Novel localized wave solutions of the (2+1)-dimensional Boiti-Leon-Manna-Pempinelli equation
本站小编 Free考研考试/2022-01-02
Li Sun, Jiaxin Qi, Hongli An,∗College of Sciences, Nanjing Agricultural University, Nanjing, 210095, China
First author contact: Author to whom any correspondence should be addressed. Received:2020-08-6Revised:2020-08-28Accepted:2020-09-23Online:2020-12-01
Abstract Based on a special transformation that we introduce, the N-soliton solution of the (2+1)-dimensional Boiti-Leon-Manna-Pempinelli equation is constructed. By applying the long wave limit and restricting certain conjugation conditions to the related solitons, some novel localized wave solutions are obtained, which contain higher-order breathers and lumps as well as their interactions. In particular, by choosing appropriate parameters involved in the N-solitons, two interaction solutions mixed by a bell-shaped soliton and one breather or by a bell-shaped soliton and one lump are constructed from the 3-soliton solution. Five solutions including two breathers, two lumps, and interaction solutions between one breather and two bell-shaped solitons, one breather and one lump, or one lump and two bell-shaped solitons are constructed from the 4-soliton solution. Five interaction solutions mixed by one breather/lump and three bell-shaped solitons, two breathers/lumps and a bell-shaped soliton, as well as mixing with one lump, one breather and a bell-shaped soliton are constructed from the 5-soliton solution. To study the behaviors that the obtained interaction solutions may have, we present some illustrative numerical simulations, which demonstrate that the choice of the parameters has a great impacts on the types of the solutions and their propagation properties. The method proposed can be effectively used to construct localized interaction solutions of many nonlinear evolution equations. The results obtained may help related experts to understand and study the interaction phenomena of nonlinear localized waves during propagations. Keywords：Boiti-Leon-Manna-Pempinelli equation;Hirota bilinear method;localized interaction wave solution
PDF (987KB)MetadataMetricsRelated articlesExportEndNote|Ris|BibtexFavorite Cite this article Li Sun, Jiaxin Qi, Hongli An. Novel localized wave solutions of the (2+1)-dimensional Boiti-Leon-Manna-Pempinelli equation. Communications in Theoretical Physics, 2020, 72(12): 125009- doi:10.1088/1572-9494/abbbd8
1. Introduction
Exact solutions of nonlinear equations play an important role in studying the physical phenomena. Not only because they can help us to understand the physical phenomena described in nature, but also because they can serve as tools to test and improve numerical codes for computing more complicated models. To seek exact solutions of nonlinear equations, lots of powerful methods have been proposed, such as the inverse scattering theory [1], Lie group method [2], the Darboux and Bäcklund transformation [3-5], the variable separation approach [6, 7], the Riemann-Hilbert method [8-13], the Hirota bilinear method [14, 15] and so on [16-23].
In the present work, our main concern is with the construction of exact solutions of the (2+1)-dimensional Boiti-Leon-Manna-Pempinelli (BLMP) equation [24]:$\begin{eqnarray}{u}_{{yt}}+{u}_{{xxxy}}-3{u}_{x}{u}_{{xy}}-3{u}_{{xx}}{u}_{y}=0.\end{eqnarray}$This equation is closely related to the KdV equation. Indeed, by taking u=P(ξ, t)=P(x+f(y), t) with f being an analytic function of y, then (1.1) reduces to the KdV equation [25]$\begin{eqnarray*}{U}_{t}-6{{UU}}_{\xi }+{U}_{\xi \xi \xi }=0,\quad \quad U={P}_{\xi },\end{eqnarray*}$which is usually used to model nonlinear shallow water waves with long wavelengths and small amplitude. Just like the KdV equation, the BLMP equation is an important integrable model and widely used in fluid, plasmas, traffic flows and nano-materials, as well as other physical and engineering phenomena [1]. Therefore, many experts have devoted themselves to the BLMP equation and many significant results have been obtained. For example, Darvishi derived the multi-wave solutions of the BLMP equation via the exp-function method [24]. Wazwaz obtained the multi-soliton solutions via the Cole-Hopf transformation and Hereman's simplified method [26]. Luo constructed some exact solutions and the Bäcklund transformation via the Bell polynomial [27]. Subsequently, some new exact solutions for the BLMP equation were constructed by Wang and Chen via the binary Bell polynomials [28]. Conservation laws were given by Yu and his coworkers via the modified CK's direct method [29]. Traveling wave solutions were obtained by Bekir and Aksoy via the G'/G expansion method [30]. Periodic wave solutions were derived by Tang and Zai via the extended homoclinic test approach [31]. Wronskian, Pfaffian and periodic wave solutions were constructed by Huang and Gao [32]. Important contributions were also made on exact solutions of the (3+1)-dimensional BLMP equation in [33-40], especially on the lump-type and breather-type solutions.
Although many results have been achieved for the (2+1)-dimensional BLMP equation, here our main goal is to investigate the localized interaction wave solutions containing higher-order lumps and breathers, as well as their interactions with solitons. As we know, little research has been carried out on the interaction solutions due to the difficulty in choosing appropriate parameters contained in the N-solitons. However, such higher-order interaction solutions, compared with the low-order ones, are more important and have wider applications in real world. Therefore, it is of interest to investigate the localized interaction wave solutions of the (2+1)-dimensional BLMP equation. What needs to be pointed out is that here we shall adopt a method of long wave limits and certain conjugate conditions to construct the interaction wave solutions, which is quite different from the positive quadratic approach and trigonometric function method used in [38-40].
The organization of the paper is arranged as follows: in section 2, we introduce a special transformation for the (2+1)-dimensional BLMP equation (1.1) and then obtain the N-soliton solutions. In section 3, by taking the long wave limit and restricting certain conjugation conditions to the 3-, 4- and 5-solitons, we construct several different types of localized interaction wave solutions. In order to study any dynamical behaviors the solutions derived may have, we present the corresponding numerical simulations, which fully show that the selection of the parameters impacts on the types of solutions and their propagation properties. Finally, a conclusion is attached.
