{{\alpha }^{2}{a}_{n}}{\alpha {b}_{n}+1}+\displaystyle \frac{{\alpha }^{2}{a}_{n+1}}{\alpha {b}_{n+2}+1}\right),\\ {V}_{21}(n) & = & \displaystyle \frac{{\lambda }^{3}}{\alpha {b}_{n}+1}+\displaystyle \frac{\lambda }{{\left(\alpha {b}_{n}+1\right)}^{2}}\left(1+\displaystyle \frac{{\alpha }^{2}{a}_{n-1}}{\alpha {b}_{n-1}+1}\right.\\ & & \left.+\,\displaystyle \frac{{\alpha }^{2}{a}_{n}}{\alpha {b}_{n+1}+1}\right),\\ {V}_{22}(n) & = & \displaystyle \frac{{\lambda }^{4}}{2\alpha }+\displaystyle \frac{\alpha {\lambda }^{2}{a}_{n}}{(\alpha {b}_{n}+1)(\alpha {b}_{n+1}+1)}-\displaystyle \frac{1}{\alpha {\left(\alpha {b}_{n}+1\right)}^{2}}\\ & & \times \,\left(1+\displaystyle \frac{{\alpha }^{2}{a}_{n-1}}{\alpha {b}_{n-1}+1}+\displaystyle \frac{{\alpha }^{2}{a}_{n}}{\alpha {b}_{n+1}+1}\right).\end{array}\end{eqnarray*}$ Here equation (18) is called the second-order RTL_($\alpha$) system which is not discussed in this paper. By continuing this process, we can get a series of discrete integrable NDDEs.
Next we will construct the Hamiltonian structures of equation (15). Before that, we first look back to some symbols used in this paper [33]. The variational derivative of the scalar function fn with regard to ui is defined as $\tfrac{\delta {f}_{n}}{\delta {u}_{i}}={\sum }_{k\in Z}{E}^{-k}\tfrac{\partial {f}_{n}}{\partial {u}_{i+k}}$. The formula $({f}_{n},{g}_{n})={\sum }_{n\in Z}^{}{\sum }_{i\,=\,0}^{p}{f}_{i,n}{g}_{i,n}$ denotes the inner product between vector functions ${f}_{n}={({f}_{1,n},{f}_{2,n},\ldots ,{f}_{p,n})}^{{\rm{T}}}$ and ${g}_{n}={({g}_{1,n},{g}_{2,n},\ldots ,{g}_{p,n})}^{{\rm{T}}}$. The Poisson bracket [33] for the Hamiltonian operator J between functions fn and gn is defined by $\{{f}_{n},{g}_{n}\}=(J\tfrac{\delta {f}_{n}}{\delta u},\tfrac{\delta {g}_{n}}{\delta u})$. The operator J* defined by (fn, J*gn) = (Jfn, gn) is called the adjoint operator of J with respective to the inner product, in which J is described as the skew-symmetric operator if J = −J*.
Next, we define $\lt U,V\gt ={tr}({UV})$, where U and V are arbitrary square matrices. If we set $\begin{eqnarray}\begin{array}{rcl}{V}_{n} & = & {P}_{n}{U}_{n}^{-1}=\left(\begin{array}{cc}{A}_{n} & {B}_{n}\\ {C}_{n} & -{A}_{n}\end{array}\right)\left(\begin{array}{cc}0 & -\tfrac{1}{\alpha \lambda }\\ \tfrac{1}{\alpha \lambda {a}_{n}} & \tfrac{{\lambda }^{2}(\alpha {b}_{n+1}+1)-1}{{\alpha }^{2}{\lambda }^{2}{a}_{n}}\end{array}\right)\\ & = & \left(\begin{array}{cc}\tfrac{{B}_{n}}{\alpha \lambda {a}_{n}} & \tfrac{[{\lambda }^{2}(\alpha {b}_{n+1}+1)-1]{B}_{n}-\alpha \lambda {a}_{n}{A}_{n}}{{\alpha }^{2}{\lambda }^{2}{a}_{n}}\\ \tfrac{-{A}_{n}}{\alpha \lambda {a}_{n}} & \tfrac{-[{\lambda }^{2}(\alpha {b}_{n+1}+1)-1]{A}_{n}-\alpha \lambda {a}_{n}{C}_{n}}{{\alpha }^{2}{\lambda }^{2}{a}_{n}}\end{array}\right),\end{array}\end{eqnarray}$ then we have $\begin{eqnarray}\begin{array}{l}\langle {V}_{n},\displaystyle \frac{\partial {U}_{n}}{\partial \lambda }\rangle =\displaystyle \frac{[{\lambda }^{2}(\alpha {b}_{n+1}+1)+1]{B}_{n}}{\alpha {\lambda }^{2}{a}_{n}},\\ \ \ \langle {V}_{n},\displaystyle \frac{\partial {U}_{n}}{\partial {a}_{n}}\rangle =-\displaystyle \frac{{A}_{n}}{{a}_{n}},\ \langle {V}_{n},\displaystyle \frac{\partial {U}_{n}}{\partial {b}_{n+1}}\rangle =\displaystyle \frac{\lambda {B}_{n}}{{a}_{n}}.\end{array}\end{eqnarray}$ By using the trace identity [33] $\begin{eqnarray}\begin{array}{l}\displaystyle \frac{\delta }{\delta u}\sum _{n\in Z}\langle {V}_{n},\displaystyle \frac{\partial {U}_{n}}{\partial \lambda }\rangle =\left({\lambda }^{-\varepsilon }\displaystyle \frac{\partial }{\partial \lambda }{\lambda }^{\varepsilon }\right)\\ \ \ \times \,\langle {V}_{n},\displaystyle \frac{\partial {U}_{n}}{\partial {u}_{i}}\rangle ,i=1,2,\end{array}\end{eqnarray}$ we arrive at $\begin{eqnarray}\displaystyle \frac{\delta }{\delta u}\sum _{n\in Z}\left[\displaystyle \frac{{\lambda }^{2}(\alpha {b}_{n+1}+1){B}_{n}+{B}_{n}}{\alpha {\lambda }^{2}{a}_{n}}\right]=({\lambda }^{-\varepsilon }\displaystyle \frac{\partial }{\partial \lambda }{\lambda }^{\varepsilon })\left(\begin{array}{c}\tfrac{-{A}_{n}}{{a}_{n}}\\ \tfrac{\lambda {B}_{n}}{{a}_{n}}\end{array}\right).\end{eqnarray}$ Comparing the coefficients of λ−2m−3 on both sides of equation (21), we have $\begin{eqnarray}\begin{array}{l}\left(\begin{array}{c}\tfrac{\delta }{\delta {a}_{n}}\\ \tfrac{\delta }{\delta {b}_{n+1}}\end{array}\right).\sum _{n\in Z}\left[\displaystyle \frac{(\alpha {b}_{n+1}+1){B}_{n}^{\left(m+1\right)}+{B}_{n}^{\left(m\right)}}{\alpha {a}_{n}}\right]\\ \ \ =(\varepsilon -2m-2)\left(\begin{array}{c}\tfrac{-{A}_{n}^{\left(m+1\right)}}{{a}_{n}}\\ \tfrac{{B}_{n}^{\left(m+1\right)}}{{a}_{n}}\end{array}\right).\end{array}\end{eqnarray}$ Setting m = 0, from equation (24) we have ϵ = 0. Let ${H}_{n}^{\left(m+1\right)}={\sum }_{n\in Z}\tfrac{(\alpha {b}_{n+1}+1){B}_{n}^{\left(m+1\right)}+{B}_{n}^{\left(m\right)}}{-2(m+1)\alpha {a}_{n}}$, then $\begin{eqnarray}\displaystyle \frac{\delta {H}_{n}^{\left(m+1\right)}}{\delta u}=\left(\begin{array}{c}\tfrac{-{A}_{n}^{\left(m+1\right)}}{{a}_{n}}\\ \tfrac{{B}_{n}^{\left(m+1\right)}}{{a}_{n}}\end{array}\right)=\left(\begin{array}{c}{f}_{n}^{\left(m+1\right)}\\ {g}_{n}^{\left(m+1\right)}\end{array}\right),\end{eqnarray}$ where ${A}_{n}^{\left(m+1\right)}=-{a}_{n}{f}_{n}^{\left(m+1\right)},$ ${B}_{n}^{\left(m+1\right)}={a}_{n}{g}_{n}^{\left(m+1\right)},$ ${C}_{n}^{\left(m+1\right)}=-{E}^{-1}{g}_{n}^{\left(m+1\right)}.$ Then equation (15) can be rewritten as the following Hamiltonian form: $\begin{eqnarray}{U}_{{t}_{m}}=\left(\begin{array}{c}{a}_{n,{t}_{m}}\\ {b}_{n,{t}_{m}}\end{array}\right)=J\displaystyle \frac{\delta {H}_{n}^{\left(m+1\right)}}{\delta u}=J\left(\begin{array}{c}{f}_{n}^{\left(m+1\right)}\\ {g}_{n}^{\left(m+1\right)}\end{array}\right),\end{eqnarray}$ in which $\begin{eqnarray}J=\left(\begin{array}{cc}{a}_{n}({a}_{n+1}E-{a}_{n-1}{E}^{-1}) & \tfrac{{a}_{n}\left[(\alpha {b}_{n+1}+1)-(\alpha {b}_{n}+1){E}^{-1}\right]}{\alpha }\\ \tfrac{(\alpha {b}_{n}+1)({a}_{n}-{a}_{n-1}{E}^{-1})}{\alpha } & 0\end{array}\right).\end{eqnarray}$ Taking $\eta =\left(\begin{array}{cc}{\eta }_{11} & {\eta }_{12}\\ {\eta }_{21} & {\eta }_{22}\end{array}\right)$ to satisfy $\tfrac{\delta {H}_{n}^{\left(m+1\right)}}{\delta u}=\eta \tfrac{\delta {H}_{n}^{\left(m\right)}}{\delta u}$, then by recursion relations (7), we can get $\begin{eqnarray*}\begin{array}{rcl}{\eta }_{11} & = & {a}_{n}^{-1}{(E-1)}^{-1}\left(\displaystyle \frac{{a}_{n+1}E-{a}_{n}}{\alpha {b}_{n+1}+1}\right),\\ {\eta }_{12} & = & \alpha {a}_{n}^{-1}{(E-1)}^{-1}\left(\displaystyle \frac{{a}_{n}{E}^{-1}-{a}_{n+1}E}{\alpha {b}_{n+1}+1}\right),\\ {\eta }_{21} & = & \alpha {(\alpha {b}_{n+1}+1)}^{-1}(E+1){(E-1)}^{-1}\,\times \,\left(\displaystyle \frac{{a}_{n}-{a}_{n+1}E}{\alpha {b}_{n+1}+1}\right),\\ {\eta }_{22} & = & {(\alpha {b}_{n+1}+1)}^{-1}\,\times \,\left[{\alpha }^{2}(E+1){(E-1)}^{-1}\left(\displaystyle \frac{{a}_{n+1}E-{a}_{n}{E}^{-1}}{\alpha {b}_{n+1}+1}\right)+1\right].\end{array}\end{eqnarray*}$ Hence, equation (24) can be rewritten as $\begin{eqnarray}\begin{array}{rcl}{U}_{{t}_{m}} & = & J\displaystyle \frac{\delta {H}_{n}^{\left(m+1\right)}}{\delta u}=J\left(\begin{array}{c}{f}_{n}^{\left(m+1\right)}\\ {g}_{n}^{\left(m+1\right)}\end{array}\right)=J\eta \left(\begin{array}{c}{f}_{n}^{\left(m\right)}\\ {g}_{n}^{\left(m\right)}\end{array}\right)\\ & = & ...=J{\eta }^{m+1}\left(\begin{array}{c}{f}_{n}^{\left(0\right)}\\ {g}_{n}^{\left(0\right)}\end{array}\right)=J{\eta }^{m+1}\left(\begin{array}{c}\tfrac{1}{2\alpha {a}_{n}}\\ -\tfrac{1}{\alpha {b}_{n+1}+1}\end{array}\right).\end{array}\end{eqnarray}$ Hence we have successfully written the lattice hierarchy (18) in the above Hamilionian form (28). It can be verified that J and J$\eta$ are skew-symmetric operators. The Hamiltonian functions ${H}_{n}^{\left(m+1\right)}\ (m\geqslant 0)$ described by equation (25) are pairwise involutory concerning Poisson bracket. So we have the following theorem: The hierarchy (15) or the Hamiltonian form (28) possesses Liouville integrability.
3. Conservation laws of equation (3)
In the previous section, we have investigated Liouville integrability of equation (3). In this section, we study its infinitely many conservation laws which will further verify the integrability of equation (3) [1, 37, 43].
