Abstract The nonlocal symmetry is derived for an equation combining the modified Calogero–Bogoyavlenskii–Schiff equation with its negative-order form from the truncated Painlevé expansion method. The nonlocal symmetries are localized to the Lie point symmetry by introducing new auxiliary dependent variables. The finite symmetry transformation and the Lie point symmetry for the prolonged system are solved directly. Many new interaction solutions among soliton and other types of interaction solutions for the modified Calogero–Bogoyavlenskii–Schiff equation with its negative-order form can be obtained from the consistent condition of the consistent tanh expansion method by selecting the proper arbitrary constants. Keywords：nonlocal symmetries;consistent tanh expansion method;interaction solutions
PDF (1446KB)MetadataMetricsRelated articlesExportEndNote|Ris|BibtexFavorite Cite this article Hengchun Hu, Feiyan Liu. New interaction solutions and nonlocal symmetry of an equation combining the modified Calogero–Bogoyavlenskii–Schiff equation with its negative-order form. Communications in Theoretical Physics, 2020, 72(6): 065002- doi:10.1088/1572-9494/ab8a2b
1. Introduction
The study of integrability and exact solutions for the nonlinear partial differential equations plays a significant role since the soliton theory became an important part of the nonlinear science. Many mathematicians and theoretical physicists have proposed a series of effective methods for solving nonlinear integrable equations, such as Darboux transformation, Bäcklund transformation, inverse scattering transformation, Painlevé analysis, the function expansion method, Hirota bilinear method and so on [1, 2, 6, 4, 5]. In addition, one can use the classical or nonclassical Lie group method to analyze and simplify many nonlinear systems with the standard procedure [6]. But it is a challenging task for the nonlocal symmetry to construct directly the finite symmetry transformation and invariant solutions. The traditional methods related to the nonlocal symmetry are the bilinear form, the negative hierarchies, the nonlinearization of Lax pair and self-consistent sources [7, 8].
Recently, abundant interaction solutions among solitons and other complicated waves including periodic cnoidal waves, Painlevé waves and Boussinesq waves for many integrable systems were obtained by residual symmetry reduction and the consistent tanh expansion (CTE) method related to the Painlevé analysis. Hinted by the results of residual symmetry reduction, Lou found that the symmetry related to the Painlevé truncated expansion is just the residue with respect to the singular manifold in the Painlevé analysis procedure, which was called residual symmetry [9–13]. On the other hand, a method called CTE is proposed to identify CTE solvable systems, which is a special simplified form of the consistent Riccati expansion method defined in [14]. It is a more generalized but much simpler method to look for new interaction solutions between a soliton and other types of nonlinear excitations [15–20]. Many new interaction solutions for integrable systems are studied extensively by means of the CTE method.
It is known that the integrable Calogero–Bogoyavlenskii–Schiff equation (CBS) was firstly constructed with the modified Lax formalism. It was also derived by Schiff from the reduction of the self-dual Yang–Mills equation and it is used to describe the interaction of a Riemann wave propagating along the y-axis with a long wave along the x-axis [21–23]. Verosky extended the work of Olver in the negative direction to obtain a sequence of equations by using the negative recursion operator and the negative CBS equation can be obtained in the same way that the authors obtained the negative KdV equation from the KdV equation. Then Wazwaz combined the modified CBS recursion operator and its inverse form to construct a new integrable equation, which is the equation combining the modified CBS equation with its negative-order form (MCBS–nMCBS) in the following [24]$\begin{eqnarray}\begin{array}{l}{u}_{x}({u}_{{xt}}+4{u}^{2}{u}_{{xy}}+12{{uu}}_{x}{u}_{y}+{u}_{{xxxy}}+{u}_{{xxxt}}\\ \quad +\,4{u}^{2}{u}_{{xt}}+12{{uu}}_{x}{u}_{t})\\ \quad -\,{u}_{{xx}}({u}_{t}+4{u}^{2}{u}_{y}+{u}_{{xxy}}+{u}_{{xxt}}+4{u}^{2}{u}_{t})=0,\end{array}\end{eqnarray}$where u=u(x, y, t) is the potential function. The Painlevé integrability of the MCBS–nMCBS equation (1) is proved and multiple-soliton solutions are obtained by means of the Hirota bilinear form [25]. The Lie algebra and similarity reductions are also studied for equation (1) from the point of view of Lie group approach in [26].