2. A special transformation to N-soliton solutions of the BLMP equation
In order to construct the localized wave solutions, we need to first derive the N-soliton solutions of the (2+1)-dimensional BLMP equation (1.1). Therefore, a special transformation is introduced via$\begin{eqnarray}u=2{\left(\mathrm{ln}f\right)}_{x}-\gamma y,\end{eqnarray}$where f=f(x, y, t) and γ is an arbitrary constant. It is seen that under the above transformation, the (2+1)-dimensional BLMP equation (1.1) is reducible to the Hirota bilinear form$\begin{eqnarray}({D}_{y}{D}_{t}+{D}_{x}^{3}{D}_{y}+3\gamma {D}_{x}^{2})f\cdot f=0,\end{eqnarray}$where D is the bilinear derivative operator defined via$\begin{eqnarray}\begin{array}{l}{D}_{x}^{\alpha }{D}_{y}^{\beta }{D}_{t}^{\gamma }(f\cdot g)={\left({\partial }_{x}-{\partial }_{x^{\prime} }\right)}^{\alpha }{\left({\partial }_{y}-{\partial }_{y^{\prime} }\right)}^{\beta }\\ \quad \times \,{\left({\partial }_{t}-{\partial }_{t^{\prime} }\right)}^{\gamma }f(x,y,t)g(x^{\prime} ,y^{\prime} ,t^{\prime} ){| }_{x^{\prime} =x,y^{\prime} =y,t^{\prime} =t}.\end{array}\end{eqnarray}$
According to the Hirota bilinear method [14], the N-soliton solutions to the (2+1)-dimensional BLMP equation (1.1) can be obtained, which takes the following form$\begin{eqnarray}f={f}_{N}=\displaystyle \sum _{\mu =0,1}\exp \left(\displaystyle \sum _{i=1}^{N}{\mu }_{i}{\chi }_{i}+\displaystyle \sum _{i\lt j}^{N}{\mu }_{i}{\mu }_{j}\mathrm{ln}{A}_{{ij}}\right),\end{eqnarray}$where$\begin{eqnarray}\begin{array}{l}{\omega }_{i}=-\left({\kappa }_{i}^{2}+\displaystyle \frac{3\gamma }{{\rho }_{i}}\right),\quad \quad {\chi }_{i}={\kappa }_{i}(x+{\rho }_{i}y+{\omega }_{i}t)+{\chi }_{i}^{0},\\ {A}_{{ij}}=\displaystyle \frac{{\rho }_{i}{\rho }_{j}({\kappa }_{i}-{\kappa }_{j})({\kappa }_{i}{\rho }_{i}-{\kappa }_{j}{\rho }_{j})-\gamma {\left({\rho }_{i}-{\rho }_{j}\right)}^{2}}{{\rho }_{i}{\rho }_{j}({\kappa }_{i}+{\kappa }_{j})({\kappa }_{i}{\rho }_{i}+{\kappa }_{j}{\rho }_{j})-\gamma {\left({\rho }_{i}-{\rho }_{j}\right)}^{2}},\\ i,j=1,2,\cdots ,\,N.\end{array}\end{eqnarray}$In the above κi, ρi, ωi and ${\chi }_{i}^{0}$ are arbitrary constants related to the amplitude and phase of the i-soliton, respectively. While ${\sum }_{\mu =\mathrm{0,1}}$ means summation over all possible combinations of μi, μj=0, 1.
3. Localized interaction wave solutions of (2+1)-dimensional BLMP equation
3.1. Two kinds of interaction solutions for N=3
When N=3, from the expression (2.4), we can readily write down f3, which is given by$\begin{eqnarray}\begin{array}{rcl}{f}_{3} & = & 1+{{\rm{e}}}^{{\chi }_{1}}+{{\rm{e}}}^{{\chi }_{2}}+{{\rm{e}}}^{{\chi }_{3}}+{A}_{12}{{\rm{e}}}^{{\chi }_{1}+{\chi }_{2}}+{A}_{23}{{\rm{e}}}^{{\chi }_{2}+{\chi }_{3}}\\ & & +{A}_{13}{{\rm{e}}}^{{\chi }_{1}+{\chi }_{3}}+{A}_{123}{{\rm{e}}}^{{\chi }_{1}+{\chi }_{2}+{\chi }_{3}},\end{array}\end{eqnarray}$where ${A}_{123}={A}_{12}{A}_{13}{A}_{23}$ and χi, Aij (i, j=1, 2, 3) are described in (2.5). In the following, we shall construct two kinds of localized interaction wave solutions by using different techniques. One is a hybrid solution between one breather and a bell-shaped soliton via the conjugate constraint method (given in case 1). The other is a hybrid solution between one lump and a bell-shaped soliton via the long wave limit method (given in case 2).
Case 1 A hybrid solution between one breather and a bell-shaped soliton
If we apply the conjugate constraints to ρ1 and ρ2 via ${\rho }_{1}={\rho }_{2}^{* }={\alpha }_{1}+{\rm{i}}{\beta }_{1}$, and set ${\kappa }_{1}={\kappa }_{2}={\delta }_{1}$, κ3=δ2, ρ3=α2 and ${\chi }_{1}^{0}={\chi }_{2}^{0}={\chi }_{3}^{0}=0$, then f3 turns into the following form$\begin{eqnarray}\begin{array}{rcl}{f}_{3} & = & 1+{A}_{12}{{\rm{e}}}^{2{\zeta }_{1}}+2{{\rm{e}}}^{{\zeta }_{1}}\cos ({\delta }_{1}{\beta }_{1}y-{\psi }_{1}t)\\ & & +{{\rm{e}}}^{{\zeta }_{2}}[1+{A}_{12}{L}_{1}^{2}{{\rm{e}}}^{2{\zeta }_{1}}+2{L}_{1}{{\rm{e}}}^{{\zeta }_{1}}\cos ({\delta }_{1}{\beta }_{1}y-{\psi }_{1}t+\phi )],\end{array}\end{eqnarray}$where$\begin{eqnarray}\begin{array}{l}{\zeta }_{1}={\delta }_{1}x+{\alpha }_{1}{\delta }_{1}y-({\delta }_{1}^{3}+\displaystyle \frac{3{\alpha }_{1}{\delta }_{1}\gamma }{{\alpha }_{1}^{2}+{\beta }_{1}^{2}})t,\quad {\psi }_{1}=-\displaystyle \frac{3{\delta }_{1}{\beta }_{1}\gamma }{{\alpha }_{1}^{2}+{\beta }_{1}^{2}},\\ {\zeta }_{2}={\delta }_{2}x+{\alpha }_{2}{\delta }_{2}y-({\delta }_{2}^{3}+\displaystyle \frac{3\gamma {\delta }_{2}}{{\alpha }_{2}})t,\quad \quad {A}_{12}=\displaystyle \frac{\gamma {\beta }_{1}^{2}}{{\alpha }_{1}{\delta }_{1}^{2}({\alpha }_{1}^{2}+{\beta }_{1}^{2})+\gamma {\beta }_{1}^{2}},\\ {A}_{13}=\displaystyle \frac{{\alpha }_{2}({\delta }_{1}-{\delta }_{2})({\alpha }_{1}+{\rm{i}}{\beta }_{1})({\delta }_{1}{\alpha }_{1}-{\delta }_{2}{\alpha }_{2}+{\rm{i}}{\delta }_{1}{\beta }_{1})-\gamma {\left({\alpha }_{1}-{\alpha }_{2}+{\rm{i}}{\beta }_{1}\right)}^{2}}{{\alpha }_{2}({\delta }_{1}+{\delta }_{2})({\alpha }_{1}+{\rm{i}}{\beta }_{1})({\delta }_{1}{\alpha }_{1}+{\delta }_{2}{\alpha }_{2}+{\rm{i}}{\delta }_{1}{\beta }_{1})-\gamma {\left({\alpha }_{1}-{\alpha }_{2}+{\rm{i}}{\beta }_{1}\right)}^{2}}={L}_{1}{{\rm{e}}}^{{\rm{i}}\phi },\\ {A}_{23}={A}_{13}^{* }={L}_{1}{{\rm{e}}}^{-{\rm{i}}\phi }.\end{array}\end{eqnarray}$In order to detect the dynamical behaviors that the solution given in (3.2) may posses, we present some illustrative numerical simulations. Here we mainly discuss three types of parameters for {κ1, ρ1}, which leads to three kinds of interaction structures of the 1-order breather in the hybrid solution (3.2):
(1) When κ1 is a pure imaginary number and ρ1 is a complex number, the breather in the hybrid solution (3.2) is shown to be only periodic along the direction which is parallel to the x-axis, see figures 1(a1) and (b1).