Setting ${\theta }_{n}=\tfrac{{\varphi }_{n}}{{\psi }_{n}}$, from the 2 × 2 matrix spectral problem (4), we have $\begin{eqnarray}\displaystyle \frac{{\varphi }_{n+1}}{{\varphi }_{n}}={\lambda }^{2}(\alpha {b}_{n+1}+1)-1+\alpha \lambda {a}_{n}{\theta }_{n}\end{eqnarray}$ and $\begin{eqnarray}[{\lambda }^{2}(\alpha {b}_{n+1}+1)-1+\alpha \lambda {a}_{n}{\theta }_{n}]{\theta }_{n+1}+\alpha \lambda =0.\end{eqnarray}$ Inserting ${\theta }_{n}={\sum }_{j=0}^{n}{\theta }_{n}^{\left(j\right)}{\lambda }^{-j}$ into (30) and collecting the coefficients of same powers of λ, we can get the recursion relations of ${\theta }_{n+1}^{\left(j\right)}$ as follows: $\begin{eqnarray*}\begin{array}{l}{\theta }_{n+1}^{(0)}=0,\ {\theta }_{n+1}^{(1)}=-\displaystyle \frac{\alpha }{\alpha {b}_{n+1}+1},\ {\theta }_{n+1}^{(2)}=0,\\ \ \ \,{\theta }_{n+1}^{(3)}=-\displaystyle \frac{\alpha ({\alpha }^{2}{a}_{n}+\alpha {b}_{n}+1)}{(\alpha {b}_{n}+1){(\alpha {b}_{n+1}+1)}^{2}},\ \ \ {\theta }_{n+1}^{(4)}=0,\\ {\theta }_{n+1}^{(5)}=-\displaystyle \frac{\alpha {({\alpha }^{2}{a}_{n}+\alpha {b}_{n}+1)}^{2}}{{(1+\alpha {b}_{n})}^{2}{(1+\alpha {b}_{n+1})}^{3}}\\ \ \ -\,\displaystyle \frac{{\alpha }^{3}{a}_{n}({\alpha }^{2}{a}_{n-1}+\alpha {b}_{n-1}+1)}{(1+\alpha {b}_{n-1}){(1+\alpha {b}_{n})}^{2}{(1+\alpha {b}_{n+1})}^{2}},\ldots ,\\ {\theta }_{n+1}^{(2j)}=0,\\ {\theta }_{n+1}^{(2j+1)}=\displaystyle \frac{{\theta }_{n+1}^{(2j-1)}-\alpha {a}_{n}{\sum }_{i=0}^{2j}{\theta }_{n}^{(i)}{\theta }_{n+1}^{(2j-i)}}{\alpha {b}_{n+1}+1},\ j\geqslant 3.\end{array}\end{eqnarray*}$ From the time part (17) of Lax pair in the previous section, we have $\begin{eqnarray}{(\mathrm{ln}{\varphi }_{n})}_{t}=-\displaystyle \frac{1}{2\alpha }{\lambda }^{2}-\displaystyle \frac{\lambda {a}_{n}}{\alpha {b}_{n+1}+1}{\theta }_{n}.\end{eqnarray}$ From (29) and (31), we can derive the following conservation laws for equation (3) as follows: $\begin{eqnarray}\begin{array}{l}{[\mathrm{ln}(\alpha {\lambda }^{2}{b}_{n+1}+{\lambda }^{2}-1+\alpha \lambda {a}_{n}{\theta }_{n})]}_{t}\\ \ \ =(E-1)(-\displaystyle \frac{{\lambda }^{2}}{2\alpha }-\displaystyle \frac{\lambda {a}_{n}}{\alpha {b}_{n+1}+1}{\theta }_{n}).\end{array}\end{eqnarray}$ Equating the same powers of λ on both sides of equation (32), we obtain infinitely many conservation laws for equation (3). The first three conservation laws are listed as follows $\begin{eqnarray}{({T}_{k})}_{t}=(E-1)({X}_{k}),\quad (k=1,2,3),\end{eqnarray}$ with $\begin{eqnarray*}\begin{array}{rcl}{T}_{1} & = & \mathrm{ln}(1+\alpha {b}_{n+1}),\quad {X}_{1}=\displaystyle \frac{\alpha {a}_{n}}{(1+\alpha {b}_{n})(1+\alpha {b}_{n+1})},\\ {T}_{2} & = & -\displaystyle \frac{{\alpha }^{2}{a}_{n}+\alpha {b}_{n}+1}{(1+\alpha {b}_{n})(1+\alpha {b}_{n+1})},\\ {X}_{2} & = & \displaystyle \frac{\alpha {a}_{n}({\alpha }^{2}{a}_{n-1}+\alpha {b}_{n-1}+1)}{(1+\alpha {b}_{n-1}){\left(1+\alpha {b}_{n}\right)}^{2}(1+\alpha {b}_{n+1})},\\ {T}_{3} & = & -\displaystyle \frac{{\alpha }^{2}{a}_{n}({\alpha }^{2}{a}_{n-1}+\alpha {b}_{n-1}+1)}{(1+\alpha {b}_{n-1}){\left(1+\alpha {b}_{n}\right)}^{2}(1+\alpha {b}_{n+1})}-\displaystyle \frac{{\alpha }^{4}{a}_{n}^{2}}{2{\left(1+\alpha {b}_{n}\right)}^{2}{\left(1+\alpha {b}_{n+1}\right)}^{2}}\\ & & -\,\displaystyle \frac{{\alpha }^{2}{a}_{n}}{(1+\alpha {b}_{n}){\left(1+\alpha {b}_{n+1}\right)}^{2}}-\displaystyle \frac{1}{2{\left(1+\alpha {b}_{n+1}\right)}^{2}},\\ {X}_{3} & = & \displaystyle \frac{\alpha {a}_{n}{\left({\alpha }^{2}{a}_{n-1}+\alpha {b}_{n-1}+1\right)}^{2}}{{\left(1+\alpha {b}_{n-1}\right)}^{2}{\left(1+\alpha {b}_{n}\right)}^{3}(1+\alpha {b}_{n+1})}\\ & & +\,\displaystyle \frac{{\alpha }^{3}{a}_{n-1}{a}_{n}({\alpha }^{2}{a}_{n-2}+\alpha {b}_{n-2}+1)}{(1+\alpha {b}_{n-2}){\left(1+\alpha {b}_{n-1}\right)}^{2}{\left(1+\alpha {b}_{n}\right)}^{2}(1+\alpha {b}_{n+1})},\end{array}\end{eqnarray*}$ where Tk and Xk denote the conserved densities and associated fluxes respectively. ${\sum }_{n=-\infty }^{+\infty }{T}_{k}\ (k=1,2,\ldots )$ are motion constants, and $-{\sum }_{n=-\infty }^{+\infty }{T}_{1}$ and ${\sum }_{n=-\infty }^{+\infty }{T}_{2}$ in physical meanings represent the total momentum and total energy of the lattice respectively. The existence of infinitely many conservation laws means that equation (3) is a discrete integrable system.
In this section, we will construct the discrete 2N-fold DT of equation (3), and then extend it to the discrete generalized (m, 2N − m)-fold DT. We first introduce the following gauge transformation: $\begin{eqnarray}{\tilde{\phi }}_{n}={T}_{n}{\phi }_{n},\end{eqnarray}$ where ${\tilde{\phi }}_{n}$ satisfies the Lax pair (4) and (17), i.e. $\begin{eqnarray}{\tilde{\phi }}_{n+1}={\tilde{U}}_{n}{\tilde{\phi }}_{n},\quad {\tilde{\phi }}_{n,t}={\tilde{V}}_{n}^{\left(0\right)}{\tilde{\phi }}_{n},\end{eqnarray}$ where ${\tilde{U}}_{n}={T}_{n+1}{U}_{n}{T}_{n}^{-1}$ and ${\tilde{V}}_{n}^{\left(0\right)}=({T}_{n,t}+{V}_{n}^{\left(0\right)}{T}_{n}){T}_{n}^{-1}$ have the same forms as Un, ${V}_{n}^{\left(0\right)}$ expect for the new potentials ${\tilde{a}}_{n},$ ${\tilde{b}}_{n}$ instead of the old potentials an, bn. To guarantee the validity of the discrete 2N-fold DT, we need to construct a special Darboux matrix Tn defined by $\begin{eqnarray}\begin{array}{l}{T}_{n}=\left(\begin{array}{cc}{F}_{n} & {G}_{n}\\ {R}_{n} & {S}_{n}\end{array}\right)=\left(\begin{array}{cc}{\sum }_{j=1}^{N}{\lambda }^{2j}{f}_{n}^{\left(2j\right)}+1 & {\sum }_{j=1}^{N}{\lambda }^{2j-1}{g}_{n}^{\left(2j-1\right)}\\ {\sum }_{j=1}^{N}{\lambda }^{2j-1}{r}_{n}^{\left(2j-1\right)} & \left(1-\tfrac{{r}_{n}^{\left(1\right)}}{\alpha }\right)+\sum _{j=1}^{N}{\lambda }^{2j}{s}_{n}^{\left(2j\right)}\end{array}\right),\end{array}\end{eqnarray}$ in which the number N is a positive integer, and ${f}_{n}^{\left(2j\right)},{g}_{n}^{\left(2j-1\right)},$ ${r}_{n}^{\left(2j-1\right)}$ and ${s}_{n}^{\left(2j\right)}\ (j=1,2,\ldots N)$ are functions of the variables n and t which are determined by the linear algebraic system T(λi)φi,n(λi) = 0 (i = 1, 2, …, 2N), where ${\phi }_{i,n}{({\lambda }_{i})={({\varphi }_{i,n}({\lambda }_{i,n}),{\psi }_{i,n}({\lambda }_{i}))}^{{\rm{T}}}\equiv ({\varphi }_{i,n},{\psi }_{i,n})}^{{\rm{T}}}$ are 2N solutions of Lax pair (4) and (17) for 2N spectral parameters λi. When the 2N distinct parameters λi (λi ≠ λj, i ≠ j) are suitably chosen so that the determinant of the coefficients of 4N functions ${f}_{n}^{\left(2j\right)},$ ${g}_{n}^{\left(2j-1\right)},$ ${r}_{n}^{\left(2j-1\right)}$ and ${s}_{n}^{\left(2j\right)}\ (j=1,2,\ldots N)$ is nonzero. Hence the Darboux matrix Tn in (36) can be uniquely determined.
According to the previous analysis, we come to the following 2N-fold DT theorem of equation (3): Let ${\phi }_{i,n}{({\lambda }_{i})=({\varphi }_{i,n},{\psi }_{i,n})}^{{\rm{T}}}$ be 2N column vector solutions of Lax pair (4) and (17) for the spectral parameters ${\lambda }_{i}\ (i=1,2,\ldots ,2N)$ with the initial solutions ${a}_{n},{b}_{n}$ of equation (3), then the 2N-fold DT of equation (3) between the old solutions ${a}_{n},$ bn and the new solutions ${\tilde{a}}_{n},{\tilde{b}}_{n}$ is given by $\begin{eqnarray}{\tilde{a}}_{n}=\displaystyle \frac{\alpha {a}_{n}+{g}_{n}^{\left(1\right)}}{\alpha -{r}_{n}^{\left(1\right)}},{\tilde{b}}_{n}=\displaystyle \frac{\alpha {b}_{n}{f}_{n}^{\left(2N\right)}-{f}_{n-1}^{\left(2N\right)}+{f}_{n}^{\left(2N\right)}}{\alpha {f}_{n-1}^{\left(2N\right)}},\end{eqnarray}$ where $\begin{eqnarray}{f}_{n}^{\left(2N\right)}=\displaystyle \frac{{\rm{\Delta }}{f}_{n}^{\left(2N\right)}}{{{\rm{\Delta }}}_{1,n}},{g}_{n}^{\left(1\right)}=\displaystyle \frac{{\rm{\Delta }}{g}_{n}^{\left(1\right)}}{{{\rm{\Delta }}}_{1,n}},{r}_{n}^{\left(1\right)}=\displaystyle \frac{{\rm{\Delta }}{r}_{n}^{\left(1\right)}}{{{\rm{\Delta }}}_{2,n}},\end{eqnarray}$ in which $\begin{eqnarray*}\begin{array}{rcl}{{\rm{\Delta }}}_{1,n} & = & \left|\begin{array}{cccccccc}{\lambda }_{1}^{2} & {\lambda }_{1}^{4} & ... & {\lambda }_{1}^{2N} & {\lambda }_{1}{\delta }_{1,n} & {\lambda }_{1}^{3}{\delta }_{1,n} & ... & {\lambda }_{1}^{2N-1}{\delta }_{1,n}\\ {\lambda }_{2}^{2} & {\lambda }_{2}^{4} & ... & {\lambda }_{2}^{2N} & {\lambda }_{2}{\delta }_{2,n} & {\lambda }_{2}^{3}{\delta }_{2,n} & ... & {\lambda }_{2}^{2N-1}{\delta }_{2,n}\\ ... & ... & ... & ... & ... & ... & ... & ...\\ {\lambda }_{2N}^{2} & {\lambda }_{2N}^{4} & ... & {\lambda }_{2N}^{2N} & {\lambda }_{2N}{\delta }_{2N,n} & {\lambda }_{2N}^{3}{\delta }_{2N,n} & ... & {\lambda }_{2N}^{2N-1}{\delta }_{2N,n}\end{array}\right|,\\ {{\rm{\Delta }}}_{2,n} & = & \left|\begin{array}{cccccccc}{\lambda }_{1}-\displaystyle \frac{1}{\alpha }{\delta }_{1,n} & {\lambda }_{1}^{3} & ... & {\lambda }_{1}^{2N-1} & {\lambda }_{1}^{2}{\delta }_{1,n} & {\lambda }_{1}^{4}{\delta }_{1,n} & ... & {\lambda }_{1}^{2N-2}\\ {\lambda }_{2}-\displaystyle \frac{1}{\alpha }{\delta }_{2,n} & {\lambda }_{2}^{3} & ... & {\lambda }_{2}^{2N-1} & {\lambda }^{2}{\delta }_{2,n} & {\lambda }_{2}^{4}{\delta }_{2,n} & ... & {\lambda }_{2}^{2N-2}\\ ... & ... & ... & ... & ... & ... & ... & ...\\ {\lambda }_{2N}-\displaystyle \frac{1}{\alpha }{\delta }_{2N,n} & {\lambda }_{2N}^{3} & ... & {\lambda }_{2N}^{2N-1} & {\lambda }_{2N}^{2}{\delta }_{2N,n} & {\lambda }_{2N}^{4}{\delta }_{2N,n} & ... & {\lambda }_{2N}^{2N-2}\end{array}\right|,\end{array}\end{eqnarray*}$ whereas ${\rm{\Delta }}{f}_{n}^{\left(2N\right)}$ and ${g}_{n}^{\left(1\right)}$ are obtained from the determinant ${{\rm{\Delta }}}_{1,n}$ by replacing the Nth and $(N+1)$th columns with the column vector $(-1,-1,\ldots -1)$, respectively, and ${\rm{\Delta }}{r}_{n}^{\left(1\right)}$ is given from the determinant ${{\rm{\Delta }}}_{2,n}$ by replacing its first column with the column vector $(-{\delta }_{1,n},-{\delta }_{2,n},\ldots ,-{\delta }_{2N,n})$. Moreover, the term ${f}_{n-1}^{\left(2N\right)}$ is derived from ${f}_{n}^{\left(2N\right)}$ by replacing the index n with $n-1$. Let ${T}_{n}^{-1}={T}_{n}^{* }/\det {T}_{n}$ and $\begin{eqnarray}{T}_{n+1}{U}_{n}{T}_{n}^{* }=\left(\begin{array}{cc}{f}_{11}(\lambda ,n) & {f}_{12}(\lambda ,n)\\ {}_{21}(\lambda ,n) & {f}_{22}(\lambda ,n)\end{array}\right).\end{eqnarray}$ By direct calculation we know that ${f}_{11}(\lambda ,n)$ is the $(4N+2)$th order polynomial in λ, ${f}_{12}(\lambda ,n)$, ${f}_{21}(\lambda ,n)$ are the $(4N+1)$th order polynomials in λ, and ${f}_{22}(\lambda ,n)$ is the $(4N)$th order polynomial in λ.
Assume ${\delta }_{i,n}=\tfrac{{\varphi }_{i,n}}{{\psi }_{i,n}}$, from equations (4) and (36) we have $\begin{eqnarray}\begin{array}{rcl}{\delta }_{i,n+1} & = & -\displaystyle \frac{\alpha {\lambda }_{i}}{{\lambda }_{i}^{2}(\alpha {b}_{n+1}+1)-1+\alpha {\lambda }_{i}{a}_{n}{\delta }_{i,n}},\\ {\delta }_{i,n-1} & = & \displaystyle \frac{-\alpha {\lambda }_{i}-{\theta }_{n}[{\lambda }_{i}^{2}(\alpha {b}_{n+1}+1)-1]}{\alpha {\lambda }_{i}{a}_{n}{\delta }_{i,n}}.\end{array}\end{eqnarray}$ Through direct calculations, we can verify that λi (i = 1, 2, …, 2N) are the roots of fj,k(λ, n) (j, k = 1, 2). Therefore, we can rewrite (39 ) as $\begin{eqnarray}{T}_{n+1}{U}_{n}{T}_{n}^{* }=\det {T}_{n}\cdot {P}_{n},\end{eqnarray}$ with $\begin{eqnarray}{P}_{n}=\left(\begin{array}{cc}{\lambda }^{2}{P}_{11}^{\left(2\right)}(n)+\lambda {P}_{11}^{\left(1\right)}(n)+{P}_{11}^{\left(0\right)}(n) & \lambda {P}_{12}^{\left(1\right)}(n)+{P}_{12}^{\left(0\right)}(n)\\ \lambda {P}_{21}^{\left(1\right)}(n)+{P}_{21}^{\left(0\right)}(n) & {P}_{22}^{\left(0\right)}(n)\end{array}\right).\end{eqnarray}$ Hence we have $\begin{eqnarray}{T}_{n+1}{U}_{n}={P}_{n}{T}_{n}.\end{eqnarray}$ Compare the coefficient of λ on both sides of equation (43) together with (36), we can get the following results as $\begin{eqnarray*}\begin{array}{rcl}{P}_{11}^{\left(2\right)}(n) & = & \displaystyle \frac{{f}_{n+1}^{\left(2\right)}(\alpha {b}_{n+1}+1)}{{f}_{n}^{\left(2\right)}}=\alpha {\tilde{b}}_{n+1}+1,\\ {P}_{11}^{\left(1\right)}(n) & = & 0,\quad \quad {P}_{11}^{\left(0\right)}(n)=-1,\\ {P}_{12}^{\left(1\right)}(n) & = & \displaystyle \frac{\alpha {a}_{n}+{g}_{n}^{\left(1\right)}}{1-\tfrac{{r}_{n}^{\left(1\right)}}{\alpha }}=\displaystyle \frac{\alpha (\alpha {a}_{n}+{g}_{n}^{\left(1\right)})}{\alpha -{r}_{n}^{\left(1\right)}}=\alpha {\tilde{a}}_{n},\ \ {P}_{12}^{\left(0\right)}(n)=0,\\ {P}_{21}^{\left(1\right)}(n) & = & -\alpha ,\ \ {P}_{21}^{\left(0\right)}(n)=0,\quad \quad {P}_{22}^{\left(0\right)}(n)=0.\end{array}\end{eqnarray*}$ Thus, we have ${P}_{n}={\tilde{U}}_{n}$. In other words, the matrices Un and ${\tilde{U}}_{n}$ have the same forms under the transformations (34) and (37).