The paper is organized as follows. In section 2, the nonlocal symmetry for the MCBS–nMCBS equation (1) is obtained from the truncated Painlevé expansion and then localized to the classical Lie point symmetry in a new prolonged system by introducing independent variables. The corresponding finite symmetry transformation is derived by solving the initial value problem of the prolonged system. The complex exact solutions are constructed from the nonzero seed solution by the infinite transformation. In section 3, new interaction solutions for the integrable MCBS–nMCBS equation (1), such as the soliton-cnodial wave solutions, the resonant solutions are studied by the CTE method through the different solutions of the consistent condition. Summary and discussion are given in the last section.
2. Nonlocal symmetries for the MCBS–nMCBS equation
In this section, the nonlocal symmetry and corresponding finite symmetry transformation for the MCBS–nMCBS equation are studied by introducing new dependent variables. Based on the truncated Painlevé expansion of the MCBS–nMCBS equation (1), the Laurent series form reads$\begin{eqnarray}u=\displaystyle \frac{{u}_{0}}{\phi }+{u}_{1},\end{eqnarray}$where u0, u1 are undetermined functions and φ=φ(x, y, t) is an arbitrary singular manifold. Substituting equation (2) into equation (1) and vanishing the coefficients of different powers of φ independently, then we have$\begin{eqnarray}{u}_{0}={\phi }_{x}{\rm{i}},\end{eqnarray}$$\begin{eqnarray}{u}_{1}=-\displaystyle \frac{1}{2}\displaystyle \frac{{\phi }_{{xx}}}{{\phi }_{x}}{\rm{i}},\end{eqnarray}$$\begin{eqnarray}{S}_{y}+{S}_{t}+{C}_{x}=0,\end{eqnarray}$where i2=−1 and $S=\displaystyle \frac{{\phi }_{{xxx}}}{{\phi }_{x}}-\displaystyle \frac{3}{2}\displaystyle \frac{{\phi }_{{xx}}^{2}}{{\phi }_{x}^{2}}$, $C=\displaystyle \frac{{\phi }_{t}}{{\phi }_{x}}$ are usual Schwarzian variables. Based on the definition of the nonlocal symmetry, we can find that the residual u0 is the symmetry corresponding to the solution u1. So the truncated Painlevé expansion$\begin{eqnarray}u=\displaystyle \frac{{\phi }_{x}}{\phi }{\rm{i}}+{u}_{1},\end{eqnarray}$is an auto-Bäcklund transformation between the solutions u and u1 if the function φ satisfies the relation (5). The nonlocal symmetry of the MCBS–nMCBS equation (1) can be read out from the Bäcklund transformation (6)$\begin{eqnarray}{\sigma }^{u}={\rm{i}}{\phi }_{x},\end{eqnarray}$which is the residual of the solution u1. This nonlocal symmetry can also be obtained from the Schwarzian form (5). The consistent condition in the Schwarzian form (5) is invariant under the Möbius transformation$\begin{eqnarray*}\phi \to \displaystyle \frac{a\phi +b}{c\phi +d},{ad}\ne {bc},\end{eqnarray*}$which means (5) possesses the symmetry σφ=−φ2 in special case a=d=1, b=0, c=ϵ. For the nonlocal symmetry σu=iφx, the corresponding initial value problem is$\begin{eqnarray}\displaystyle \frac{{\rm{d}}\bar{u}}{{\rm{d}}\varepsilon }={\rm{i}}{\bar{\phi }}_{x},\quad \,\bar{u}{| }_{\varepsilon =0}=u,\end{eqnarray}$with ϵ being an infinitesimal parameter. It is difficult to solve the initial value problem of the Lie’s first principle due to the intrusion of the function φ and its differentiation [12]. In order to solve the initial value problem (8), we introduce a new dependent variable by requiring$\begin{eqnarray}{\phi }_{x}=f.