Figure 1.
New window|Download| PPT slide Figure 1.Three types of interaction behavior of the hybrid solution between one breather and a bell-shaped soliton given in (3.2). The parameters are chosen as follows: (a1)(b1): ${\delta }_{1}=\tfrac{{\rm{i}}}{4}$, δ2=2, ${\alpha }_{1}=\tfrac{1}{4}$, α2=2, β1=1; (a2)(b2): ${\delta }_{1}=\tfrac{1}{4}$, δ2=1.8, ${\alpha }_{1}=\tfrac{1200}{4121}$, α2=1.8, ${\beta }_{1}=\tfrac{15360}{4121};$ (a3)(b3): ${\delta }_{1}=\tfrac{2}{3}+\tfrac{3{\rm{i}}}{5}$, δ2=1.8, ${\alpha }_{1}=\tfrac{2}{3}$, α2=1.8, ${\beta }_{1}=\tfrac{3}{5}$.
(2) When κ1 is a pure real number and ρ1 is a complex number, the breather in the hybrid solution (3.2) is shown to be only periodic along the direction which is perpendicular to the x-axis, see figures 1(a2) and (b2).
(3) When both κ1 and ρ1 are complex numbers, the breather is periodic along the direction which is intersectant to the x-axis, see figures 1(a3) and (b3).
Case 2 A hybrid solution between one lump and a bell-shaped soliton
If we set the parameters in (3.1) as ${\kappa }_{s}={l}_{s}\epsilon \ (s=1,2)$, ${\chi }_{1}^{0}={\chi }_{2}^{0* }={\rm{i}}\pi $, ${\chi }_{3}^{0}=0$ and take the long wave limit as $\epsilon \to 0$, then f3 in (3.1) becomes$\begin{eqnarray}\begin{array}{rcl}{f}_{3} & = & ({a}_{12}+{\theta }_{1}{\theta }_{2}){l}_{1}{l}_{2}{\epsilon }^{2}+({\theta }_{1}{\theta }_{2}+{a}_{23}{\theta }_{1}+{a}_{13}{\theta }_{2}+{a}_{12}\\ & & +{a}_{13}{a}_{23}){{\rm{e}}}^{{\chi }_{3}}{l}_{1}{l}_{2}{\epsilon }^{2}+O({\epsilon }^{3}),\end{array}\end{eqnarray}$where$\begin{eqnarray}\begin{array}{rcl}{\theta }_{s} & = & x+{\rho }_{s}y-\displaystyle \frac{3\gamma }{{\rho }_{s}}t,\quad {a}_{12}=\displaystyle \frac{2{\rho }_{1}{\rho }_{2}({\rho }_{1}+{\rho }_{2})}{\gamma {\left({\rho }_{1}-{\rho }_{2}\right)}^{2}},\\ {a}_{s3} & = & \displaystyle \frac{2{\rho }_{s}{\rho }_{3}{\kappa }_{3}({\rho }_{s}+{\rho }_{3})}{-{\rho }_{s}{\kappa }_{3}^{2}{\rho }_{3}^{2}+\gamma {\left({\rho }_{s}-{\rho }_{3}\right)}^{2}}\quad (s=1,2).\end{array}\end{eqnarray}$Inserting (3.4) into the special transformation (2.1) and taking ${\rho }_{1}={\rho }_{2}^{* }={\alpha }_{1}+{\rm{i}}{\beta }_{1}$, the solution u will become a kind of mixed solution between one lump and a bell-shaped soliton. In order to study the behaviors this solution may have, we take ${\alpha }_{1}=\tfrac{3}{2}$, ${\beta }_{1}=\tfrac{1}{2}$, ${\rho }_{3}=\tfrac{5}{2}$, ${\kappa }_{3}=\tfrac{5}{2}$ and γ=−2. Numerical simulations are presented in figure 2. From the figures, we can see that the lump intersects to the corresponding soliton with time evolution.
Figure 2.
New window|Download| PPT slide Figure 2.The interaction behaviors of the solution mixed by one lump and a bell-shaped soliton given in (3.4). The parameters are chosen as ${\alpha }_{1}=\tfrac{3}{2}$, ${\beta }_{1}=\tfrac{1}{2}$, ${\rho }_{3}=\tfrac{5}{2}$, ${\kappa }_{3}=\tfrac{5}{2}$ and γ=−2.
3.2. Five kinds of interaction solutions for N=4
When N=4, we can easily write down the 4-soliton solution from the expression (2.4). On restricting certain conjugate conditions and a long wave limit, we can obtain several kinds of reduced localized interaction wave solutions. Details are as follows.