Next, we try to prove that the matrix ${V}_{n}^{\left(0\right)}$ has the same form as ${\tilde{V}}_{n}^{\left(0\right)}$ under the transformations (34) and (37). Let ${T}_{n}^{-1}={T}_{n}^{* }/\det {T}_{n}$ and $\begin{eqnarray}({T}_{n,t}+{T}_{n}{V}_{n}^{\left(0\right)}){T}_{n}^{* }=\left(\begin{array}{cc}{g}_{11}(\lambda ,n) & {g}_{12}(\lambda ,n)\\ {g}_{21}(\lambda ,n) & {g}_{22}(\lambda ,n)\end{array}\right).\end{eqnarray}$ Through a straightforward calculation, we know that g11(λ, n), g22(λ, n) are (4N + 2)th order polynomials in λ, g12(λ, n), g21(λ, n) are (4N + 1)th order polynomials in λ.
From (17) and ${\delta }_{i,n}=\tfrac{{\varphi }_{i,n}}{{\psi }_{i,n}}$, we can get $\begin{eqnarray}\begin{array}{rcl}{\delta }_{i,n,t} & = & \displaystyle \frac{{\lambda }_{i}}{\alpha {b}_{n}+1}+\left(\displaystyle \frac{{b}_{n}}{\alpha {b}_{n}+1}+\displaystyle \frac{{\lambda }_{i}^{2}-1}{\alpha }\right){\delta }_{i,n}\,+\,\displaystyle \frac{{\lambda }_{i}{a}_{n}}{\alpha {b}_{n+1}+1}{\delta }_{i,n}^{2},\ \ i=1,2,\end{array}\end{eqnarray}$ from which we can verify that g11(λi, n), g12(λi, n), g21(λi, n) and g22(λi, n) are all zeroes, so we have $\begin{eqnarray}({T}_{n,t}+{T}_{n}{M}_{n}){T}_{n}^{* }=\det {T}_{n}\cdot {R}_{n},\end{eqnarray}$ with $\begin{eqnarray}{R}_{n}=\left(\begin{array}{cc}{\lambda }^{2}{R}_{11}^{\left(2\right)}(n)+\lambda {R}_{11}^{\left(1\right)}(n)+{R}_{11}^{\left(0\right)}(n) & \lambda {R}_{12}^{\left(1\right)}(n)+{R}_{12}^{\left(0\right)}(n)\\ \lambda {R}_{21}^{\left(1\right)}(n)+{R}_{21}^{\left(0\right)}(n) & {\lambda }^{2}{R}_{22}^{\left(2\right)}(n)+\lambda {R}_{22}^{\left(1\right)}(n)+{R}_{22}^{\left(0\right)}(n)\end{array}\right).\end{eqnarray}$ Therefore we obtain $\begin{eqnarray}{T}_{n,t}+{T}_{n}{M}_{n}={R}_{n}{T}_{n}.\end{eqnarray}$ Expanding and comparing the coefficient of λ on both sides of equation (48) together with (36), we can get the following results as $\begin{eqnarray*}\begin{array}{rcl}{R}_{11}^{\left(2\right)}(n) & = & -\displaystyle \frac{1}{2\alpha },\quad \quad {R}_{11}^{\left(1\right)}(n)=0,\quad \quad {R}_{11}^{\left(0\right)}(n)=0,\\ {R}_{12}^{\left(1\right)}(n) & = & -\displaystyle \frac{{a}_{n}}{\alpha {b}_{n+1}+1}+\displaystyle \frac{{g}_{n}^{\left(1\right)}}{\alpha {f}_{n}^{\left(2\right)}}=-\displaystyle \frac{{\tilde{a}}_{n}}{\alpha {\tilde{b}}_{n+1}+1},\\ {R}_{12}^{\left(0\right)}(n) & = & 0,\quad \quad {R}_{21}^{\left(1\right)}(n)=\displaystyle \frac{{r}_{n}^{\left(1\right)}}{\alpha {f}_{n}^{\left(2\right)}}+\displaystyle \frac{1}{\alpha {b}_{n}+1}=\displaystyle \frac{1}{\alpha {\tilde{b}}_{n}+1},\\ {R}_{21}^{\left(0\right)}(n) & = & 0,\quad \quad {R}_{22}^{\left(2\right)}(n)=\displaystyle \frac{1}{2\alpha },\\ {R}_{22}^{\left(1\right)}(n) & = & 0,\quad \quad {R}_{22}^{\left(0\right)}(n)=-\displaystyle \frac{{R}_{21}^{\left(1\right)}(n)}{\alpha }=-\displaystyle \frac{1}{\alpha (\alpha {\tilde{b}}_{n}+1)}.\end{array}\end{eqnarray*}$ Therefore, we obtain ${R}_{n}={\tilde{V}}_{n}^{\left(0\right)}$, that is to say, the matrices ${V}_{n}^{\left(0\right)}$ and ${\tilde{V}}_{n}^{\left(0\right)}$ have the same forms under the transformations (34) and (37). The proof is completed.
The transformations (34) and (37) using 2N spectral parameters λi (i = 1, 2, …, 2N) are usually called the 2N-fold DT of Lax pair (4) and (17) of equation (3). Here the number 2N represents the order of DT. For the 2N-fold DT, we need 2N spectral parameters to construct multi-soliton solutions. In [27-29], one of the authors in this paper has presented a discrete generalized (m, N − m)-fold DT with no more than N spectral parameters (i.e. 1 ≤ m < N) to give the rogue wave and rational soltion solutions of several integrable NDDEs. Next, we shall extend this idea to equation (3) for seeking some new rational, semi-rational and mixed solutions by using no more than 2N spectral parameters. To this aim, we must reduce the number of spectral parameter λ, in other words, we use the less m (1 ≤ m < 2N) spectral parameters. We know that the condition Tn(λi)φn(λi) = 0 (i = 1, 2, …, m) yields the linear algebraic system with only 2m algebraic equations so that we can not determine 4N unknown variables ${a}_{n}^{\left(j\right)},{b}_{n}^{\left(j\right)},{c}_{n}^{\left(j\right)},{d}_{n}^{\left(j\right)}$. To obtain the linear algebraic system with 4N algebraic equations of 4N unknown variables ${a}_{n}^{\left(j\right)},{b}_{n}^{\left(j\right)},{c}_{n}^{\left(j\right)},{d}_{n}^{\left(j\right)}$, for every λi, we expand $\begin{eqnarray}\begin{array}{l}T({\lambda }_{i}+\varepsilon ){\phi }_{i,n}({\lambda }_{i}+\varepsilon )\,=\,\sum _{K=0}^{N-1}\sum _{j=0}^{k}{T}^{\left(j\right)}({\lambda }_{i}){\phi }_{i,n}^{\left(k-j\right)}({\lambda }_{i}){\varepsilon }^{k},\end{array}\end{eqnarray}$ where ${\varphi }_{n}^{\left(k\right)}({\lambda }_{i})=\tfrac{1}{k!}\tfrac{{\partial }^{k}}{\partial {\lambda }_{i}^{k}}{\varphi }_{n}({\lambda }_{i})$, and ϵ is a small parameter. In the expression (49), we make binomial expansions for Tn(λi + ϵ) as $T({\lambda }_{i}+\varepsilon )={T}_{n}^{\left(0\right)}+{T}_{n}^{\left(1\right)}\varepsilon +\cdots +{T}_{n}^{({m}_{i})}{\varepsilon }^{{m}_{i}}$, while for φn(λi + ϵ) we utilize the Taylor series around ϵ = 0 as ${\phi }_{i,n}({\lambda }_{i}+\varepsilon )={\phi }_{i,n}^{\left(0\right)}({\lambda }_{i})+{\phi }_{i,n}^{\left(1\right)}({\lambda }_{i})\varepsilon +{\phi }_{i,n}^{\left(2\right)}({\lambda }_{i}){\varepsilon }^{2}+{\phi }_{i,n}^{\left(3\right)}({\lambda }_{i}){\varepsilon }^{3}+\cdots $. Thus the formula ${\mathrm{lim}}_{\varepsilon \to 0}\tfrac{{T}_{n}({\lambda }_{i}+\varepsilon ){\phi }_{n}({\lambda }_{i}+\varepsilon )}{{\varepsilon }^{{k}_{i}}}=0$ ($i=1,2,\ldots ,m,{k}_{i}=0,1,\ldots ,{v}_{i},2N=m+{\sum }_{i=1}^{m}{v}_{i}$) can produce 4N algebraic equations for 4N unknown functions ${a}_{n}^{\left(j\right)},{b}_{n}^{\left(j\right)},{c}_{n}^{\left(j\right)},{d}_{n}^{\left(j\right)}$, i.e. $\begin{eqnarray*}\left\{\begin{array}{l}{T}^{\left(0\right)}({\lambda }_{i}){\phi }_{i,n}^{\left(0\right)}({\lambda }_{i})=0,\\ {T}^{\left(0\right)}({\lambda }_{i}){\phi }_{i,n}^{\left(1\right)}({\lambda }_{i})+{T}^{\left(1\right)}({\lambda }_{i}){\phi }_{i,n}^{\left(0\right)}({\lambda }_{i})=0,\\ {T}_{n}^{\left(0\right)}({\lambda }_{i}){\phi }_{i,n}^{\left(2\right)}({\lambda }_{i})+{T}_{n}^{\left(1\right)}({\lambda }_{i}){\phi }_{i,n}^{\left(1\right)}({\lambda }_{i})+{T}_{n}^{\left(2\right)}({\lambda }_{i}){\phi }_{i,n}^{\left(0\right)}({\lambda }_{i})=0,\\ \cdots \cdots ,\\ \sum _{j=0}^{{v}_{i}}{T}^{\left(j\right)}({\lambda }_{i}){\phi }_{i,n}^{\left({v}_{i}-j\right)}({\lambda }_{i})=0,\end{array}\right.\end{eqnarray*}$ from which the determinant of the coefficients for system (4) is nonzero when the m spectral parameters λi are suitably chosen so that ${a}_{n}^{\left(j\right)},{b}_{n}^{\left(j\right)},{c}_{n}^{\left(j\right)},{d}_{n}^{\left(j\right)}$ in the Darboux matrix Tn are uniquely determined by (4). Moreover, theorem 2 still holds for the Darboux matrix Tn with new ${a}_{n}^{\left(j\right)},{b}_{n}^{\left(j\right)},{c}_{n}^{\left(j\right)},{d}_{n}^{\left(j\right)}$ given by the new system (4). Due to new distinct functions ${a}_{n}^{\left(j\right)},{b}_{n}^{\left(j\right)},{c}_{n}^{\left(j\right)},{d}_{n}^{\left(j\right)}$ obtained in the Darboux matrix Tn in theorem 2, we can derive the new DT with m spectral parameters λi. Here the transformations (34) and (37) with m spectral parameters related to new functions ${a}_{n}^{\left(j\right)},{b}_{n}^{\left(j\right)},{c}_{n}^{\left(j\right)},{d}_{n}^{\left(j\right)}$ given by (4) are called the discrete generalized (m, 2N − m)-fold DT of equation (3) which is summarized as the following theorem: Let ${\phi }_{i,n}{({\lambda }_{i})=({\varphi }_{i,n},{\psi }_{i,n})}^{{\rm{T}}}$ be m column vector solutions of Lax pair (4) and (17) for the spectral parameters ${\lambda }_{i}\ (i=1,2,\ldots ,m)$ with the initial solutions ${a}_{n},{b}_{n}$ of equation (3), then the generalized $(m,2N-m)$-fold DT of equation (3) from the old solutions ${a}_{n},$ bn to the new solutions ${\tilde{a}}_{n},$ ${\tilde{b}}_{n}$ is given by $\begin{eqnarray}{\tilde{a}}_{n}=\displaystyle \frac{\alpha {a}_{n}+{g}_{n}^{\left(1\right)}}{\alpha -{r}_{n}^{\left(1\right)}},{\tilde{b}}_{n}=\displaystyle \frac{\alpha {b}_{n}{f}_{n}^{\left(2N\right)}-{f}_{n-1}^{\left(2N\right)}+{f}_{n}^{\left(2N\right)}}{\alpha {f}_{n-1}^{\left(2N\right)}},\end{eqnarray}$ where $\begin{eqnarray*}{f}_{n}^{\left(2N\right)}=\displaystyle \frac{{\rm{\Delta }}{f}_{n}^{\left(2N\right)}}{{{\rm{\Delta }}}_{1}},{g}_{n}^{\left(1\right)}=\displaystyle \frac{{\rm{\Delta }}{g}_{n}^{\left(1\right)}}{{{\rm{\Delta }}}_{1}},{r}_{n}^{\left(1\right)}=\displaystyle \frac{{\rm{\Delta }}{r}_{n}^{\left(1\right)}}{{{\rm{\Delta }}}_{2}},\end{eqnarray*}$ with ${{\rm{\Delta }}}_{1}=({{\rm{\Delta }}}_{1}^{\left(1\right)}$, ${{\rm{\Delta }}}_{1}^{\left(2\right)},\ldots ,{{\rm{\Delta }}}_{1}^{\left(m\right)}{)}^{{\rm{T}}}$, ${{\rm{\Delta }}}_{2}=({{\rm{\Delta }}}_{2}^{\left(1\right)}$, ${{\rm{\Delta }}}_{2}^{\left(2\right)},\ldots ,{{\rm{\Delta }}}_{2}^{\left(m\right)}{)}^{{\rm{T}}}$, ${{\rm{\Delta }}}_{1}^{\left(i\right)}={({{\rm{\Delta }}}_{1,j,s}^{\left(i\right)})}_{2({v}_{i}+1)\times 2N}$, ${{\rm{\Delta }}}_{2}^{\left(i\right)}={({{\rm{\Delta }}}_{2,j,s}^{\left(i\right)})}_{2({v}_{i}+1)\times 2N}$, in which ${{\rm{\Delta }}}_{1,j,s}^{\left(i\right)}$, ${{\rm{\Delta }}}_{2,j,s}^{\left(i\right)}(1\leqslant j\leqslant 2({v}_{i}+1)$, $1\leqslant s\leqslant 2N,i=1,2,\ldots ,m)$ are given as $\begin{eqnarray*}\begin{array}{rcl}{{\rm{\Delta }}}_{1,j,s}^{\left(i\right)} & = & \left\{\begin{array}{ll}{\sum }_{k=0}^{j-1}{C}_{2N-2s}^{k}{\lambda }_{i}^{2N-2s-k}{\varphi }_{i,n}^{\left(j-1-k\right)} & \mathrm{for}1\leqslant j\leqslant {v}_{i}+1,1\leqslant s\leqslant N,\\ {\sum }_{k=0}^{j-1}{C}_{4N-2s-1}^{k}{\lambda }_{i}^{4N-2s-k-1}{\psi }_{i,n}^{\left(j-1-k\right)} & \mathrm{for}1\leqslant j\leqslant {v}_{i}+1,N+1\leqslant s\leqslant 2N,\end{array}\right.\\ {{\rm{\Delta }}}_{2,j,s}^{\left(i\right)} & = & \left\{\begin{array}{ll}{\sum }_{k=0}^{j-(N+1)}{C}_{2N-2s-1}^{k}{\lambda }_{i}^{2N-2s-k-1}{\varphi }_{i,n}^{\left(j-N-1-k\right)} & \mathrm{for}\quad {v}_{i}+1\leqslant j\leqslant 2({v}_{i}+1),1\leqslant s\leqslant N,\\ \sum _{k=0}^{j-(N+1)}{C}_{4N-2s}^{k}{\lambda }_{i}^{4N-2s-k}{\psi }_{i,n}^{\left(j-N-1-k\right)} & \mathrm{for}\quad {v}_{i}+1\leqslant j\leqslant 2({v}_{i}+1),N+1\leqslant s\leqslant 2N,\end{array}\right.\end{array}\end{eqnarray*}$ where ${\rm{\Delta }}{f}_{n}^{\left(2N\right)}$ and ${g}_{n}^{\left(1\right)}$ are given from the determinant ${{\rm{\Delta }}}_{1}$ by replacing their Nth and $(N+1)$ th columns by the column vector $({f}_{1}^{\left(1\right)},{f}_{2}^{\left(1\right)},\ldots ,{f}_{({v}_{1}+1)}^{\left(1\right)},\ldots ,{f}_{1}^{\left(i\right)},{f}_{2}^{\left(i\right)},\ldots ,{f}_{({v}_{i}+1)}^{\left(i\right)},\ldots ,{f}_{1}^{\left(m\right)},{f}_{2}^{\left(m\right)},\ldots ,{f}_{({v}_{m}+1)}^{\left(m\right)})$ with ${f}_{j}^{\left(i\right)}=-{\varphi }_{i,n}^{\left(j-1\right)}(1\leqslant j\leqslant ({v}_{i}+1),1\leqslant i\leqslant m)$ respectively, while ${\rm{\Delta }}{r}_{n}^{\left(1\right)}$ is obtained from the determinant ${{\rm{\Delta }}}_{2}$ by replacing the first columns by the column vector $({r}_{1}^{\left(1\right)},{r}_{2}^{\left(1\right)},\ldots ,{r}_{({v}_{1}+1)}^{\left(1\right)},\ldots ,{r}_{1}^{\left(i\right)},{r}_{2}^{\left(i\right)},\ldots ,{r}_{({v}_{i}+1)}^{\left(i\right)},\ldots ,{r}_{1}^{\left(m\right)},{r}_{2}^{\left(m\right)},\ldots ,{r}_{({v}_{m}+1)}^{\left(m\right)})$ with ${r}_{j}^{\left(i\right)}=-{\psi }_{i,n}^{\left(j-1\right)}(1\leqslant j\leqslant ({v}_{i}+1),1\leqslant i\leqslant m)$. Here the transformations (34) and (50) are called the discrete generalized $(m,2N-m)$-fold DT of equation (3). Note that m in the notation '$(m,2N-m)$'denotes the number of the distinct spectral parameter we use, 2N denotes the order number of DT. In theorem 3, vi means the order number of the highest derivative in Taylor series expansion for every ${\phi }_{i,n}({\lambda }_{i})$, and $2N-m={\sum }_{i=1}^{m}{v}_{i}$ is the order number sum of the highest derivative of Darboux matrix Tn or the vector eigenfunction ${\phi }_{i,n}({\lambda }_{i})$. Notice that if $m=2N$ and ${m}_{i}=0$, theorem 3 can reduce to the discrete generalized $(2N,0)$-fold DT including the discrete 2N-fold DT when we do not make Taylor series expansion for every ${\phi }_{i,n}({\lambda }_{i})$. If m = 1 and ${m}_{i}=2N-1$, theorem 3 can reduce to the discrete generalized $(1,2N-1)$-fold DT which is used to derive higher-order rational and semi-rational solutions of equation (3). If m = 2 and ${m}_{i}=2N-2$, theorem 3 can reduce to the discrete generalized $(2,2N-2)$-fold DT which is used to obtain mixed solutions of usual soliton solutions and rational or semi-rational solutions. If $2\lt m\lt 2N$, theorem 3 can reduce to the other discrete generalized DTs which can give the new discrete mixed solutions. In the next section, the discrete generalized $(m,2N-m)$-fold DT with $m=2N,1,2$ will be used to derive usual soliton solutions, rational or semi-rational solutions and their mixed solutions for equation (3).