\end{eqnarray}$Then it is not difficult to verify that the local Lie point symmetries of the linearized system (1), (4) and (9) have the form$\begin{eqnarray}{\sigma }^{u}=f{\rm{i}},{\sigma }^{\phi }=-{\phi }^{2},\quad {\sigma }^{f}=-2f\phi ,\end{eqnarray}$and the corresponding initial value problem becomes$\begin{eqnarray}\displaystyle \frac{{\rm{d}}\bar{u}}{{\rm{d}}\varepsilon }=\bar{f}{\rm{i}},\quad \bar{u}{| }_{\varepsilon =0}=u,\end{eqnarray}$$\begin{eqnarray}\displaystyle \frac{{\rm{d}}\bar{\phi }}{{\rm{d}}\varepsilon }=-{\bar{\phi }}^{2},\quad \bar{\phi }{| }_{\varepsilon =0}=\phi ,\end{eqnarray}$$\begin{eqnarray}\displaystyle \frac{{\rm{d}}\bar{f}}{{\rm{d}}\varepsilon }=-2\bar{f}\bar{\phi },\quad \bar{f}{| }_{\varepsilon =0}=f.\end{eqnarray}$The solution for the initial value problem (11)–(13) of the enlarged system (1), (4) and (9) can be written as$\begin{eqnarray}\bar{u}=u+\displaystyle \frac{{\rm{i}}\varepsilon f}{1+\varepsilon \phi },\ \ \bar{\phi }=\displaystyle \frac{\phi }{1+\varepsilon \phi },\ \ \bar{f}=\displaystyle \frac{f}{{\left(1+\varepsilon \phi \right)}^{2}}.\end{eqnarray}$
Using the finite symmetry transformation, one can obtain solitary wave solution for equation (1) with some exact solutions φ of the consistent condition. It is easy to see that the consistent condition (5) has a simple solution$\begin{eqnarray}\phi ={kx}+{ly}+{ct},\end{eqnarray}$with k, l, c being arbitrary constants and the complex rational solution for equation (1) is as follows$\begin{eqnarray}u=\displaystyle \frac{{\rm{i}}k}{{kx}+{ly}+{ct}}.\end{eqnarray}$It is also known that the MCBS–nMCBS equation(1) has a solitary solution in [24]$\begin{eqnarray}{u}_{0}=\displaystyle \frac{2{{ke}}^{\xi }}{1+{e}^{2\xi }},\end{eqnarray}$where$\begin{eqnarray}{rcl}\xi ={kx}+{ly}-\displaystyle \frac{{k}^{2}{lt}}{1+{k}^{2}},\end{eqnarray}$and if we select the solution (17) as the seed solution for equation (1), the function φ can be solved easily from equation (4)$\begin{eqnarray}\phi =x+\displaystyle \frac{4(1+{\rm{i}}{e}^{\xi })}{k(1+{e}^{2\xi })}+f(y),\end{eqnarray}$with k, l being arbitrary constants and the function f(y) is an arbitrary function of the independent variable y. Then we can get a new solution for the MCBS–nMCBS equation (1) from (4) and (19)$\begin{eqnarray}{u}_{1}={u}_{0}+\displaystyle \frac{{\rm{i}}\varepsilon {\phi }_{x}}{1+\varepsilon \phi }.\end{eqnarray}$Since the arbitrariness of the function f(y) in equation (19), one can obtain much more complex exact solutions of equation (1) by selecting proper function and constants in (19). Here we give two concrete examples for simplicity and omit the complicated calculations. The first complex exact solution for equation (1) with nontrivial seed solution equation (17) can be obtained by substituting equations (17)–(19) into (20) and the real and imaginary parts of the complex solution are given in figure 1 with the arbitrary function and constant selection$\begin{eqnarray}f(y)=\sinh (y),k=1,l=3,\varepsilon =0.5.\end{eqnarray}$From the figure 1, the real and imaginary parts of the first type of complex exact solution can be regarded as the interaction of two solitons at the direction y=0.