Case 1 Two solutions: two breathers and a solution mixed by one breather and two bell-shaped solitons
Here we set ρ1, ρ2 and ρ3, ρ4 to satisfy the conjugate conditions of ${\rho }_{1}={\rho }_{2}^{* }={\alpha }_{1}+{\rm{i}}{\beta }_{1}$ and ${\rho }_{3}={\rho }_{4}^{* }={\alpha }_{2}+{\rm{i}}{\beta }_{2}$. Meanwhile, we choose κ1=κ2=δ1, κ3=κ4=δ2 and ${\chi }_{s}^{0}=0\ ({\text{}}s=1,2,3,4)$, so that the 4-soliton solution is reducible to:$\begin{eqnarray}\begin{array}{rcl}{f}_{4} & = & 1+{A}_{12}{{\rm{e}}}^{2{\zeta }_{1}}+{A}_{34}{{\rm{e}}}^{2{\zeta }_{2}}+2{{\rm{e}}}^{{\zeta }_{1}}\cos ({\delta }_{1}{\beta }_{1}y-{\psi }_{1}t)\\ & & +2{{\rm{e}}}^{{\zeta }_{2}}\cos ({\delta }_{2}{\beta }_{2}y-{\psi }_{2}t)+2{L}_{1}{e}^{{\zeta }_{1}+{\zeta }_{2}}\\ & & \cos ({\delta }_{1}{\beta }_{1}y-{\psi }_{1}t+{\delta }_{2}{\beta }_{2}y-{\psi }_{2}t+{\phi }_{1})\\ & & +2{L}_{2}{{\rm{e}}}^{{\zeta }_{1}+{\zeta }_{2}}\cos ({\delta }_{1}{\beta }_{1}y-{\psi }_{1}t-{\delta }_{2}{\beta }_{2}y+{\psi }_{2}t-{\phi }_{2})\\ & & +2{A}_{12}{{\rm{e}}}^{2{\zeta }_{1}+{\zeta }_{2}}\cos ({\delta }_{2}{\beta }_{2}\xi -{\psi }_{2}t+{\phi }_{1}+{\phi }_{2})\\ & & +2{A}_{34}{{\rm{e}}}^{{\zeta }_{1}+2{\zeta }_{2}}\cos ({\delta }_{1}{\beta }_{1}y-{\psi }_{1}t+{\phi }_{1}-{\phi }_{2})\\ & & +{A}_{12}{A}_{34}{L}_{1}^{2}{L}_{2}^{2}{{\rm{e}}}^{2{\zeta }_{1}+2{\zeta }_{2}},\end{array}\end{eqnarray}$where$\begin{eqnarray}\begin{array}{l}{\zeta }_{s}={\delta }_{s}x+{\alpha }_{s}{\delta }_{s}y-({\delta }_{s}^{3}+\displaystyle \frac{3{\alpha }_{s}{\delta }_{s}\gamma }{{\alpha }_{s}^{2}+{\beta }_{s}^{2}})t,\quad {\psi }_{s}=-\displaystyle \frac{3{\delta }_{s}{\beta }_{s}\gamma }{{\alpha }_{s}^{2}+{\beta }_{s}^{2}},\quad (s=1,2),\\ {A}_{12}=\displaystyle \frac{\gamma {\beta }_{1}^{2}}{{\alpha }_{1}{\delta }_{1}^{2}({\alpha }_{1}^{2}+{\beta }_{1}^{2})+\gamma {\beta }_{1}^{2}},\quad {A}_{34}=\displaystyle \frac{\gamma {\beta }_{2}^{2}}{{\alpha }_{1}{\delta }_{2}^{2}({\alpha }_{2}^{2}+{\beta }_{2}^{2})+\gamma {\beta }_{2}^{2}},\\ {A}_{14}=\displaystyle \frac{({\alpha }_{2}+{\rm{i}}{\beta }_{2})({\alpha }_{1}-{\rm{i}}{\beta }_{1})({\delta }_{1}-{\delta }_{2})[{\delta }_{1}({\alpha }_{1}-{\rm{i}}{\beta }_{1})-{\delta }_{2}({\alpha }_{2}+{\rm{i}}{\beta }_{2})]-\gamma {\left({\alpha }_{1}-{\rm{i}}{\beta }_{1}-{\alpha }_{2}-{\rm{i}}{\beta }_{2}\right)}^{2}}{({\alpha }_{2}+{\rm{i}}{\beta }_{2})({\alpha }_{1}-{\rm{i}}{\beta }_{1})({\delta }_{1}+{\delta }_{2})[{\delta }_{1}({\alpha }_{1}-{\rm{i}}{\beta }_{1})+{\delta }_{2}({\alpha }_{2}+{\rm{i}}{\beta }_{2})]-\gamma {\left({\alpha }_{1}-{\rm{i}}{\beta }_{1}-{\alpha }_{2}-{\rm{i}}{\beta }_{2}\right)}^{2}}={L}_{1}{{\rm{e}}}^{{\rm{i}}{\phi }_{1}},\\ {A}_{24}=\displaystyle \frac{({\alpha }_{2}+{\rm{i}}{\beta }_{2})({\alpha }_{1}+{\rm{i}}{\beta }_{1})({\delta }_{1}-{\delta }_{2})[{\delta }_{1}({\alpha }_{1}+{\rm{i}}{\beta }_{1})-{\delta }_{2}({\alpha }_{2}+{\rm{i}}{\beta }_{2})]-\gamma {\left({\alpha }_{1}+{\rm{i}}{\beta }_{1}-{\alpha }_{2}-{\rm{i}}{\beta }_{2}\right)}^{2}}{({\alpha }_{2}+{\rm{i}}{\beta }_{2})({\alpha }_{1}+{\rm{i}}{\beta }_{1})({\delta }_{1}+{\delta }_{2})[{\delta }_{1}({\alpha }_{1}+{\rm{i}}{\beta }_{1})+{\delta }_{2}({\alpha }_{2}+{\rm{i}}{\beta }_{2})]-\gamma {\left({\alpha }_{1}+{\rm{i}}{\beta }_{1}-{\alpha }_{2}-{\rm{i}}{\beta }_{2}\right)}^{2}}={L}_{2}{{\rm{e}}}^{{\rm{i}}{\phi }_{2}},\\ {A}_{14}={A}_{23}^{* },\quad {A}_{24}={A}_{13}^{* }.\end{array}\end{eqnarray}$Interestingly, we find that the solution u given by (3.6) corresponds to two different types of solutions according to the values of the parameters {κi, ρi} (i=1, 2, 3, 4) selected, which are described as follows:
(1). It represents a kind of 2-order breather solution when ${\rho }_{3}={\rho }_{4}^{* }$ and κ3=κ4 are complex. Such a 2-order breather exhibits three different types of dynamical behavior which can be seen in figure 3. For example, when κi are pure imaginary numbers and ρi are complex numbers, the two 1-order breathers in the solution u (3.6) are both periodic along the x-axis, see figures 3(a1) and (b1). When κi are complex numbers and ρi are pure imaginary numbers, the two 1-order breathers turn to periodic along the direction which is perpendicular to each other, see figures 3(a2) and (b2). When κ1, ρ1 are pure real numbers and κ3, ρ3 are pure imaginary numbers, they are shown to be periodic along the direction intersecting at a certain angle, see figures 3(a3) and (b3).
Figure 3.
New window|Download| PPT slide Figure 3.Three types of interaction behavior of two breathers given in (3.6). The parameters are chosen as follows: (a1)(b1): κ1=i, ${\kappa }_{3}=\tfrac{{\rm{i}}}{2}$, ${\rho }_{1}=1+2{\rm{i}}$, ${\rho }_{3}=\tfrac{4}{5}+\tfrac{1}{2}{\rm{i}};$ (a2)(b2): ${\kappa }_{1}=1+\tfrac{{\rm{i}}}{3}$, ${\kappa }_{3}=\tfrac{2}{3}+\tfrac{3}{5}{\rm{i}}$, ${\rho }_{1}=\tfrac{1}{2}{\rm{i}}$, ${\rho }_{3}=\tfrac{1}{2}{\rm{i}};$ (a3)(b3): ${\kappa }_{1}=\tfrac{2}{3}$, κ3=i, ρ1=1, ${\rho }_{3}=\tfrac{4}{5}{\rm{i}}$.