5. Explicit exact solutions and asymptotic state analysis of equation (3)
In this section, we will use the discrete generalized (m, 2N − m)-fold DT with three cases m = 2N, 1, 2 to discrete soliton solutions, discrete rational and semi-rational solutions, and their mixed solutions.
5.1. Multi-soliton solutions via the discrete generalized (2N, 0)-fold DT
When m = 2N, the discrete generalized (m, 2N − m)-fold DT reduces to the discrete generalized (2N, 0)-fold DT which includes the usual 2N-fold DT. If we do not make Taylor series expansion for every eigenfunction φi,n(λi) (i = 1, 2, …, 2N), the discrete generalized (2N, 0)-fold DT is just the usual 2N-fold DT, while if we do Taylor series expansion for one of eigenfunctions φi,n(λi) (i = 1, 2, …, 2N), in fact, we can give some mixed solutions of usual soliton solutions and rational or semi-rational solutions. In this subsection, we shall use the usual 2N-fold DT to give multi-soliton solutions of equation (3). Substituting the initial seed solutions an = 1, bn = 0 into (4) and (17) gives one basic solution with λ = λi (i = 1, 2, …, 2N) as follows: $\begin{eqnarray}\begin{array}{l}{\phi }_{i,n}={\left(\begin{array}{c}{\varphi }_{i,n,}{\psi }_{i,n}\end{array}\right)}^{{\rm{T}}}=\left(\begin{array}{c}{C}_{i1}{\tau }_{i1}^{n}{{\rm{e}}}^{{\rho }_{i1}t}+{C}_{i2}{\tau }_{i2}^{n}{{\rm{e}}}^{{\rho }_{i2}t}\\ -\alpha {\lambda }_{i}{C}_{i1}{\tau }_{i1}^{n-1}{{\rm{e}}}^{{\rho }_{i1}t}-\alpha {\lambda }_{i}{C}_{i2}{\tau }_{i2}^{n-1}{{\rm{e}}}^{{\rho }_{i2}t}\end{array}\right),\end{array}\end{eqnarray}$ with $\begin{eqnarray*}\begin{array}{rcl}{\tau }_{i1} & = & \displaystyle \frac{1}{2}{\lambda }_{i}^{2}-\displaystyle \frac{1}{2}+\displaystyle \frac{1}{2}\sqrt{-4{\alpha }^{2}{\lambda }_{i}^{2}+{\lambda }_{i}^{4}-2{\lambda }_{i}^{2}+1},\\ {\tau }_{i2} & = & \displaystyle \frac{1}{2}{\lambda }_{i}^{2}-\displaystyle \frac{1}{2}-\displaystyle \frac{1}{2}\sqrt{-4{\alpha }^{2}{\lambda }_{i}^{2}+{\lambda }_{i}^{4}-2{\lambda }_{i}^{2}+1},\\ {\rho }_{i1} & = & \displaystyle \frac{{\tau }_{i1}+(1-{\lambda }_{i})}{-\alpha }=\displaystyle \frac{1}{2}\displaystyle \frac{{\lambda }_{i}^{2}-1-\sqrt{-4{\alpha }^{2}{\lambda }_{i}^{2}+{\lambda }_{i}^{4}-2{\lambda }_{i}^{2}+1}}{\alpha },\\ {\rho }_{i2} & = & \displaystyle \frac{{\tau }_{i2}+(1-{\lambda }_{i})}{-\alpha }=\displaystyle \frac{1}{2}\displaystyle \frac{{\lambda }_{i}^{2}-1+\sqrt{-4{\alpha }^{2}{\lambda }_{i}^{2}+{\lambda }_{i}^{4}-2{\lambda }_{i}^{2}+1}}{\alpha }.\end{array}\end{eqnarray*}$ According to (37), we can obtain exact 2N-soliton solutions of equation (3). To understand them, we plot their structures with N = 1, 2 as shown in figures 2-6.
Figure 2.
New window|Download| PPT slide Figure 2.(Color online) One-soliton solutions with parameters ${\lambda }_{1}=\tfrac{3}{5},{\lambda }_{2}=2,\alpha =\tfrac{8}{15},{C}_{11}=1,{C}_{12}=\tfrac{1}{2},{C}_{21}=1,{C}_{22}=2$. (a1) The component ${\tilde{a}}_{n}$. (a2) The component ${\tilde{b}}_{n}$. The propagation processes for $(b1)\ {\tilde{a}}_{n}$ and $(b2)\ {\tilde{b}}_{n}$ at t = −10 (long-dash line), t = 0 (dash-dot line) and t = 10 (dot line).
(I) When N = 1, λ = λi (i = 1, 2), based on the 2-fold DT in theorem 2, the transformation (37) produces the two-fold exact solutions as $\begin{eqnarray}{\tilde{a}}_{n}=\displaystyle \frac{\alpha +{g}_{n}^{\left(1\right)}}{\alpha -{r}_{n}^{\left(1\right)}},{\tilde{b}}_{n}=\displaystyle \frac{{f}_{n}^{\left(2\right)}-{f}_{n-1}^{\left(2\right)}}{\alpha {f}_{n-1}^{\left(2\right)}},\end{eqnarray}$ where $\begin{eqnarray*}{g}_{n}^{\left(1\right)}=\displaystyle \frac{{\rm{\Delta }}{g}_{n}^{\left(1\right)}}{{{\rm{\Delta }}}_{1,n}},\quad {f}_{n}^{\left(2\right)}=\displaystyle \frac{{\rm{\Delta }}{f}_{n}^{\left(2\right)}}{{{\rm{\Delta }}}_{1,n}},\quad {r}_{n}^{\left(1\right)}=\displaystyle \frac{{\rm{\Delta }}{r}_{n}^{\left(1\right)}}{{{\rm{\Delta }}}_{2,n}},\end{eqnarray*}$ with $\begin{eqnarray*}\begin{array}{rcl}{{\rm{\Delta }}}_{1,n} & = & \left|\begin{array}{cc}{\lambda }_{1}^{2} & {\lambda }_{1}{\delta }_{1,n}\\ {\lambda }_{2}^{2} & {\lambda }_{2}{\delta }_{2,n}\end{array}\right|,\\ {{\rm{\Delta }}}_{2,n} & = & \left|\begin{array}{cc}{\lambda }_{1}-\displaystyle \frac{1}{\alpha }{\delta }_{1,n} & {\lambda }_{1}^{2}{\delta }_{1,n}\\ {\lambda }_{2}-\displaystyle \frac{1}{\alpha }{\delta }_{2,n} & {\lambda }_{2}^{2}{\delta }_{2,n}\end{array}\right|,\\ {\rm{\Delta }}{f}_{n}^{\left(2\right)} & = & \left|\begin{array}{cc}-1 & {\lambda }_{1}{\delta }_{1,n}\\ -1 & {\lambda }_{2}{\delta }_{2,n}\end{array}\right|,{\rm{\Delta }}{g}_{n}^{\left(1\right)}=\left|\begin{array}{cc}{\lambda }_{1}^{2} & -1\\ {\lambda }_{2}^{2} & -1\end{array}\right|,\\ {\rm{\Delta }}{r}_{n}^{\left(1\right)} & = & \left|\begin{array}{cc}-{\delta }_{1,n} & {\lambda }_{1}^{2}{\delta }_{1,n}\\ -{\delta }_{2,n} & {\lambda }_{2}^{2}{\delta }_{2,n}\end{array}\right|.\end{array}\end{eqnarray*}$ The solutions (52) may be one-soliton or two-soliton solutions when the parameters are suitably chosen.
Case (a) When one of the two parameters λ1, λ2 is equal to $-\alpha +\sqrt{{\alpha }^{2}+1}$, the solutions (52) are one-soliton solutions. Here we set ${\lambda }_{1}=-\alpha +\sqrt{{\alpha }^{2}+1}$, then the analytical expressions of solutions ${\tilde{a}}_{n},$ ${\tilde{b}}_{n}$ are given as follows: $\begin{eqnarray}\begin{array}{rcl}{\tilde{a}}_{n} & = & \displaystyle \frac{\alpha +{g}_{n}^{\left(1\right)}}{\alpha -{r}_{n}^{\left(1\right)}}=\displaystyle \frac{{A}_{1}}{{B}_{1}},\\ {\tilde{b}}_{n} & = & -\displaystyle \frac{1}{\alpha }+\displaystyle \frac{{f}_{n}^{\left(2\right)}}{\alpha {f}_{n-1}^{\left(2\right)}}=-\displaystyle \frac{1}{\alpha }+\displaystyle \frac{{C}_{1}}{{D}_{1}},\end{array}\end{eqnarray}$ in which $\begin{eqnarray*}\begin{array}{l}{A}_{1}=[({\alpha }^{2}{\lambda }_{2}^{2}{A}^{2}+\displaystyle \frac{1}{2}{\lambda }_{2}^{2}{A}^{2}-\displaystyle \frac{1}{2}{\lambda }_{2}^{2}-\displaystyle \frac{1}{2}{A}^{4}+\displaystyle \frac{1}{2}{A}^{2})\\ \ \ \times \,\cosh {\xi }_{2}-\alpha {\lambda }_{2}{A}^{2}(\displaystyle \frac{1}{2}{A}^{2}-\displaystyle \frac{1}{2})\\ \ \ \times \,\cosh ({\xi }_{2}-{X}_{2})][\alpha {A}^{3}{\lambda }_{2}\cosh {\xi }_{2}\\ \ \ +\,({A}^{3}-\displaystyle \frac{1}{2}{A}^{3}{\lambda }_{2}^{2}-{\lambda }_{2}^{2}A+\displaystyle \frac{1}{2}A{\lambda }_{2}^{2})\cosh ({\xi }_{2}-{X}_{2})],\\ {B}_{1}=\alpha {A}^{2}{\lambda }_{2}^{2}\left[\alpha {\lambda }_{2}A\cosh {\xi }_{2}-A\left(\displaystyle \frac{1}{2}{A}^{2}-\displaystyle \frac{1}{2}\right)\right.\\ \ \ \left.\times \,\cosh ({\xi }_{2}-{X}_{2})\right]\left[\alpha {A}^{2}\cosh {\xi }_{2}-{\lambda }_{2}\left(\displaystyle \frac{1}{2}{A}^{2}-\displaystyle \frac{1}{2}\right)\cosh ({\xi }_{2}-{X}_{2})\right],\\ {C}_{1}=\left[{\lambda }_{2}(\displaystyle \frac{1}{2}{A}^{2}-\displaystyle \frac{1}{2})\cosh ({\xi }_{2}-{X}_{2})-\alpha {A}^{2}\cosh {\xi }_{2}\right]\\ \ \times \,\left[\alpha {\lambda }_{2}\cosh ({\xi }_{2}-{X}_{2})-\left(\displaystyle \frac{1}{2}{A}^{2}-\displaystyle \frac{1}{2}\right)\cosh ({\xi }_{2}-2{X}_{2})\right],\\ {D}_{1}=\alpha \left[\alpha {\lambda }_{2}\cosh {\xi }_{2}-(\displaystyle \frac{1}{2}{A}^{2}-\displaystyle \frac{1}{2})\cosh ({\xi }_{2}-{X}_{2})\right]\\ \ \times \,\left[{\lambda }_{2}\left(\displaystyle \frac{1}{2}{A}^{2}-\displaystyle \frac{1}{2}\right)\cosh ({\xi }_{2}-2{X}_{2})-\alpha {A}^{2}\cosh ({\xi }_{2}-{X}_{2})\right],\end{array}\end{eqnarray*}$ with $\begin{eqnarray*}\begin{array}{rcl}{\xi }_{2} & = & \displaystyle \frac{1}{2}[(\mathrm{ln}{\tau }_{21}-\mathrm{ln}{\tau }_{22})n+({\rho }_{21}-{\rho }_{22})t\\ & & +\,(\mathrm{ln}{C}_{21}-\mathrm{ln}{C}_{22})],\\ {X}_{2} & = & \displaystyle \frac{1}{2}(\mathrm{ln}{\tau }_{21}-\mathrm{ln}{\tau }_{22}),\ A=-\alpha +\sqrt{{\alpha }^{2}+1}.\end{array}\end{eqnarray*}$ The wave structures of one-soliton solutions (52) are shown in figure 2. Figure 2 (a1)(b1) present the bell-shaped soliton structures of the component ${\tilde{a}}_{n}$ on nonzero seed background. Figure 2 (a2)(b2) display the anti-bell-shaped soliton structures of the component ${\tilde{b}}_{n}$ on zero seed background. From figure 2, both one-soliton solutions ${\tilde{a}}_{n}$ and ${\tilde{b}}_{n}$ propagate stably with the same amplitude and remain their shapes and velocities unchanged during the propagation.