Figure 1.
New window|Download| PPT slide Figure 1.The first type of complex exact solution (20) with (17)–(19) and (21) of the MCBS–nMCBS equation (1) at y=0. (a) Real part; (b) imaginary part.
In a similar way, the second type of the complex exact solution for equation (1) can be obtained by substituting equations (17)–(19) into (20) with the constant selection in figure 2$\begin{eqnarray}f(y)=\mathrm{sn}(y,m),k=2,l=-1,m=0.8,\varepsilon =1,\end{eqnarray}$where sn is the Jacobi elliptic sine function.
Figure 2.
New window|Download| PPT slide Figure 2.The module of the second type of the complex exact solution (20) with (17)–(19) and (22) of the MCBS–nMCBS equation (1) at y=0.
When f(y)=sn(y, m), the module of the second type of the complex exact solution can be regarded as the interaction of two solitons with different heights. If selecting other different forms of the arbitrary function in (19), we can construct more exact complex solutions for equation (1).
3. Interaction solutions of the MCBS–nMCBS equation
In this section, the CTE method is developed to find the interaction solutions between solitons and other types of nonlinear waves such as soliton-cnoidal waves and resonant wave solutions. By the leading order analysis of equation (1), we can take the following truncated tanh function expansion$\begin{eqnarray}u={u}_{0}+{u}_{1}\tanh (\omega ),\end{eqnarray}$where u0, u1 and ω are undetermined functions of x, y, t. Substituting equation (23) into equation (1) and vanishing the coefficients of different powers of $\tanh (\omega )$, we have$\begin{eqnarray}{u}_{1}={\omega }_{x}{\rm{i}},\quad {u}_{0}=-\displaystyle \frac{{\omega }_{{xx}}}{2{\omega }_{x}}{\rm{i}},\end{eqnarray}$and the function ω only needs to satisfy$\begin{eqnarray}{S}_{y}+{S}_{t}+{C}_{x}-4{\omega }_{{xt}}{\omega }_{x}-4{\omega }_{{xy}}{\omega }_{x}=0,\end{eqnarray}$where $S=\displaystyle \frac{{\omega }_{{xxx}}}{{\omega }_{x}}-\displaystyle \frac{3{\omega }_{{xx}}^{2}}{2{\omega }_{x}^{2}}$, $C=\displaystyle \frac{{\omega }_{t}}{{\omega }_{x}}$ are usual Schwarzian variables. If ω is the solution of the equation (25), then$\begin{eqnarray}u=-\displaystyle \frac{{\omega }_{{xx}}}{2{\omega }_{x}}{\rm{i}}+{\omega }_{x}\tanh (\omega ){\rm{i}},\end{eqnarray}$is a solution of equation (1). That is to say, once the solution of equation (25) is known, the corresponding expression u in (26) can be obtained and the new solutions also satisfy equation (1). In the following section, we will list the explicit solutions of equation (1) by selecting the different forms of the function ω in the consistent condition (25).