(2). It denotes a kind of solution mixed by one breather and two bell-shaped solitons when ${\rho }_{3}={\rho }_{4}^{* }$ and κ3=κ4 are real numbers. Such a hybrid solution also shows three different types of dynamical behavior which can be seen in figure 4. As previously discussed for solution (3.2), when κ1 is a pure imaginary number and ρ1 is a complex number, the breather in the mixed solution (3.6) is only periodic along the x-axis, see figures 4(a1) and (b1). When κ1 is a pure real number and ρ1 is a complex number, the breather in (3.6) is only periodic along the direction which is perpendicular to the x-axis, see figures 4(a2) and (b2). When both κ1 and ρ1 are complex numbers, the breather is periodic along the direction which is intersectant to the x-axis, see figures 4(a3) and (b3).
Figure 4.
New window|Download| PPT slide Figure 4.Three types of interaction behavior of the hybrid solution between one breather and two bell-shaped solitons (3.6) with κ3=1 and ${\rho }_{3}=\tfrac{5}{4}$. The other parameters are chosen as follows: (a1)(b1): ${\kappa }_{1}=\tfrac{{\rm{i}}}{4}$, ${\rho }_{1}=\tfrac{1}{4}+{\rm{i}};$ (a2)(b2): ${\kappa }_{1}=\tfrac{1}{4}$, ${\rho }_{1}=\tfrac{1200}{4121}+\tfrac{15360}{4121}{\rm{i}};$ (a3)(b3): κ1=1+i, ρ1=1+i.
Case 2 Two hybrid solutions: mixed by one lump and one breather/two bell-shaped solitons
(1). Setting ${\rho }_{1}={\rho }_{2}^{* }$, ${\rho }_{3}={\rho }_{4}^{* }$ and κ3=κ4, the solution u given by (3.8) represents a kind of solution mixed by one lump and one breather whose dynamical behaviors are shown in figure 5. From the first column of figure 5, we can see that the breather is periodic along the x-axis and it turns to be periodic perpendicular to the x-axis in the second column. While in the last column, the breather is shown to be periodic along the direction intersecting the x-axis.
Figure 5.
New window|Download| PPT slide Figure 5.Three types of of interaction behavior of the hybrid solution between one lump and one breather given in (3.8). The parameters are chosen as: (a1)(b1): ${\kappa }_{3}=\tfrac{{\rm{i}}}{4}$, ${\rho }_{1}=\tfrac{1}{2}+\tfrac{1}{2}{\rm{i}}$, ${\rho }_{3}=\tfrac{1}{4}+{\rm{i}};$ (a2)(b2): ${\kappa }_{3}=\tfrac{1}{4}$, ${\rho }_{1}=\tfrac{1}{2}+\tfrac{1}{2}{\rm{i}}$, ${\rho }_{3}=\tfrac{1200}{4121}+\tfrac{15360}{4121}{\rm{i}};$ (a3)(b3): ${\kappa }_{3}=\tfrac{1}{4}$, ${\rho }_{1}=\tfrac{1}{2}+\tfrac{1}{2}{\rm{i}}$, ${\rho }_{3}=\tfrac{1}{4}+{\rm{i}}$.
(2). Setting ${\rho }_{1}={\rho }_{2}^{* }$ and ρ3, ρ4, κ3, κ4 as real, the solution u given by (3.8) denotes a kind of solution mixed by one lump and two bell-shaped solitons whose dynamical behaviors are exhibited in figure 6.
Figure 6.
New window|Download| PPT slide Figure 6.Interaction behaviors of the hybrid solution between one lump and two bell-shaped solitons given in (3.8). The parameters are chosen as ${\rho }_{1}=\tfrac{1}{2}+\tfrac{{\rm{i}}}{2}$, ${\rho }_{3}=\tfrac{3}{2}$, ${\rho }_{4}=\tfrac{5}{2}$, ${\kappa }_{3}=\tfrac{3}{2}$ and ${\kappa }_{4}=\tfrac{5}{2}$.
Case 3 A 2-order lump solution
Setting ${\kappa }_{s}={l}_{s}\epsilon \ (s=1,2,3,4)$ and ${\chi }_{1}^{0}={\chi }_{2}^{0* }={\chi }_{3}^{0}\,={\chi }_{4}^{0* }={\rm{i}}\pi $, and applying the long wave limit ε → 0 to the 4-soliton solution, we obtain$\begin{eqnarray}\begin{array}{rcl}{f}_{4} & = & ({\theta }_{1}{\theta }_{2}{\theta }_{3}{\theta }_{4}+{a}_{34}{\theta }_{1}{\theta }_{2}+{a}_{24}{\theta }_{1}{\theta }_{3}+{a}_{23}{\theta }_{1}{\theta }_{4}+{a}_{14}{\theta }_{2}{\theta }_{3}\\ & & +{a}_{13}{\theta }_{2}{\theta }_{4}+{a}_{12}{\theta }_{3}{\theta }_{4}+{a}_{12}{a}_{34}+{a}_{13}{a}_{24}\\ & & +{a}_{14}{a}_{23}){l}_{1}{l}_{2}{l}_{3}{l}_{4}{\epsilon }^{4}+O({\epsilon }^{5}),\end{array}\end{eqnarray}$where$\begin{eqnarray}\begin{array}{rcl}{\theta }_{s} & = & x+{\rho }_{s}y-\displaystyle \frac{3\gamma }{{\rho }_{s}}t,\\ {a}_{{sj}} & = & \displaystyle \frac{2{\rho }_{s}{\rho }_{j}({\rho }_{s}+{\rho }_{j})}{\gamma {\left({\rho }_{s}-{\rho }_{j}\right)}^{2}}\quad (1\leqslant s\lt j\leqslant 4).\end{array}\end{eqnarray}$On setting ${\rho }_{1}={\rho }_{3}^{* }={\alpha }_{1}+{\beta }_{1}{\rm{i}}$, ${\rho }_{2}={\rho }_{4}^{* }={\alpha }_{2}+{\beta }_{2}{\rm{i}}$, asj>0 and inserting (3.10) into the expression (2.1), then we get a 2-order lump solution. Here we take α1=1, α2=0.9, ${\beta }_{1}=\tfrac{1}{2}$ and ${\beta }_{2}=\tfrac{1}{3}$ and dynamical behaviors of the 2-order lump are shown in figure 7.
Figure 7.