Case (b) When both of the two parameters λ1, λ2 are not equal to $-\alpha +\sqrt{{\alpha }^{2}+1}$, the solutions (52) are two-soliton solutions expressed as $\begin{eqnarray}\begin{array}{rcl}{\tilde{a}}_{n} & = & \displaystyle \frac{\alpha +{g}_{n}^{\left(1\right)}}{\alpha -{r}_{n}^{\left(1\right)}}=\displaystyle \frac{{A}_{2}}{{B}_{2}},\\ {\tilde{b}}_{n} & = & -\displaystyle \frac{1}{\alpha }+\displaystyle \frac{{f}_{n}^{\left(2\right)}}{\alpha {f}_{n-1}^{\left(2\right)}}=-\displaystyle \frac{1}{\alpha }+\displaystyle \frac{{C}_{2}}{{D}_{2}},\end{array}\end{eqnarray}$ where $\begin{eqnarray*}\begin{array}{rl}{A}_{2} & =\,[\alpha {\lambda }_{1}{\lambda }_{2}^{2}\cosh ({\xi }_{1}-{X}_{1})\cosh {\xi }_{2}-\alpha {\lambda }_{1}^{2}{\lambda }_{2}\cosh {\xi }_{1}\\ & \times \,\cosh \,({\xi }_{2}-{X}_{2})+({\lambda }_{2}^{2}-{\lambda }_{1}^{2})\cosh {\xi }_{1}\cosh {\xi }_{2}]\\ & \times [\alpha {\lambda }_{1}^{2}{\lambda }_{2}\cosh ({\xi }_{1}-{X}_{1})\cosh {\xi }_{2}-\alpha {\lambda }_{1}{\lambda }_{2}^{2}\cosh {\xi }_{1}\\ & \times \,\cosh \,({\xi }_{2}-{X}_{2})+({\lambda }_{1}^{2}-{\lambda }_{2}^{2})\cosh ({\xi }_{1}-{X}_{1})\cosh ({\xi }_{2}-{X}_{2})],\\ {B}_{2} & =\,{\alpha }^{2}{\lambda }_{1}^{2}{\lambda }_{2}^{2}[{\lambda }_{2}\cosh ({\xi }_{1}-{X}_{1})\cosh {\xi }_{2}-{\lambda }_{1}\cosh {\xi }_{1}\\ & \times \,\cosh ({\xi }_{2}-{X}_{2})][{\lambda }_{1}\cosh ({\xi }_{1}-{X}_{1})\cosh {\xi }_{2}\\ & -{\lambda }_{2}\cosh {\xi }_{1}\cosh ({\xi }_{2}-{X}_{2})],\\ {C}_{2} & =\,[{\lambda }_{2}\cosh {\xi }_{1}\cosh ({\xi }_{2}-{X}_{2})-{\lambda }_{1}\cosh ({\xi }_{1}-{X}_{1})\\ & \times \,\cosh {\xi }_{2}][{\lambda }_{2}\cosh ({\xi }_{1}-2{X}_{1})\cosh ({\xi }_{2}-{X}_{2})\\ & -{\lambda }_{1}\cosh ({\xi }_{1}-{X}_{1})\cosh ({\xi }_{2}-2{X}_{2})],\\ {D}_{2} & =\,\alpha [{\lambda }_{2}\cosh ({\xi }_{1}-{X}_{1})\cosh {\xi }_{2}-{\lambda }_{1}\cosh \,{\xi }_{1}\\ & \times \,\cosh ({\xi }_{2}-{X}_{2})][{\lambda }_{2}\cosh ({\xi }_{1}-{X}_{1})\cosh ({\xi }_{2}-2{X}_{2})\\ & -\,{\lambda }_{1}\cosh ({\xi }_{1}-2{X}_{1})\cosh ({\xi }_{2}-{X}_{2})]\end{array}\end{eqnarray*}$ with $\begin{eqnarray*}\begin{array}{l}{\xi }_{i}=\displaystyle \frac{1}{2}[(\mathrm{ln}{\tau }_{i1}-\mathrm{ln}{\tau }_{i2})n+({\rho }_{i1}-{\rho }_{i2})t\\ \ \,+\,(\mathrm{ln}{C}_{i1}-\mathrm{ln}{C}_{i2})],\ \\ {X}_{i}=\displaystyle \frac{1}{2}(\mathrm{ln}{\tau }_{i1}-\mathrm{ln}{\tau }_{i2}),i=1,2.\end{array}\end{eqnarray*}$ To find whether the interaction between two solitons is elastic, we carry out the asymptotic analysis for solutions (54), which yields the following eight asymptotic expressions of ${\tilde{a}}_{n}$ and ${\tilde{b}}_{n}$.
From the above analysis, we can see that the interactions between two solitons for the solutions (54) are elastic. When the parameters are properly chosen, the elastic interaction structures of two-soliton solutions (54) are shown in figure 3. Figure 3 (a1)(b1) present the headon elastic interaction between two bell-shaped solitons structures of the component ${\tilde{a}}_{n}$ on nonzero seed background. Figure 3 (a2)(b2) display the head-on elastic interaction between one bell-shaped bright soliton and one anti-bell-shaped soliton of the component ${\tilde{b}}_{n}$ on zero seed background. From figure 3, we can clearly see that the shapes of two solitons remain the same before and after the interactions such that their interactions are elastic, which are consistent with our asymptotic analysis above.
Figure 3.
New window|Download| PPT slide Figure 3.(Color online) Head-on elastic interaction between two solitons via the solutions (54) with parameters ${\lambda }_{1}=\tfrac{1}{3},{\lambda }_{2}=3,\alpha ={C}_{11}={C}_{12}={C}_{21}=1,{C}_{22}=\tfrac{1}{2}$. (a1) The component ${\tilde{a}}_{n}$. (a2) The component ${\tilde{b}}_{n}$. The propagation processes for $(b1)\ {\tilde{a}}_{n}$ and $(b2)\ {\tilde{b}}_{n}$ at t = − 10 (longdash line), t = 0 (dashdot line) and t = 10 (dot line).
(II) When N = 2, λ = λi (i = 1, 2, 3, 4), based on the 4-fold DT in theorem 2, the transformation (37) produces the four-fold exact solutions as follows: $\begin{eqnarray}{\tilde{a}}_{n}=\displaystyle \frac{\alpha +{g}_{n}^{\left(1\right)}}{\alpha -{r}_{n}^{\left(1\right)}},{\tilde{b}}_{n}=\displaystyle \frac{{f}_{n}^{\left(4\right)}-{f}_{n-1}^{\left(4\right)}}{\alpha {f}_{n-1}^{\left(4\right)}},\end{eqnarray}$ where $\begin{eqnarray*}{g}_{n}^{\left(1\right)}=\displaystyle \frac{{\rm{\Delta }}{g}_{n}^{\left(1\right)}}{{{\rm{\Delta }}}_{1,n}},\quad \quad {f}_{n}^{\left(4\right)}=\displaystyle \frac{{\rm{\Delta }}{f}_{n}^{\left(4\right)}}{{{\rm{\Delta }}}_{1,n}},\quad \quad {r}_{n}^{\left(1\right)}=\displaystyle \frac{{\rm{\Delta }}{r}_{n}^{\left(1\right)}}{{{\rm{\Delta }}}_{2,n}},\end{eqnarray*}$ with $\begin{eqnarray*}\begin{array}{rcl}{{\rm{\Delta }}}_{1,n} & = & \left|\begin{array}{cccc}{\lambda }_{1}^{2} & {\lambda }_{1}^{4} & {\lambda }_{1}{\delta }_{1,n} & {\lambda }_{1}^{3}{\delta }_{1,n}\\ {\lambda }_{2}^{2} & {\lambda }_{2}^{4} & {\lambda }_{2}{\delta }_{1,n} & {\lambda }_{2}^{3}{\delta }_{2,n}\\ {\lambda }_{3}^{2} & {\lambda }_{3}^{4} & {\lambda }_{3}{\delta }_{3,n} & {\lambda }_{3}^{3}{\delta }_{3,n}\\ {\lambda }_{4}^{2} & {\lambda }_{4}^{4} & {\lambda }_{4}{\delta }_{4,n} & {\lambda }_{4}^{3}{\delta }_{4,n}\end{array}\right|,\\ {{\rm{\Delta }}}_{2,n} & = & \left|\begin{array}{cccc}{\lambda }_{1}-\tfrac{1}{\alpha }{\delta }_{1,n} & {\lambda }_{1}^{3} & {\lambda }_{1}^{2}{\delta }_{1,n} & {\lambda }_{1}^{4}{\delta }_{1,n}\\ {\lambda }_{2}-\tfrac{1}{\alpha }{\delta }_{2,n} & {\lambda }_{2}^{3} & {\lambda }_{2}^{2}{\delta }_{2,n} & {\lambda }_{2}^{4}{\delta }_{2,n}\\ {\lambda }_{3}-\tfrac{1}{\alpha }{\delta }_{3,n} & {\lambda }_{3}^{3} & {\lambda }_{3}^{2}{\delta }_{3,n} & {\lambda }_{3}^{4}{\delta }_{3,n}\\ {\lambda }_{4}-\tfrac{1}{\alpha }{\delta }_{4,n} & {\lambda }_{4}^{3} & {\lambda }_{4}^{2}{\delta }_{4,n} & {\lambda }_{4}^{4}{\delta }_{4,n}\end{array}\right|,\\ {\rm{\Delta }}{f}_{n}^{\left(4\right)} & = & \left|\begin{array}{cccc}{\lambda }_{1}^{2} & -1 & {\lambda }_{1}{\delta }_{1,n} & {\lambda }_{1}^{3}{\delta }_{1,n}\\ {\lambda }_{2}^{2} & -1 & {\lambda }_{2}{\delta }_{1,n} & {\lambda }_{2}^{3}{\delta }_{2,n}\\ {\lambda }_{3}^{2} & -1 & {\lambda }_{3}{\delta }_{3,n} & {\lambda }_{3}^{3}{\delta }_{3,n}\\ {\lambda }_{4}^{2} & -1 & {\lambda }_{4}{\delta }_{4,n} & {\lambda }_{4}^{3}{\delta }_{4,n}\end{array}\right|,\\ {\rm{\Delta }}{g}_{n}^{\left(1\right)} & = & \left|\begin{array}{cccc}{\lambda }_{1}^{2} & {\lambda }_{1}^{4} & -1 & {\lambda }_{1}^{3}{\delta }_{1,n}\\ {\lambda }_{2}^{2} & {\lambda }_{2}^{4} & -1 & {\lambda }_{2}^{3}{\delta }_{2,n}\\ {\lambda }_{3}^{2} & {\lambda }_{3}^{4} & -1 & {\lambda }_{3}^{3}{\delta }_{3,n}\\ {\lambda }_{4}^{2} & {\lambda }_{4}^{4} & -1 & {\lambda }_{4}^{3}{\delta }_{4,n}\end{array}\right|,\\ {\rm{\Delta }}{r}_{n}^{\left(1\right)} & = & \left|\begin{array}{cccc}-{\delta }_{1,n} & {\lambda }_{1}^{3} & {\lambda }_{1}^{2}{\delta }_{1,n} & {\lambda }_{1}^{4}{\delta }_{1,n}\\ -{\delta }_{2,n} & {\lambda }_{2}^{3} & {\lambda }_{2}^{2}{\delta }_{2,n} & {\lambda }_{2}^{4}{\delta }_{2,n}\\ -{\delta }_{3,n} & {\lambda }_{3}^{3} & {\lambda }_{3}^{2}{\delta }_{3,n} & {\lambda }_{3}^{4}{\delta }_{3,n}\\ -{\delta }_{4,n} & {\lambda }_{4}^{3} & {\lambda }_{4}^{2}{\delta }_{4,n} & {\lambda }_{4}^{4}{\delta }_{4,n}\end{array}\right|.\end{array}\end{eqnarray*}$ When the parameters are suitably chosen, the solutions (55) may be three-soliton or four-soliton solutions whose corresponding evolution plots are shown in figures 4 and 5. Similar to two-soliton solutions'asymptotic analysis, we can also analyze the three-soliton and four-soliton solutions which are cumbersome and not presented here. Figure 4 (a1)(b1) present the headon elastic interactions among two unidirectional overtaking bell-shaped solitons and one opposite bell-shaped soliton of the component ${\tilde{a}}_{n}$ on nonzero seed background. Figure 4 (a2)(b2) display the headon elastic interactions among two unidirectional overtaking anti-bell-shaped solitons and one opposite bell-shaped bright soliton of the component ${\tilde{b}}_{n}$ on zero seed background. From figure 4, we can clearly see that three solitons preserve their shapes and amplitudes before and after the interactions. Figure 5 (a1)(b1) present the headon elastic interactions among three unidirectional overtaking bell-shaped solitons and one opposite bell-shaped soliton of the component ${\tilde{a}}_{n}$ on nonzero seed background. Figure 5 (a2)(b2) display the headon elastic interactions among three unidirectional overtaking anti-bell-shaped solitons and one opposite bell-shaped bright soliton of the component bn on zero seed background. Note that we plot the absolute value of solution $\tilde{{b}_{n}}$ for the sake of a better view here. From figure 5, we can clearly see that four solitons keep their shapes and amplitudes before and after the interactions. In order to better understand the properties of equation (3) more comprehensively, we plot the structures for the combined potential term ${\tilde{a}}_{n}{\tilde{b}}_{n}$ as illustrated in figure 6. From figure 6, we can distinctly see that the combined potential term ${\tilde{a}}_{n}{\tilde{b}}_{n}$ presents the stable one-, two-, three- and four-soliton structures which also shows that the total energy of the system is conserved from another aspect. The point here is that the solutions (37) are the even $(2N)$-soliton solutions if all ${\lambda }_{j}\ (j=1,2,\ldots ,2N)$ are not equal to $-\alpha +\sqrt{{\alpha }^{2}+1}$, however, if only one of ${\lambda }_{j}\ (j=1,2,\ldots ,2N)$ is equivalent to $-\alpha +\sqrt{{\alpha }^{2}+1}$, the solutions (37) are the odd $(2N-1)$-order soliton solutions. The multi-soliton solutions of equation (3) are composed of the even $(2N)$-soliton and odd $(2N-1)$-soliton solutions which is a very interesting feature worthy of further study.
Figure 4.
New window|Download| PPT slide Figure 4.(Color online) Three-soliton elastic interaction structures via the solutions (55) with parameters ${\lambda }_{1}=\tfrac{1}{2},{\lambda }_{2}=\tfrac{5}{2},{\lambda }_{3}\,=3,{\lambda }_{4}=\tfrac{1}{3},\alpha =\tfrac{3}{4},{C}_{11}=1,{C}_{12}=2,{C}_{21}=1,{C}_{22}=1,{C}_{31}\,=-1,{C}_{32}=1,{C}_{41}=-1,{C}_{42}=1$. (a1) The component ${\tilde{a}}_{n}$. (a2) The component ${\tilde{b}}_{n}$. The propagation processes for $(b1)\ {\tilde{a}}_{n}$ and $(b2)\ {\tilde{b}}_{n}$ at t = − 6 (longdash line), t = 0 (dashdot line) and t = 6 (dot line).
Figure 5.
New window|Download| PPT slide Figure 5.(Color online) Four-soliton elastic interaction structures via the solutions (55) with parameters ${\lambda }_{1}=\tfrac{1}{3},{\lambda }_{2}=\tfrac{5}{2},{\lambda }_{3}\,=3,{\lambda }_{4}=4,\alpha =\tfrac{3}{4},{C}_{11}=1,{C}_{12}=2,{C}_{21}=1,{C}_{22}=1,{C}_{31}=-1,{C}_{32}=1,{C}_{41}=1,{C}_{42}=2$. (a1) The component ${\tilde{a}}_{n}$. (a2) The component ${\tilde{b}}_{n}$. The propagation processes for $(b1)\ {\tilde{a}}_{n}$ and $(b2)\ | {\tilde{b}}_{n}| $ at t = − 6 (longdash line), t = 0 (dashdot line) and t = 6 (dot line).