3.1. Simple soliton solution
A quite trivial straight line solution of the consistent condition (25) has the form$\begin{eqnarray}\omega ={kx}+{ly}+{ct}+d,\end{eqnarray}$where k, l, c, d are the free constants. Substituting the trivial line solution (27) into the expression (26), the complex simple traveling wave solution of the MCBS–nMCBS equation (1) yields$\begin{eqnarray}u={\rm{i}}k\tanh ({kx}+{lt}+{hy}+d).\end{eqnarray}$
3.2. Interaction solutions between the soliton and the cnodial periodic waves
In order to find more interaction solutions for the MCBS–nMCBS equation (1), one can select the proper function ω which satisfies the consistent condition (25). The first type of the soliton-cnoidal wave solution for equation (1) is to fix the function ω as$\begin{eqnarray}\begin{array}{rcl}\omega & = & {k}_{0}x+{l}_{0}y+{c}_{0}t\\ & & +\,A\mathrm{arctanh}[\mathrm{sn}({k}_{1}x+{l}_{1}y+{c}_{1}t,m)],\end{array}\end{eqnarray}$where sn is the usual Jacobi elliptic sine function. Then substituting equation (29) into the consistent condition (25), we can find$\begin{eqnarray}{l}_{1}=-{c}_{1},{c}_{0}=\displaystyle \frac{{c}_{1}{k}_{0}}{{k}_{1}},\end{eqnarray}$and the constants A, k0, k1, l0, c1, m are arbitrary constants. So the first type of soliton-cnoidal interaction solution of equation (1) can be obtained by substituting (29) into (26) with the constant constraint (30).
If we choose the function ω as$\begin{eqnarray}\begin{array}{rcl}\omega & = & {k}_{0}x+{l}_{0}y+{c}_{0}t\\ & & +\,A{{E}}_{\pi }(\mathrm{sn}({k}_{1}x+{l}_{1}y+{c}_{1}t,m),n,m),\end{array}\end{eqnarray}$where Eπ is the third type of incomplete elliptic integral, then the second type of the soliton-cnoidal wave interaction solution for equation (1) should be constructed by substituting (31) into (25) to find the constant relation. One of the constant constraints is the same as (30) and other constants in (31) are arbitrary. We omit the complicated expression for u in (26) with (31) because of its complexity and different soliton-cnoidal wave interaction solutions for equation (1) can be obtained by selecting different constants.
3.3. Resonant soliton solution
One special resonant soliton solution for equation (1) can be constructed by selecting$\begin{eqnarray}\begin{array}{rcl}\omega & = & {k}_{1}x+{l}_{1}y+{c}_{1}t\\ & & +\,c\mathrm{ln}[1+\exp ({k}_{2}x+{l}_{2}y+{c}_{2}t)],\end{array}\end{eqnarray}$then substituting equation (32) into the consistent condition (25), we find the constant constraint$\begin{eqnarray}{c}_{2}=-{l}_{2},{c}_{1}=\displaystyle \frac{{k}_{1}{c}_{2}}{{k}_{2}},\end{eqnarray}$with k1, l1, c, k2, l2 being arbitrary constants.
In this section, the complex traveling wave solution, the soliton-cnoidal interaction solution and the resonant solution for the MCBS–nMCBS equation (1) are given out from the CTE method with arbitrary constants.
4. Summary and discussion
In summary, the nonlocal symmetries of the MCBS–nMCBS equation are obtained with the truncated Painlevé expansion method. In order to solve the initial value problem related to the nonlocal symmetries, we prolong the MCBS–nMCBS equation such that the nonlocal symmetries will become the local Lie point symmetries for the enlarged system. The finite symmetry transformation of the enlarged MCBS–nMCBS system is derived by using the Lie’s first principle and the corresponding symmetry group is given out explicitly with the infinitesimal parameter. The complex exact solutions with nonzero seed solution for the MCBS–nMCBS equation from the finite transformation are studied analytically and graphically in detail. These complex exact solutions can be regarded as the interaction solution of two solitons with different heights. Meanwhile, the MCBS–nMCBS equation is proved to be CTE solvable and we find abundant interaction solutions between the soliton and cnoidal-periodic waves with arbitrary constants. These new interaction solutions obtained in the last section are firstly proposed for the integrable MCBS–nMCBS equation to our knowledge. Furthermore, the integrable nonlinear equations with their negative-order form are of great importance in soliton theory because of their mathematical and physical applications. Many more integrable properties such as the Darboux transformation, symmetry reduction and Hirota bilinear form for different nonlinear equations with their negative-order form are worthy of study in future.
Acknowledgments
The work is supported by National Natural Science Foundation of China (No. 11471215), Shanghai Natural Science Foundation (No. 18ZR142600).
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,, Feiyan LiuCollege of Science, 
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