New window|Download| PPT slide Figure 7.Interaction behaviors of the 2-order lump solution given by (3.10) with α1=1, α2=0.9, ${\beta }_{1}=\tfrac{1}{2}$ and ${\beta }_{2}=\tfrac{1}{3}$.
3.3. Five kinds of interaction solutions for N=5
When N=5, we can easily write down the 5-soliton solution from expression (2.4). On restricting certain conjugate conditions and a long wave limit, we shall obtain several kinds of localized wave solutions. Details are described as follows.
Case 1 Two solutions mixed by i-order breather and (5-2i)-bell-shaped solitons (i=1, 2)
Here, we shall first discuss the solution mixed by a 2-order breather and a bell-shaped soliton. In order to obtain such a solution, we introduce the conjugate conditions to ρi (i=1, ⋯ 4) via$\begin{eqnarray}{\rho }_{1}={\rho }_{2}^{* }={\alpha }_{1}+{\rm{i}}{\beta }_{1},\quad {\rho }_{3}={\rho }_{4}^{* }={\alpha }_{2}+{\rm{i}}{\beta }_{2}.\end{eqnarray}$Meanwhile, we set$\begin{eqnarray}{\rho }_{5}={\alpha }_{3},{\kappa }_{1}={\kappa }_{2}={\delta }_{1},{\kappa }_{3}={\kappa }_{4}={\delta }_{2},{\kappa }_{5}={\delta }_{3},\end{eqnarray}$and ${\chi }_{s}^{0}=0\ (s=1,\cdots ,\,5)$. Inserting the above conditions into (2.4), so that we obtain a hybrid solution consisting of a 2-order breather and a bell-shaped soliton, in which f5 takes a form of$\begin{eqnarray}\begin{array}{rcl}{f}_{5} & = & 1+{A}_{12}{{\rm{e}}}^{2{\zeta }_{1}}+{A}_{34}{{\rm{e}}}^{2{\zeta }_{2}}+2{e}^{{\zeta }_{1}}\cos ({\delta }_{1}{\beta }_{1}y-{\psi }_{1}t)\\ & & +2{{\rm{e}}}^{{\zeta }_{2}}\cos ({\delta }_{2}{\beta }_{2}y-{\psi }_{2}t)+{{\rm{e}}}^{{\zeta }_{3}}\\ & & +{{\rm{e}}}^{{\zeta }_{1}+{\zeta }_{2}}[2{L}_{1}\cos ({\delta }_{1}{\beta }_{1}y-{\psi }_{1}t+{\delta }_{2}{\beta }_{2}y-{\psi }_{2}t+{\phi }_{1})\\ & & +2{L}_{2}\cos ({\delta }_{1}{\beta }_{1}y-{\psi }_{1}t-{\delta }_{2}{\beta }_{2}y+{\psi }_{2}t-{\phi }_{2})\\ & & +2{A}_{12}{L}_{1}{L}_{2}{{\rm{e}}}^{{\zeta }_{1}}\cos ({\delta }_{2}{\beta }_{2}y-{\psi }_{2}t+{\phi }_{1}+{\phi }_{2})\\ & & +2{A}_{34}{L}_{1}{L}_{2}{{\rm{e}}}^{{\zeta }_{2}}\\ & & \times \cos ({\delta }_{1}{\beta }_{1}y-{\psi }_{1}t+{\phi }_{1}-{\phi }_{2})+{A}_{12}{A}_{34}{L}_{1}^{2}{L}_{2}^{2}{{\rm{e}}}^{{\zeta }_{1}+{\zeta }_{2}}]\\ & & +{{\rm{e}}}^{{\zeta }_{1}+{\zeta }_{3}}[{A}_{12}{L}_{3}^{2}{{\rm{e}}}^{{\zeta }_{1}}+2{L}_{3}\cos ({\delta }_{1}{\beta }_{1}y-{\psi }_{2}t+{\phi }_{3})]\\ & & +{{\rm{e}}}^{{\zeta }_{2}+{\zeta }_{3}}[{A}_{34}{L}_{4}^{2}{{\rm{e}}}^{{\zeta }_{2}}+2{L}_{4}\cos ({\delta }_{2}{\beta }_{2}y-{\psi }_{2}t+{\phi }_{4})]\\ & & +{L}_{3}{L}_{4}{{\rm{e}}}^{{\zeta }_{1}+{\zeta }_{2}+{\zeta }_{3}}[2{L}_{1}\cos ({\delta }_{1}{\beta }_{1}y-{\psi }_{1}t+{\delta }_{2}{\beta }_{2}y\\ & & -{\psi }_{2}t+{\phi }_{1}+{\phi }_{3}+{\phi }_{4})+2{L}_{2}\cos ({\delta }_{1}{\beta }_{1}y\\ & & -{\psi }_{1}t-{\delta }_{2}{\beta }_{2}y+{\psi }_{2}t-{\phi }_{2}+{\phi }_{3}-{\phi }_{4})\\ & & +2{A}_{12}{L}_{1}{L}_{2}{L}_{3}{{\rm{e}}}^{{\zeta }_{1}}\cos ({\delta }_{2}{\beta }_{2}y+{\psi }_{2}t+{\phi }_{1}+{\phi }_{2}+{\phi }_{4})\\ & & +2{A}_{34}{L}_{1}{L}_{2}{L}_{4}{{\rm{e}}}^{{\zeta }_{2}}\cos ({\delta }_{1}{\beta }_{1}y-{\psi }_{1}t+{\phi }_{1}-{\phi }_{2}+{\phi }_{3})\\ & & +{A}_{12}{A}_{34}{L}_{1}^{2}{L}_{2}^{2}{L}_{3}{L}_{4}{{\rm{e}}}^{{\zeta }_{1}+{\zeta }_{2}}],\end{array}\end{eqnarray}$where ${\zeta }_{1},{\zeta }_{2},{\psi }_{1},{\psi }_{2},{A}_{{sj}}\ (1\leqslant s\lt j\leqslant 4)$ are given in (3.7), and$\begin{eqnarray}\begin{array}{rcl}{\zeta }_{3} & = & {\delta }_{3}x+{\delta }_{3}{\alpha }_{3}y-({\delta }_{3}^{3}+\displaystyle \frac{3{\delta }_{3}\gamma }{{\alpha }_{3}})t,\\ {A}_{15} & = & {A}_{25}^{* }\\ & = & \displaystyle \frac{{\alpha }_{3}({\alpha }_{1}-{\rm{i}}{\beta }_{1})({\delta }_{1}-{\delta }_{3})({\delta }_{1}{\alpha }_{1}-{\delta }_{3}{\alpha }_{3}-{\rm{i}}{\delta }_{1}{\beta }_{1})-\gamma {\left({\alpha }_{1}-{\alpha }_{3}-{\rm{i}}{\beta }_{1}\right)}^{2}}{{\alpha }_{3}({\alpha }_{1}-{\rm{i}}{\beta }_{1})({\delta }_{1}+{\delta }_{3})({\delta }_{1}{\alpha }_{1}+{\delta }_{3}{\alpha }_{3}-{\rm{i}}{\delta }_{1}{\beta }_{1})-\gamma {\left({\alpha }_{1}-{\alpha }_{3}-{\rm{i}}{\beta }_{1}\right)}^{2}}={L}_{3}{{\rm{e}}}^{{\rm{i}}{\phi }_{3}},\\ {A}_{45} & = & {A}_{35}^{* }\\ & = & \displaystyle \frac{{\alpha }_{3}({\alpha }_{2}+{\rm{i}}{\beta }_{2})({\delta }_{1}-{\delta }_{3})({\delta }_{1}{\alpha }_{2}-{\delta }_{3}{\alpha }_{3}+{\rm{i}}{\delta }_{1}{\beta }_{2})-\gamma {\left({\alpha }_{2}-{\alpha }_{3}+{\rm{i}}{\beta }_{2}\right)}^{2}}{{\alpha }_{3}({\alpha }_{2}+{\rm{i}}{\beta }_{2})({\delta }_{1}+{\delta }_{3})({\delta }_{1}{\alpha }_{2}+{\delta }_{3}{\alpha }_{3}+{\rm{i}}{\delta }_{1}{\beta }_{2})-\gamma {\left({\alpha }_{2}-{\alpha }_{3}+{\rm{i}}{\beta }_{2}\right)}^{2}}={L}_{4}{{\rm{e}}}^{{\rm{i}}{\phi }_{4}}.