Figure 6.
New window|Download| PPT slide Figure 6.(Color online) Soltion structures of the combined potential term ${\tilde{a}}_{n}{\tilde{b}}_{n}$. (a1) (a2) One-soliton structures with the same parameters as in figure 1. (b1) (b2) Two-soliton structures with the same parameters as in figure 2. (c1) (c2) Three-soliton structures with the same parameters as in figure 3. (d1) (d2) Four-soliton structures with the same parameters as in figure 4.
5.2. Rational and semi-rational solutions via the discrete (1, 2N − 1)-fold DT
In this subsection, we will use the discrete generalized (1, 2N − 1)-fold DT with single eigenvalue to investigate some rational solutions and semi-rational solutions of equation (3) when m = 1. First of all, we fix the spectral parameter λ = λ1 + ϵ. Then expanding the vector function φ1,n in (51) as two Taylor series around ϵ = 0 and choosing $\alpha =\tfrac{3}{4},$ C11 = C12 = 1, we arrive at $\begin{eqnarray}{\phi }_{1,n}(\varepsilon )={\phi }_{1,n}^{\left(0\right)}+{\phi }_{1,n}^{\left(1\right)}\varepsilon +{\phi }_{1,n}^{\left(2\right)}{\varepsilon }^{2}+{\phi }_{1,n}^{\left(3\right)}{\varepsilon }^{3}+...,\end{eqnarray}$ where $\begin{eqnarray*}\begin{array}{rcl}{\phi }_{1,n}^{\left(0\right)} & = & \left(\begin{array}{c}{\varphi }_{1,n}^{\left(0\right)}\\ {\psi }_{1,n}^{\left(0\right)}\end{array}\right)=\left(\begin{array}{c}2{\left(-\tfrac{3}{8}\right)}^{n}{{\rm{e}}}^{-\tfrac{1}{2}t}\\ 2{\left(-\tfrac{3}{8}\right)}^{n}{{\rm{e}}}^{-\tfrac{1}{2}t}\end{array}\right),\\ {\phi }_{1,n}^{\left(1\right)} & = & \left(\begin{array}{c}{\varphi }_{1,n}^{\left(1\right)}\\ {\psi }_{1,n}^{\left(1\right)}\end{array}\right)=\left(\begin{array}{c}-\left(\tfrac{1}{3}\right){\left(-\tfrac{3}{8}\right)}^{n}{{\rm{e}}}^{-\tfrac{1}{2}t}(20{n}^{2}+20{nt}+5{t}^{2}-12n-4t)\\ -\left(\tfrac{1}{3}\right){\left(-\tfrac{3}{8}\right)}^{n}{{\rm{e}}}^{-\tfrac{1}{2}t}(20{n}^{2}+20{nt}+5{t}^{2}-52n-24t+20)\end{array}\right),\\ {\phi }_{1,n}^{\left(2\right)} & = & \left(\begin{array}{c}{\varphi }_{1,n}^{\left(2\right)}\\ {\psi }_{1,n}^{\left(2\right)}\end{array}\right),\quad {\phi }_{1,n}^{\left(3\right)}=\left(\begin{array}{c}{\varphi }_{1,n}^{\left(3\right)}\\ {\psi }_{1,n}^{\left(3\right)}\end{array}\right),\end{array}\end{eqnarray*}$ with $\begin{eqnarray*}\begin{array}{l}{\varphi }_{1,n}^{\left(2\right)}=(\displaystyle \frac{1}{108}){\left(-\displaystyle \frac{3}{8}\right)}^{n}{{\rm{e}}}^{-\tfrac{1}{2}t}(400{n}^{4}+800{n}^{3}t+600{n}^{2}{t}^{2}\\ \ \ +\,200{{nt}}^{3}+25{t}^{4}-1440{n}^{3}-1920{n}^{2}t-840{{nt}}^{2}\\ \ \ -\,120{t}^{3}+1184{n}^{2}+400{nt}-84{t}^{2}-432n+144t),\\ {\psi }_{1,n}^{\left(2\right)}=(\displaystyle \frac{1}{108}){\left(-\displaystyle \frac{3}{8}\right)}^{n}{{\rm{e}}}^{-\tfrac{1}{2}t}(400{n}^{4}+800{n}^{3}t+600{n}^{2}{t}^{2}\\ \ \ +\,200{{nt}}^{3}+25{t}^{4}-3040{n}^{3}-4320{n}^{2}t-2040{{nt}}^{2}\\ -320{t}^{3}+6464{n}^{2}+5200{nt}+996{t}^{2}\\ \ \ -\,4976n-1248t+1152),\\ {\varphi }_{1,n}^{\left(3\right)}=-(\displaystyle \frac{1}{1944}){\left(-\displaystyle \frac{3}{8}\right)}^{n}{{\rm{e}}}^{-\tfrac{1}{2}t}(1600{n}^{6}+4800{n}^{5}t\\ \ \ +\,6000{n}^{4}{t}^{2}+4000{n}^{3}{t}^{3}+1500{n}^{2}{t}^{4}+300{{nt}}^{5}+25{t}^{6}\\ -14400{n}^{5}-33600{n}^{4}t-31200{n}^{3}{t}^{2}\\ \ \ -\,14400{n}^{2}{t}^{3}-3300{{nt}}^{4}-300{t}^{5}+40960{n}^{4}+60480{n}^{3}t\\ +30000{n}^{2}{t}^{2}+5120{{nt}}^{3}+60{t}^{4}-58176{n}^{3}\\ \ \ -\,35520{n}^{2}t+3840{{nt}}^{2}+3552{t}^{3}+40384{n}^{2}\\ +3840{nt}-3456{t}^{2}-10368n),\end{array}\end{eqnarray*}$
$\begin{eqnarray*}\begin{array}{l}{\psi }_{1,n}^{\left(3\right)}=-(\displaystyle \frac{1}{1944}){\left(-\displaystyle \frac{3}{8}\right)}^{n}{{\rm{e}}}^{-\tfrac{1}{2}t}(1600{n}^{6}+4800{n}^{5}t\\ \ \ +\,6000{n}^{4}{t}^{2}+4000{n}^{3}{t}^{3}+1500{n}^{2}{t}^{4}+300{{nt}}^{5}+25{t}^{6}\\ -24000{n}^{5}-57600{n}^{4}t-55200{n}^{3}{t}^{2}-26400{n}^{2}{t}^{3}\\ \ \ -\,6300{{nt}}^{4}-600{t}^{5}+122560{n}^{4}+214080{n}^{3}t\\ +138000{n}^{2}{t}^{2}+38720{{nt}}^{3}+3960{t}^{4}-288576{n}^{3}\\ \ \ -\,311040{n}^{2}t-100320{{nt}}^{2}-8448{t}^{3}+344128{n}^{2}\\ +175680{nt}+11088{t}^{2}-197184n-31104t+41472),\end{array}\end{eqnarray*}$ and the other ${({\varphi }_{1,n}^{\left(i\right)},{\psi }_{1,n}^{\left(i\right)})}^{{\rm{T}}}(i=4,5,\cdots )$ are omitted here. From (50), we can obtain the new rational solutions of equation (3). Next we shall discuss three cases: N = 1, 2, 3.
(I) When N = 1, based on the discrete generalized (1, 1)-fold DT in theorem 3, we can obtain the first-order rational solutions of equation (3) as $\begin{eqnarray}{\tilde{a}}_{n}=\displaystyle \frac{\alpha +{g}_{n}^{\left(1\right)}}{\alpha -{r}_{n}^{\left(1\right)}},\quad \quad \quad {\tilde{b}}_{n}=\displaystyle \frac{{f}_{n}^{\left(2\right)}-{f}_{n-1}^{\left(2\right)}}{\alpha {f}_{n-1}^{\left(2\right)}},\end{eqnarray}$ in which $\begin{eqnarray*}{g}_{n}^{\left(1\right)}=\displaystyle \frac{{\rm{\Delta }}{g}_{n}^{\left(1\right)}}{{{\rm{\Delta }}}_{1,n}},\quad \quad {f}_{n}^{\left(2\right)}=\displaystyle \frac{{\rm{\Delta }}{f}_{n}^{\left(2\right)}}{{{\rm{\Delta }}}_{1,n}},\quad \quad {r}_{n}^{\left(1\right)}=\displaystyle \frac{{\rm{\Delta }}{r}_{n}^{\left(1\right)}}{{{\rm{\Delta }}}_{2,n}},\end{eqnarray*}$ with $\begin{eqnarray*}\begin{array}{l}{{\rm{\Delta }}}_{1,n}=\left|\begin{array}{cc}{\lambda }_{1}^{2}{\varphi }_{1,n}^{\left(0\right)} & {\lambda }_{1}{\psi }_{1,n}^{\left(0\right)}\\ {\lambda }_{1}^{2}{\varphi }_{1,n}^{\left(1\right)}+2{\lambda }_{1}{\varphi }_{1,n}^{\left(0\right)} & {\lambda }_{1}{\psi }_{1,n}^{\left(1\right)}+{\psi }_{1,n}^{\left(0\right)}\end{array}\right|,\\ {{\rm{\Delta }}}_{2,n}=\left|\begin{array}{cc}{\lambda }_{1}{\varphi }_{1,n}^{\left(0\right)}-\displaystyle \frac{1}{\alpha }{\psi }_{1,n}^{\left(0\right)} & {\lambda }_{1}^{2}{\psi }_{1,n}^{\left(0\right)}\\ {\lambda }_{1}{\varphi }_{1,n}^{\left(1\right)}+{\varphi }_{1,n}^{\left(0\right)}-\displaystyle \frac{1}{\alpha }{\psi }_{1,n}^{\left(1\right)} & {\lambda }_{1}^{2}{\psi }_{1,n}^{\left(1\right)}+{\psi }_{1,n}^{\left(0\right)}\end{array}\right|,\\ {\rm{\Delta }}{f}_{n}^{\left(2\right)}=\left|\begin{array}{cc}-{\varphi }_{1,n}^{\left(0\right)} & {\lambda }_{1}{\psi }_{1,n}^{\left(0\right)}\\ -{\varphi }_{1,n}^{\left(1\right)} & {\lambda }_{1}{\psi }_{1,n}^{\left(1\right)}+{\psi }_{1,n}^{\left(0\right)}\end{array}\right|,\ {\rm{\Delta }}{g}_{n}=\left|\begin{array}{cc}{\lambda }_{1}^{2}{\varphi }_{1,n}^{\left(0\right)} & -{\varphi }_{1,n}^{\left(0\right)}\\ {\lambda }_{1}^{2}{\varphi }_{1,n}^{\left(1\right)}+2{\lambda }_{1}{\varphi }_{1,n}^{\left(0\right)} & -{\varphi }_{1,n}^{\left(1\right)}\end{array}\right|,\\ {\rm{\Delta }}{r}_{n}=\left|\begin{array}{cc}-{\psi }_{1,n}^{\left(0\right)} & {\lambda }_{1}^{2}{\psi }_{1,n}^{\left(0\right)}\\ -{\psi }_{1,n}^{\left(1\right)} & {\lambda }_{1}^{2}{\psi }_{1,n}^{\left(1\right)}+{\psi }_{1,n}^{\left(0\right)}\end{array}\right|.\end{array}\end{eqnarray*}$ Through a direct calculation, the simplification analytical expressions of solutions (57) are listed as follows: $\begin{eqnarray}\begin{array}{rcl}{\tilde{a}}_{n} & = & 1-\displaystyle \frac{160}{(10n+5t-8)(10n+5t-2)},\\ {\tilde{b}}_{n} & = & -\displaystyle \frac{80}{(10n+5t-8)(10n+5t-12)},\end{array}\end{eqnarray}$ from which we can see that ${\tilde{a}}_{n}$ possesses singularity at two paralleled straight lines 10n + 5t − 8 = 0 and 10n + 5t − 2 = 0, while ${\tilde{b}}_{n}$ has singularity at two paralleled straight lines 10n + 5t − 8 = 0 and 10n + 5t − 12 = 0. Moreover, we can conclude that ${\tilde{a}}_{n}\to 1,{\tilde{b}}_{n}\to 0$ as n → ±∞ or t → ±∞.
It is important to point out that we have derived the previous rational solutions composed of the polynomials of variables n, t if we expand φ1,n in (51) around ${\lambda }_{1}=\tfrac{1}{2}$. When ${\lambda }_{1}\ne \tfrac{1}{2}$, through the similar process like above, we can obtain the semi-rational solutions which are made up of polynomial functions and exponential functions. For instance, we fix the spectral parameter λ = λ1 + ϵ with λ1 = 3 in (51), based on the generalized (1, 1)-fold DT, from (57) we can derive the first-order semi-rational solutions whose simplification forms are listed as follows: $\begin{eqnarray}{\tilde{a}}_{n}=\displaystyle \frac{{A}_{z}}{{A}_{m}},\ \ {\tilde{b}}_{n}=\displaystyle \frac{{B}_{z}}{{B}_{m}},\end{eqnarray}$ where $\begin{eqnarray*}\begin{array}{l}{A}_{z}=-\displaystyle \frac{1}{113400}(16\sqrt{7}-35){\left(12496+4715\sqrt{7}\right)}^{n}\\ \ \times \,{\left(\displaystyle \frac{12496}{531441}+\displaystyle \frac{4715}{531441}\sqrt{7}\right)}^{n}(627\sqrt{7}-3020n+24915t\\ \ \ +1510){{\rm{e}}}^{\tfrac{10}{3}\sqrt{7}t}+\displaystyle \frac{1}{1814400}{{\rm{e}}}^{\tfrac{20}{3}\sqrt{7}t}\\ \ \times \,{\left(431+160\sqrt{7}\right)}^{n}{\left(\displaystyle \frac{431}{6561}+\displaystyle \frac{160}{6561}\sqrt{7}\right)}^{n}(1814400{n}^{2}-29937600{nt}\\ +123492600{t}^{2}-1814400n+14968800t-870167)\\ \ +\,\displaystyle \frac{9}{44800}(431+160\sqrt{7}){{\rm{e}}}^{\tfrac{40}{3}\sqrt{7}t}-\displaystyle \frac{1}{113400}(35\\ \ +\,16\sqrt{7}){\left(\displaystyle \frac{16}{81}+\displaystyle \frac{5}{81}\sqrt{7}\right)}^{n}{\left(16+5\sqrt{7}\right)}^{n}(627\sqrt{7}\\ \ +\,3020n-24915t-1510){{\rm{e}}}^{10\sqrt{7}t}-\displaystyle \frac{9}{44800}(160\sqrt{7}\\ -431){\left(364961+137920\sqrt{7}\right)}^{n}{\left(\displaystyle \frac{364961}{43046721}+\displaystyle \frac{137920}{43046721}\sqrt{7}\right)}^{n},\end{array}\end{eqnarray*}$
$\begin{eqnarray*}\begin{array}{l}{A}_{m}=\left[\displaystyle \frac{1}{20}{{\rm{e}}}^{\tfrac{10}{3}\sqrt{7}t}{\left(16+5\sqrt{7}\right)}^{n}{\left(\displaystyle \frac{16}{81}+\displaystyle \frac{5}{81}\sqrt{7}\right)}^{n}\right.\\ \ \times \,(20n-165t-18)-\displaystyle \frac{9}{560}(28+11\sqrt{7})\\ \ \times \,{{\rm{e}}}^{\tfrac{20}{3}\sqrt{7}t}+\displaystyle \frac{9}{560}(-28\\ \ \ +\left.\,11\sqrt{7}){\left(431+160\sqrt{7}\right)}^{n}{\left(\displaystyle \frac{431}{6561}+\displaystyle \frac{160}{6561}\sqrt{7}\right)}^{n}]\\ \ \times \,\left[\displaystyle \frac{1}{20}{{\rm{e}}}^{\tfrac{10}{3}\sqrt{7}t}{\left(16+5\sqrt{7}\right)}^{n}{\left(\displaystyle \frac{16}{81}+\displaystyle \frac{5}{81}\sqrt{7}\right)}^{n}(20n-165t\right.\\ -2)-\displaystyle \frac{1}{560}{{\rm{e}}}^{\tfrac{20}{3}\sqrt{7}t}(29\sqrt{7}+28)+\displaystyle \frac{1}{560}(-28+29\sqrt{7})\\ \ \times \,{\left(431+160\sqrt{7}\right)}^{n}{\left(\displaystyle \frac{431}{6561}+\displaystyle \frac{160}{6561}\sqrt{7}\right)}^{n}\right],\\ {B}_{z}=\displaystyle \frac{431}{243}\left[\displaystyle \frac{1}{60340}(431\sqrt{7}-1120){{\rm{e}}}^{-\tfrac{10}{3}\sqrt{7}t}\right.\\ \ \times \,{\left(12496+4715\sqrt{7}\right)}^{n}\\ \ \times \,{\left(\displaystyle \frac{12496}{531441}+\displaystyle \frac{4715}{531441}\sqrt{7}\right)}^{n}(64\sqrt{7}-140n+1155t\\ \ +\,140)+\displaystyle \frac{2592}{2155}{\left(431+160\sqrt{7}\right)}^{n}{\left(\displaystyle \frac{431}{6561}+\displaystyle \frac{160}{6561}\sqrt{7}\right)}^{n}\\ \ +\,\displaystyle \frac{1}{60340}(1120+431\sqrt{7}){{\rm{e}}}^{\tfrac{10}{3}\sqrt{7}t}{\left(16+5\sqrt{7}\right)}^{n}\left(\displaystyle \frac{16}{81}\right.\\ +\left.{\left.\displaystyle \frac{5}{81}\sqrt{7}\right)}^{n}(140n-1155t-140+64\sqrt{7})\right]{{\rm{e}}}^{\tfrac{20}{3}\sqrt{7}t},\\ {B}_{m}=\left[\displaystyle \frac{1}{20}{{\rm{e}}}^{\tfrac{10}{3}\sqrt{7}t}{\left(16+5\sqrt{7}\right)}^{n}{\left(\displaystyle \frac{16}{81}+\displaystyle \frac{5}{81}\sqrt{7}\right)}^{n}\right.\\ \ \times \,(20n-165t-2)-\displaystyle \frac{1}{560}(29\sqrt{7}+28)\\ \ \times \,{{\rm{e}}}^{\tfrac{20}{3}\sqrt{7}t}+\displaystyle \frac{1}{560}(-28\\ +29\left.\sqrt{7}){\left(431+160\sqrt{7}\right)}^{n}{\left(\displaystyle \frac{431}{6561}+\displaystyle \frac{160}{6561}\sqrt{7}\right)}^{n}\right]\\ \ \times \,\left[\displaystyle \frac{1}{5040}(9221\sqrt{7}-24388){\left(431+160\sqrt{7}\right)}^{n}\left(\displaystyle \frac{431}{6561}\right.\right.\\ +{\left.\displaystyle \frac{160}{6561}\sqrt{7}\right)}^{n}+\displaystyle \frac{1}{20}{{\rm{e}}}^{\tfrac{10}{3}\sqrt{7}t}{\left(16+5\sqrt{7}\right)}^{n}{\left(\displaystyle \frac{16}{81}+\displaystyle \frac{5}{81}\sqrt{7}\right)}^{n}\\ \ \times \,(20n-165t-38)-\displaystyle \frac{1}{5040}(9221\sqrt{7}\\ \ \left.+24388){{\rm{e}}}^{\tfrac{20}{3}\sqrt{7}t}\right].\end{array}\end{eqnarray*}$ From the above expressions, we can clearly see that the solutions (59) consist of polynomial functions and exponential functions. Here we call this kind of solutions the semi-rational solutions relative to the rational solutions. With the help of symbolic computation, we can easily verify that the solutions (59) are correct by inserting them into equation (3).
$\begin{eqnarray*}\begin{array}{l}{H}_{n}=(8000{n}^{6}+24000{n}^{5}t+30000{n}^{4}{t}^{2}+20000{n}^{3}{t}^{3}\\ \ +\,7500{n}^{2}{t}^{4}+1500{{nt}}^{5}-57600{n}^{5}-144000{n}^{4}t-72000{n}^{2}{t}^{3}\\ +125{t}^{6}-144000{n}^{3}{t}^{2}-18000{{nt}}^{4}-1800{t}^{5}+152000{n}^{4}\\ +312000{n}^{3}t+240000{n}^{2}{t}^{2}+82000{{nt}}^{3}+10500{t}^{4}\\ -180480{n}^{3}-299520{n}^{2}t-164160{{nt}}^{2}-29760{t}^{3}\\ \ +\,95744{n}^{2}+127104{nt}+38016{t}^{2}-22272n-19584t\\ +4608)(8000{n}^{6}+24000{n}^{5}t+30000{n}^{4}{t}^{2}+20000{n}^{3}{t}^{3}\\ \ +\,7500{n}^{2}{t}^{4}+1500{{nt}}^{5}-38400{n}^{5}-96000{n}^{4}t\\ -96000{n}^{3}{t}^{2}-48000{n}^{2}{t}^{3}-12000{{nt}}^{4}-1200{t}^{5}\\ \ +\,56000{n}^{4}+120000{n}^{3}t+96000{n}^{2}{t}^{2}-11520{n}^{3}+125{t}^{6}\\ \ +\,4500{t}^{4}+34000{{nt}}^{3}-36480{n}^{2}t-27840{{nt}}^{2}-6240{t}^{3}\\ \ -\,27136{n}^{2}-14976{nt}-2304{t}^{2}+13056n+3456t).\end{array}\end{eqnarray*}$ Next, we implement the asymptotic analysis to study the rational solutions ${\tilde{a}}_{n}$ and ${\tilde{b}}_{n}$. Let ${\xi }_{1}=2n+t+{\left(\tfrac{3}{10}\sqrt{5}-\tfrac{1}{2}\right)}^{\tfrac{1}{3}}{t}^{\tfrac{1}{3}}$, ${\xi }_{2}=2n+t-{\left(\tfrac{3}{10}\sqrt{5}+\tfrac{1}{2}\right)}^{\tfrac{1}{3}}{t}^{\tfrac{1}{3}}$ and $c={\left(\tfrac{3}{10}\sqrt{5}-\tfrac{1}{2}\right)}^{\tfrac{1}{3}}+{\left(\tfrac{3}{10}\sqrt{5}+\tfrac{1}{2}\right)}^{\tfrac{1}{3}}$,then we can find that the solutions ${\tilde{a}}_{n}$ and ${\tilde{b}}_{n}$ have two different asymptotic states when ∣t∣ → ∞ , which are listed as follows:
(i) If ${\xi }_{1}=2n+t+{\left(\tfrac{3}{10}\sqrt{5}-\tfrac{1}{2}\right)}^{\tfrac{1}{3}}{t}^{\tfrac{1}{3}}=O(1)$, from ${\xi }_{2}={\xi }_{1}-{{ct}}^{\tfrac{1}{3}}$ we have ξ2 → ∓ ∞ when t → ± ∞ , then calculating the limit states of solutions ${\tilde{a}}_{n}$ and ${\tilde{b}}_{n}$ in (60) gives the following asymptotic expressions as $\begin{eqnarray}\begin{array}{rcl}{\tilde{a}}_{n} & \to & {a}_{1}^{\pm }=1-\displaystyle \frac{160}{25{\xi }_{1}^{2}-50{\xi }_{1}+16},\\ {\tilde{b}}_{n} & \to & {b}_{1}^{\pm }=-\displaystyle \frac{80}{25{\xi }_{1}^{2}-100{\xi }_{1}+96}.\end{array}\end{eqnarray}$ (ii) If ${\xi }_{2}=2n+t-{\left(\tfrac{3}{10}\sqrt{5}+\tfrac{1}{2}\right)}^{\tfrac{1}{3}}{t}^{\tfrac{1}{3}}$, from ${\xi }_{1}={\xi }_{2}+{{ct}}^{\tfrac{1}{3}}$ we have ξ1 → ± ∞ when t → ± ∞ , then calculating the limits of solutions an and bn in (60) produces the following asymptotic expressions in the form $\begin{eqnarray}\begin{array}{rcl}{\tilde{a}}_{n} & \to & {a}_{2}^{\pm }=1-\displaystyle \frac{160}{25{\xi }_{2}^{2}-50{\xi }_{2}+16},\\ {\tilde{b}}_{n} & \to & {b}_{2}^{\pm }=-\displaystyle \frac{80}{25{\xi }_{2}^{2}-100{\xi }_{2}+96}.\end{array}\end{eqnarray}$ It can be seen that ${a}_{1}^{\pm }$ and ${a}_{2}^{\pm }$ possess singularity at four curves $5{\xi }_{1}-2=0,5{\xi }_{1}-8=0,5{\xi }_{2}-2\,=0,5{\xi }_{2}-8=0$, which also are the four center trajectories of solution ${\tilde{a}}_{n}$, while ${b}_{1}^{\pm }$ and ${b}_{2}^{\pm }$ possess singularity at four curves $5{\xi }_{1}-12=0,5{\xi }_{1}-8=0,5{\xi }_{2}-12=0,5{\xi }_{2}-8=0$, which are also the four center trajectories of solution ${\tilde{b}}_{n}$. From the asymptotic expressions (62) and (63) we can clearly that the asymptotic expressions of second-order rational solutions are formally consistent with the first-order rational solutions, but the main difference is that the trajectories of the first-order rational solutions are straight lines, whereas the trajectories of the higher-order rational solutions are curves.
If we expand φ1,n in (51) around λ1 = 3, based on the generalized (1, 3)-fold DT, from (60) we can derive the second-order semi-rational solutions whose simplification analytical forms are very cumbersome, and so not presented here.
whereas ${\rm{\Delta }}{f}_{n}^{\left(6\right)}$ and ${\rm{\Delta }}{g}_{n}^{\left(1\right)}$ are produced from Δ1 by replacing its third and fourth columns with $(-{\varphi }_{1,n}^{\left(0\right)},-{\varphi }_{1,n}^{\left(1\right)}$, $-{\varphi }_{1,n}^{\left(2\right)},-{\varphi }_{1,n}^{\left(3\right)}$, ${\left.-{\varphi }_{1,n}^{\left(4\right)},-{\varphi }_{1,n}^{\left(5\right)}\right)}^{{\rm{T}}}$, and ${\rm{\Delta }}{r}_{n}^{\left(1\right)}$ is given from Δ2 by replacing its first column with $(-{\psi }_{1,n}^{\left(0\right)}$, $-{\psi }_{1,n}^{\left(1\right)},-{\psi }_{1,n}^{\left(2\right)}$, $-{\psi }_{1,n}^{\left(3\right)}$, ${\left.-{\psi }_{1,n}^{\left(4\right)},-{\psi }_{1,n}^{\left(5\right)}\right)}^{{\rm{T}}}$.
It should be noted that the analytical expressions of (64) are also very complicated and not listed here. If we expand φ1,n in (51) around λ1 = 3, based on the generalized (1, 5)-fold DT, from (64) we can also derive the third-order semi-rational solutions whose simplification analytical forms are very cumbersome, and so not presented here.
Next, we summarize some mathematical features of the previous discrete rational solutions for equation (3) listed in tables 1 and 2. In two tables, the first column shows the order number of the rational solutions, the second and third columns show the highest power in the numerator and denominator polynomials involved in the solution an respectively, the fourth and fifth columns show the highest power in the numerator and denominator polynomials involved in the solution bn respectively, the sixth column means the background level of the solution an, while the last column provides the background level of the solution bn. From table 1, we can easily that for the rational solution an of order j, the highest powers in the numerator and denominator polynomials are both 2j(2j − 1), while for the rational solution bn of order j, the highest powers in the numerator and denominator polynomials are 2j(2j − 1) − 2 and 2j(2j − 1) respectively.
Table 1. Table 1.Main mathematical features of rational solutions an and bn of order j.
j
HPN(an)
HPD(an)
HPN(bn)
HPD(bn)
Background(an)
Background(bn)
1
2
2
0
2
1
0
2
12
12
10
12
1
0
3
30
30
28
30
1
0
⋯
⋯
⋯
⋯
⋯
⋯
⋯
j
2j(2j − 1)
2j(2j − 1)
2j(2j − 1) − 2
2j(2j − 1)
1
0
Note. HPN denotes the highest power in the numerator, while HPD denotes the highest power in the denominator.
It is particularly worth pointing out that we can derive some new rational solutions from (57), (60) and (64) if we fix the spectral parameter λ = λ1 + ϵ with $\alpha =\tfrac{3}{4}$ and choose ${C}_{\mathrm{1,1}}=-{C}_{\mathrm{2,1}}=\tfrac{1}{\varepsilon }$ in (51) and (56). Moreover, we here omit their analytical expressions and only sum up their mathematical properties listed in table 2.
5.3. Mixed solutions via the discrete generalized (2, 2N − 2)-fold DT
In the previous two subsections, we have used the discrete 2N-fold DT with 2N spectral parameters to derive multi-solutions of equation (3), and also used the discrete generalized (1, 2N − 1)-fold DT with only one spectral parameter to derive the rational and semi-rational solutions of equation (3). In this subsection we will employ the discrete generalized (2, 2N − 2)-fold DT with two spectral parameters to give some mixed solutions of usual soliton solutions and rational or semi-rational solutions of equation (3). To save space, we only discuss the discrete generalized (2, 2)-fold DT (i.e. the discrete generalized (2, 2N − 2)-fold DT with N = 2). In what follows, we will only list a type of mixed solution of usual one-soliton solution and rational solution via the discrete generalized (2, 2)-fold DT.