\end{array}\end{eqnarray}$In order to shed light on the dynamical behaviors that the above solution may posses, we give some illustrative numerical simulations in figure 8. It can be seen from the figures that the solution u given in (3.14) exhibits three types of interaction behaviors due to different values of {κi, ρi} (i=1, ⋯, 5). When the parameters are chosen as κ1=i, ${\kappa }_{3}=\tfrac{{\rm{i}}}{2}$, κ5=2.2, ${\rho }_{1}=1+2{\rm{i}}$, ${\rho }_{3}=\tfrac{4}{5}+{\rm{i}}$ and ρ5=2.2, the solution represents the interactions among two parallel 1-order breathers and a soliton as shown in figures 8(a1) and (b1). When the parameters are chosen as ${\kappa }_{1}=\tfrac{2}{3}$, κ3=i, κ5=2.2, ${\rho }_{1}=\tfrac{1350}{829}+\tfrac{3645}{829}{\rm{i}}$, ρ3=i and ρ5=2.2, the solution denotes the interactions among two perpendicular 1-order breathers and a bell-shaped soliton as shown in figures 8(a2) and (b2). Meanwhile, when the parameters are chosen as ${\kappa }_{1}=\tfrac{2}{3}$, κ3=i, κ5=2.2, ${\rho }_{1}=\tfrac{4}{5}+{\rm{i}}$, ρ3=i and ρ5=2.2, the solution represents the interactions among two cross 1-order breathers and a soliton as seen in figures 8(a3) and (b3).
Figure 8.
New window|Download| PPT slide Figure 8.Three types of of interaction behavior of the solution mixed by a 2-order breather and a bell-shaped soliton. The parameters are selected as follows: (a1)(b1) κ1=i, ${\kappa }_{3}=\tfrac{{\rm{i}}}{2}$, κ5=2.2, ${\rho }_{1}=1+2{\rm{i}}$, ${\rho }_{3}=\tfrac{4}{5}+{\rm{i}}$, ρ5=2.2; (a2)(b2) ${\kappa }_{1}=\tfrac{2}{3}$, κ3=i, κ5=2.2, ${\rho }_{1}=\tfrac{1350}{829}+\tfrac{3645{\rm{i}}}{829}$, ρ3=i, ρ5=2.2; (a3)(b3) ${\kappa }_{1}=\tfrac{2}{3}$, κ3=i, κ5=2.2, ${\rho }_{1}=\tfrac{4}{5}+{\rm{i}}$, ρ3=i, ρ5=2.2.
In the above we have discussed the solution mixed by a 2-order breather and a bell-shaped soliton via two conjugate conditions as introduced in (3.12). However, when only one conjugate condition ${\rho }_{1}={\rho }_{2}^{* }={\alpha }_{1}+{\rm{i}}{\beta }_{1}$ is introduced into (2.4), then we can obtain an interaction solution between a 1-order breather and three bell-shaped solitons. Since the expression of the solution is too complicated, we omit the detailed expression and only give its numerical simulations in figure 9. From the figures, we can see that just as the solution given in (3.12), the solution mixed by a 1-order breather and three bell-shaped solitons also has three different structures. When we take ρ1=1+i and κ1=i, the breather is shown to be periodic along the x-axis as seen in figures 9(a1) and (b1). When we take ${\rho }_{1}=\tfrac{1350}{829}+\tfrac{3645}{829}{\rm{i}}$ and ${\kappa }_{1}=\tfrac{2}{3}$, the breather is periodic along the direction which is perpendicular to the x-axis as seen in figures 9(a2) and (b2). Meanwhile, when we take ${\rho }_{1}=\tfrac{4}{5}+{\rm{i}}$ and ${\kappa }_{1}=\tfrac{2}{3}$, it turns to periodic along the direction which is intersecting the x-axis at a certain angle as seen in figures 9(a3) and (b3).
Figure 9.
New window|Download| PPT slide Figure 9.Three types of interaction behavior of the solution mixed by one breather and three bell-shaped solitons. The parameters are selected as follows: (a1)(b1): κ1=i, κ3=1.2, κ4=1.8, κ5=2.2, ρ1=1+i, ρ3=1.2, ρ4=1.8, ρ5=2.2; (a2)(b2): ${\kappa }_{1}=\tfrac{2}{3}$, κ3=1.2, κ4=1.6, κ5=2.2, ${\rho }_{1}=\tfrac{1350}{829}+\tfrac{3645}{829}{\rm{i}}$, ρ3=1.2, ρ4=1.6, ρ5=2.2; (a3)(b3): ${\kappa }_{1}=\tfrac{2}{3}$, κ3=1.8, κ4=2, κ5=2.2, ${\rho }_{1}=\tfrac{4}{5}+{\rm{i}}$, ρ3=1.8, ρ4=2, ρ5=2.2.