First of all, when N = 2, we need to use two spectral parameters, here we set the parameters ${\lambda }_{1}=\tfrac{1}{2}\ ($ i.e. $\alpha =\tfrac{3}{4})$ and ${\lambda }_{2}\ne \tfrac{1}{2}$ (e.g. λ2 = 3), then we let the spectral parameter λ in equation (51) as λ = λ1 + ϵ, and expand the vector function φi,n in (51) as Taylor series around ϵ = 0 by choosing C11 = C12 = C21 = C22 = 1, based on the discrete generalized (2, 2)-fold DT, we can get the mixed solutions of usual one-soliton solution and rational solution as $\begin{eqnarray}{\tilde{a}}_{n}=\displaystyle \frac{\alpha +{g}_{n}^{\left(1\right)}}{\alpha -{r}_{n}^{\left(1\right)}},\quad \quad \quad {\tilde{b}}_{n}=\displaystyle \frac{{f}_{n}^{\left(4\right)}-{f}_{n-1}^{\left(4\right)}}{\alpha {f}_{n-1}^{\left(4\right)}},\end{eqnarray}$ where ${g}_{n}^{\left(1\right)}$, ${r}_{n}^{\left(1\right)}$, ${f}_{n-1}^{\left(4\right)}$ and ${f}_{n}^{\left(4\right)}$ can be given by $\begin{eqnarray*}{g}_{n}^{\left(1\right)}=\displaystyle \frac{{\rm{\Delta }}{g}_{n}^{\left(1\right)}}{{{\rm{\Delta }}}_{1,n}},\quad \quad {f}_{n}^{\left(4\right)}=\displaystyle \frac{{\rm{\Delta }}{f}_{n}^{\left(4\right)}}{{{\rm{\Delta }}}_{1,n}},\quad \quad {r}_{n}^{\left(1\right)}=\displaystyle \frac{{\rm{\Delta }}{r}_{n}^{\left(1\right)}}{{{\rm{\Delta }}}_{2,n}},\end{eqnarray*}$ with $\begin{eqnarray*}\begin{array}{rcl}{\rm{\Delta }}{f}_{n}^{\left(4\right)} & = & \left|\begin{array}{cccc}{\lambda }_{1}^{2}{\varphi }_{1,n}^{\left(0\right)} & -{\varphi }_{1,n}^{\left(0\right)} & {\lambda }_{1}{\psi }_{1,n}^{\left(0\right)} & {\lambda }_{1}^{3}{\psi }_{1,n}^{\left(0\right)}\\ {\lambda }_{1}^{2}{\varphi }_{1,n}^{\left(1\right)}+2{\lambda }_{1}{\varphi }_{1,n}^{\left(0\right)} & -{\varphi }_{1,n}^{\left(1\right)} & {\lambda }_{1}{\psi }_{1,n}^{\left(1\right)}+{\psi }_{1,n}^{\left(0\right)} & {\lambda }_{1}^{3}{\psi }_{1,n}^{\left(1\right)}+3{\lambda }_{1}^{2}{\psi }_{1,n}^{\left(0\right)}\\ {\lambda }_{1}^{2}{\varphi }_{1,n}^{\left(2\right)}+2{\lambda }_{1}{\varphi }_{1,n}^{\left(1\right)}+{\varphi }_{1,n}^{\left(0\right)} & -{\varphi }_{1,n}^{\left(2\right)} & {\lambda }_{1}{\psi }_{1,n}^{\left(2\right)}+{\psi }_{1,n}^{\left(1\right)} & {\lambda }_{1}^{3}{\psi }_{1,n}^{\left(2\right)}+3{\lambda }_{1}^{2}{\psi }_{1,n}^{\left(1\right)}+3{\lambda }_{1}{\psi }_{1,n}^{\left(0\right)}\\ {\lambda }_{2}^{2}{\varphi }_{2,n} & -{\varphi }_{2,n} & {\lambda }_{2}{\psi }_{2,n} & {\lambda }_{2}^{3}{\psi }_{2,n}\end{array}\right|,\\ {\rm{\Delta }}{g}_{n}^{\left(1\right)} & = & \left|\begin{array}{cccc}{\lambda }_{1}^{2}{\varphi }_{1,n}^{\left(0\right)} & {\lambda }_{1}^{4}{\varphi }_{1,n}^{\left(0\right)} & -{\varphi }_{1,n}^{\left(0\right)} & {\lambda }_{1}^{3}{\psi }_{1,n}^{\left(0\right)}\\ {\lambda }_{1}^{2}{\varphi }_{1,n}^{\left(1\right)}+2{\lambda }_{1}{\varphi }_{1,n}^{\left(0\right)} & {\lambda }_{1}^{4}{\varphi }_{1,n}^{\left(1\right)}+4{\lambda }_{1}^{3}{\varphi }_{1,n}^{\left(0\right)} & -{\varphi }_{1,n}^{\left(1\right)} & {\lambda }_{1}^{3}{\psi }_{1,n}^{\left(1\right)}+3{\lambda }_{1}^{2}{\psi }_{1,n}^{\left(0\right)}\\ {\lambda }_{1}^{2}{\varphi }_{1,n}^{\left(2\right)}+2{\lambda }_{1}{\varphi }_{1,n}^{\left(1\right)}+{\varphi }_{1,n}^{\left(0\right)} & {\lambda }_{1}^{4}{\varphi }_{1,n}^{\left(2\right)}+4{\lambda }_{1}^{3}{\varphi }_{1,n}^{\left(1\right)}+6{\lambda }_{1}^{2}{\varphi }_{1,n}^{\left(0\right)} & -{\varphi }_{1,n}^{\left(2\right)} & {\lambda }_{1}^{3}{\psi }_{1,n}^{\left(2\right)}+3{\lambda }_{1}^{2}{\psi }_{1,n}^{\left(1\right)}+3{\lambda }_{1}{\psi }_{1,n}^{\left(0\right)}\\ {\lambda }_{2}^{2}{\varphi }_{2,n} & {\lambda }_{2}^{4}{\varphi }_{2,n} & -{\varphi }_{2,n} & {\lambda }_{2}^{3}{\psi }_{2,n}\end{array}\right|,\\ {\rm{\Delta }}{r}_{n}^{\left(1\right)} & = & \left|\begin{array}{cccc}-{\psi }_{1,n}^{\left(0\right)} & {\lambda }_{1}^{3}{\varphi }_{1,n}^{\left(0\right)} & {\lambda }_{1}^{2}{\psi }_{1,n}^{\left(0\right)} & {\lambda }_{1}^{4}{\psi }_{1,n}^{\left(0\right)}\\ -{\psi }_{1,n}^{\left(1\right)} & {\lambda }_{1}^{3}{\varphi }_{1,n}^{\left(1\right)}+3{\lambda }_{1}^{2}{\varphi }_{1,n}^{\left(0\right)} & {\lambda }_{1}^{2}{\psi }_{1,n}^{\left(1\right)}+2{\lambda }_{1}{\psi }_{1,n}^{\left(0\right)} & {\lambda }_{1}^{4}{\psi }_{1,n}^{\left(1\right)}+4{\lambda }_{1}^{3}{\psi }_{1,n}^{\left(0\right)}\\ -{\psi }_{1,n}^{\left(2\right)} & {\lambda }_{1}^{3}{\varphi }_{1,n}^{\left(2\right)}+3{\lambda }_{1}^{2}{\varphi }_{1,n}^{\left(1\right)}+3{\lambda }_{1}{\varphi }_{1,n}^{\left(0\right)} & {\lambda }_{1}^{2}{\psi }_{1,n}^{\left(2\right)}+2{\lambda }_{1}{\psi }_{1,n}^{\left(1\right)}+{\psi }_{1,n}^{\left(0\right)} & {\lambda }_{1}^{4}{\psi }_{1,n}^{\left(2\right)}+4{\lambda }_{1}^{3}{\psi }_{1,n}^{\left(1\right)}+6{\lambda }_{1}^{2}{\psi }_{1,n}^{\left(0\right)}\\ -{\psi }_{2,n} & {\lambda }_{2}^{3}{\varphi }_{2,n} & {\lambda }_{2}^{2}{\psi }_{2,n} & {\lambda }_{2}^{4}{\psi }_{2,n}\end{array}\right|,\\ {{\rm{\Delta }}}_{1,n} & = & \left|\begin{array}{cccc}{\lambda }_{1}^{2}{\varphi }_{1,n}^{\left(0\right)} & {\lambda }_{1}^{4}{\varphi }_{1,n}^{\left(0\right)} & {\lambda }_{1}{\psi }_{1,n}^{\left(0\right)} & {\lambda }_{1}^{3}{\psi }_{1,n}^{\left(0\right)}\\ {\lambda }_{1}^{2}{\varphi }_{1,n}^{\left(1\right)}+2{\lambda }_{1}{\varphi }_{1,n}^{\left(0\right)} & {\lambda }_{1}^{4}{\varphi }_{1,n}^{\left(1\right)}+4{\lambda }_{1}^{3}{\varphi }_{1,n}^{\left(0\right)} & {\lambda }_{1}{\psi }_{1,n}^{\left(1\right)}+{\psi }_{1,n}^{\left(0\right)} & {\lambda }_{1}^{3}{\psi }_{1,n}^{\left(1\right)}+3{\lambda }_{1}^{2}{\psi }_{1,n}^{\left(0\right)}\\ {\lambda }_{1}^{2}{\varphi }_{1,n}^{\left(2\right)}+2{\lambda }_{1}{\varphi }_{1,n}^{\left(1\right)}+{\varphi }_{1,n}^{\left(0\right)} & {\lambda }_{1}^{4}{\varphi }_{1,n}^{\left(2\right)}+4{\lambda }_{1}^{3}{\varphi }_{1,n}^{\left(1\right)}+6{\lambda }_{1}^{2}{\varphi }_{1,n}^{\left(0\right)} & {\lambda }_{1}{\psi }_{1,n}^{\left(2\right)}+{\psi }_{1,n}^{\left(1\right)} & {\lambda }_{1}^{3}{\psi }_{1,n}^{\left(2\right)}+3{\lambda }_{1}^{2}{\psi }_{1,n}^{\left(1\right)}+3{\lambda }_{1}{\psi }_{1,n}^{\left(0\right)}\\ {\lambda }_{2}^{2}{\varphi }_{2,n} & {\lambda }_{2}^{4}{\varphi }_{2,n} & {\lambda }_{2}{\psi }_{2,n} & {\lambda }_{2}^{3}{\psi }_{2,n}\end{array}\right|,\\ {{\rm{\Delta }}}_{2,n} & = & \left|\begin{array}{cccc}{\lambda }_{1}{\varphi }_{1,n}^{\left(0\right)}-\displaystyle \frac{1}{\alpha }{\psi }_{1,n}^{\left(0\right)} & {\lambda }_{1}^{3}{\varphi }_{1,n}^{\left(0\right)} & {\lambda }_{1}^{2}{\psi }_{1,n}^{\left(0\right)} & {\lambda }_{1}^{4}{\psi }_{1,n}^{\left(0\right)}\\ {\lambda }_{1}{\varphi }_{1,n}^{\left(1\right)}+{\varphi }_{1,n}^{\left(0\right)}-\displaystyle \frac{1}{\alpha }{\psi }_{1,n}^{\left(1\right)} & {\lambda }_{1}^{3}{\varphi }_{1,n}^{\left(1\right)}+3{\lambda }_{1}^{2}{\varphi }_{1,n}^{\left(0\right)} & {\lambda }_{1}^{2}{\psi }_{1,n}^{\left(1\right)}+2{\lambda }_{1}{\psi }_{1,n}^{\left(0\right)} & {\lambda }_{1}^{4}{\psi }_{1,n}^{\left(1\right)}+4{\lambda }_{1}^{3}{\psi }_{1,n}^{\left(0\right)}\\ {\lambda }_{1}{\varphi }_{1,n}^{\left(2\right)}+{\varphi }_{1,n}^{\left(1\right)}-\displaystyle \frac{1}{\alpha }{\psi }_{1,n}^{\left(2\right)} & {\lambda }_{1}^{3}{\varphi }_{1,n}^{\left(2\right)}+3{\lambda }_{1}^{2}{\varphi }_{1,n}^{\left(1\right)}+3{\lambda }_{1}{\varphi }_{1,n}^{\left(0\right)} & {\lambda }_{1}^{2}{\psi }_{1,n}^{\left(2\right)}+2{\lambda }_{1}{\psi }_{1,n}^{\left(1\right)}+{\psi }_{1,n}^{\left(0\right)} & {\lambda }_{1}^{4}{\psi }_{1,n}^{\left(2\right)}+4{\lambda }_{1}^{3}{\psi }_{1,n}^{\left(1\right)}+6{\lambda }_{1}^{2}{\psi }_{1,n}^{\left(0\right)}\\ {\lambda }_{2}{\varphi }_{2,n}-\displaystyle \frac{1}{\alpha }{\psi }_{2,n} & {\lambda }_{2}^{3}{\varphi }_{2,n} & {\lambda }_{2}^{2}{\psi }_{2,n} & {\lambda }_{2}^{4}{\psi }_{2,n}\end{array}\right|,\end{array}\end{eqnarray*}$ where ${\varphi }_{1,n}^{\left(0\right)},$ ${\psi }_{1,n}^{\left(0\right)},$ ${\varphi }_{1,n}^{\left(1\right)},$ ${\psi }_{1,n}^{\left(1\right)},$ ${\varphi }_{1,n}^{\left(2\right)},$ ${\psi }_{1,n}^{\left(2\right)}$ are the same as ones in the previous subsection when i = 1, 2. The analytical expressions of solutions (65) are are very cumbersome, and so not presented here.
The mixed solutions (65) are also composed of polynomial functions and exponential functions which completely differs from the semi-rational solutions (59). The mixed solutions (65) need to use two spectral parameters, while the semi-rational solutions (59) only need one spectral parameter. With the help of symbolic computation, one can verify the mixed solutions (65) by substituting them into equation (3), and we will not draw their evolution structures due to their singularity here. Similarly, some more complicated mixed solutions can be given by means of the discrete generalized $(2,2N-2)$-fold DT when $N\gt 2$, whose simplified expressions are too complicated and not presented here.
6. Dynamical behaviors of soliton solutions
In this section, we use numerical simulations to illustrate the dynamical behaviors of the previous soliton solutions of equation (3) by using finite difference method [44]. Figure 7-9 exhibit dynamical behaviors of the exact one-, two-, and three-soliton solutions respectively. In figures 7-9, it is notable that the first columns show the exact soliton solutions corresponding to figures 2-4 respectively, the second columns present the numerical solutions with no noise by means of exact soliton solutions as initial conditions of the difference scheme algorithm, while the last two columns present the perturbed numerical solutions through adding 2% and 8% small noises to the exact solutions as initial conditions respectively.
Figure 7.
New window|Download| PPT slide Figure 7.(Color online) One-soliton solutions (52) with the same parameters as figure 2. (a1) (a2) Exact solutions. (b1) (b2) Numerical solutions with no noise. (c1) (c2) Numerical solutions with a 2% noise. (d1) (d2) Numerical solutions with a 8% noise.
Figure 8.
New window|Download| PPT slide Figure 8.(Color online) Two-soliton solution (54) with the same parameters as figure 3. (a1) (a2) Exact solutions. (b1) (b2) Numerical solutions with no noise. (c1) (c2) Numerical solutions with a 2% noise. (d1) (d2) Numerical solutions with a 8% noise.
Figure 9.
New window|Download| PPT slide Figure 9.(Color online) Three-soliton solutions (55) with the same parameters as figure 4. (a1) (a2) Exact solutions. (b1) (b2) Numerical solutions with no noise. (c1) (c2) Numerical solutions with a 2% noise. (d1) (d2) Numerical solutions with a 8% noise.
From figures 7-9 (a1) (b1)-(a2) (b2), we can clearly see that the wave propagations of soliton solutions with on noise are almost identical to the corresponding exact soliton solutions in each case which also show the accuracy of our numerical scheme. When a 2% small noise is added to both the initial exact solutions, the time evolution profiles are also almost their same as the corresponding exact soliton solutions in a short time (see figures 7-9 (c1)-(c2)). However, if a 8% noise is added to the initial exact solutions, the wave propagations only have a weak oscillation at a later time (see figures 7-9 (d1)-(d2)). Numerical results in figures 7-9 show that these soliton solutions have stable evolutions against a small noise.
7. Conclusions and discussions
In this paper, we have studied equation (3) which may describe particle vibrations in lattices with an exponential interaction force. First of all we have constructed a discrete integrable hierarchy (15) related to equation (3), from which some related properties such as Hamiltonian structures, Liouville integrability and infinitely many conservation laws have been discussed. Secondly, the discrete generalized (m, 2N − m)-fold DT (34) and (50) of equation (3) has been constructed in detail to derive its multi-soliton solutions with m = 2N case, rational and semi-rational solutions with m = 1 case and their mixed solutions with m = 2 case. The asymptotic state analysis for multi-soliton solutions and rational solutions are investigated in detail, from which some elastic interaction phenomena also have been discussed (see figures 2-6). Moreover, we find that the even soliton solutions may reduce to odd soliton solutions by choosing special spectral parameters, and together they make up the multi-soliton solution of equation (3). We also have summarized a few mathematical features of different-order rational solutions of equation (3) (see tables 1 and 2). At Last, numerical simulations are utilized to illustrate the dynamical behaviors of one-, two- and three-soliton solutions, showing that the evolutions of such soliton solutions are robust against a small noise (see figures 7-9). The results given in this paper might provide a few new views about understanding the particle vibrations in lattices described by equation (3).
In theory, we can use the discrete generalized (m, 2N − m)-fold DT to give more new exact solutions of equation (3). But in fact, the calculations are quite complicated, and how to discuss these novel results is worthy of further investigation in this research area. We believe that the key technique used in this paper can also extended to solve other Lax integrable NDDEs.
Acknowledgments
This work is supported by National Natural Science Foundation of China (Grant No. 12 071 042) and Beijing Natural Science Foundation (Grant No. 1 202 006).
This article was updated on 20 May 2021 to amend the typesetting of equation 54 and to correct the spacing of equation 60.
,∗, Cui-Lian YuanSchool of Applied Science, 
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