Case 2 Two solutions: mixed by one lump and three bell-shaped solitons and by a lump, breather and soliton
(1). Taking ${\rho }_{1}={\rho }_{2}^{* }$, ${\rho }_{3}={\rho }_{4}^{* }$ and ${\kappa }_{3}={\kappa }_{4}^{* }$, the solution u given by (3.16) represents an interaction solution mixed by one lump, one breather and a bell-shaped soliton. To exhibit the dynamical behaviors, we choose ${\rho }_{1}=\tfrac{4}{5}+{\rm{i}}$, ρ3=1+i, ${\rho }_{5}=\tfrac{1}{2}$, ${\kappa }_{3}=\tfrac{2}{3}$ and ${\kappa }_{5}=\tfrac{1}{2}$, the interactions of one lump, one breather and a bell-shaped soliton are shown in figure 10.
Figure 10.
New window|Download| PPT slide Figure 10.Interaction behaviors of the solution consisting of a lump, a breather and a bell-shaped soliton. The parameters are selected as ${\rho }_{1}=\tfrac{4}{5}+{\rm{i}}$, ρ3=1+i, ${\rho }_{5}=\tfrac{1}{2}$, ${\kappa }_{3}=\tfrac{2}{3}$, ${\kappa }_{5}=\tfrac{1}{2}$.
(2). Taking ${\rho }_{1}={\rho }_{2}^{* }$ and ${\kappa }_{3}={\kappa }_{4}^{* }$, the solution u given by (3.16) denotes an interaction solution mixed by a lump and three bell-shaped solitons. Here we take ${\rho }_{1}=\tfrac{1}{2}+\tfrac{{\rm{i}}}{2}$, ${\rho }_{3}=\tfrac{3}{2}$, ρ4=2, ρ5=1.6, ${\kappa }_{3}=\tfrac{3}{2}$ and κ5=1.6, the interactions of a lump and three bell-shaped solitons are simulated in figure 11.
Figure 11.
New window|Download| PPT slide Figure 11.The interactions of the solution consisting of a lump and three bell-shaped solitons. The parameters are chosen as: ${\rho }_{1}=\tfrac{1}{2}+\tfrac{{\rm{i}}}{2}$, ${\rho }_{3}=\tfrac{3}{2}$, ρ4=2, ρ5=1.6, ${\kappa }_{3}=\tfrac{3}{2}$ and κ5=1.6.
Case 3 A hybrid solution between two lumps and a bell-shaped soliton
Setting ${\kappa }_{s}={l}_{s}\epsilon \ (s=1,2,3,4)$, ${\chi }_{1}^{0}={\chi }_{2}^{0* }={\chi }_{3}^{0}\,={\chi }_{4}^{0* }={\rm{i}}\pi $, ${\chi }_{5}^{0}=0$ and applying the long wave limit ε → 0 to the 5-soliton solution, then we can get$\begin{eqnarray}\begin{array}{rcl}{f}_{5} & = & ({\theta }_{1}{\theta }_{2}{\theta }_{3}{\theta }_{4}+{a}_{34}{\theta }_{1}{\theta }_{2}+{a}_{24}{\theta }_{1}{\theta }_{3}+{a}_{23}{\theta }_{1}{\theta }_{4}+{a}_{14}{\theta }_{2}{\theta }_{3}\\ & & +{a}_{13}{\theta }_{2}{\theta }_{4}+{a}_{12}{\theta }_{3}{\theta }_{4}+{a}_{34}{\theta }_{1}{\theta }_{2}+{a}_{12}{a}_{34}\\ & & +{a}_{13}{a}_{24}+{a}_{14}{a}_{23}){l}_{1}{l}_{2}{l}_{3}{l}_{4}{\epsilon }^{4}+\{{\theta }_{1}{\theta }_{2}{\theta }_{3}{\theta }_{4}\\ & & +{a}_{45}{\theta }_{1}{\theta }_{2}{\theta }_{3}+{a}_{35}{\theta }_{1}{\theta }_{2}{\theta }_{4}+{a}_{25}{\theta }_{1}{\theta }_{3}{\theta }_{4}\\ & & +{a}_{15}{\theta }_{2}{\theta }_{3}{\theta }_{4}+{\theta }_{1}[{\theta }_{2}({a}_{35}{a}_{45}+{a}_{34})\\ & & +{\theta }_{3}({a}_{25}{a}_{45}+{a}_{24})+{\theta }_{4}({a}_{25}{a}_{35}+{a}_{23})]\\ & & +{\theta }_{2}[{\theta }_{3}({a}_{15}{a}_{45}+{a}_{14})+{\theta }_{4}({a}_{15}{a}_{35}+{a}_{13})]\\ & & +{\theta }_{3}{\theta }_{4}({a}_{15}{a}_{25}+{a}_{12})\\ & & +{\theta }_{1}[({a}_{25}{a}_{35}+{a}_{23}){a}_{45}+{a}_{24}{a}_{35}+{a}_{25}{a}_{34}]\\ & & +{\theta }_{2}[({a}_{15}{a}_{35}+{a}_{13}){a}_{45}+{a}_{14}{a}_{35}+{a}_{15}{a}_{34}]\\ & & +{\theta }_{3}[({a}_{15}{a}_{25}+{a}_{12}){a}_{45}+{a}_{14}{a}_{25}+{a}_{15}{a}_{24}]\\ & & +{\theta }_{4}[({a}_{15}{a}_{25}+{a}_{12}){a}_{35}+{a}_{13}{a}_{25}+{a}_{15}{a}_{23}]\\ & & +{a}_{13}{a}_{24}+{a}_{23}{a}_{14}+({a}_{15}{a}_{25}+{a}_{34}){a}_{12}\\ & & +({a}_{14}{a}_{25}+{a}_{15}{a}_{24}){a}_{35}\\ & & +[({a}_{15}{a}_{25}+{a}_{12}){a}_{35}+{a}_{13}{a}_{25}\\ & & +{a}_{15}{a}_{23}]{a}_{45}\}{{\rm{e}}}^{{\chi }_{5}}{l}_{1}{l}_{2}{l}_{3}{l}_{4}{\epsilon }^{4}+O({\epsilon }^{5}),\end{array}\end{eqnarray}$where$\begin{eqnarray}\begin{array}{rcl}{\theta }_{s} & = & x+{\rho }_{s}y-\displaystyle \frac{3\gamma }{{\rho }_{s}}t,\\ {a}_{s5} & = & \displaystyle \frac{2{\rho }_{s}{\rho }_{j}{\kappa }_{5}({\rho }_{s}+{\rho }_{j})}{-{\rho }_{s}{\kappa }_{5}^{2}{\rho }_{j}^{2}+\gamma {\left({\rho }_{s}-{\rho }_{j}\right)}^{2}}\quad (s=1,2,3,4),\\ {a}_{{sj}} & = & \displaystyle \frac{2{\rho }_{s}{\rho }_{j}({\rho }_{s}+{\rho }_{j})}{\gamma {\left({\rho }_{s}-{\rho }_{j}\right)}^{2}}\quad (1\leqslant s\lt j\leqslant 4).\end{array}\end{eqnarray}$Here we take ${\rho }_{1}={\rho }_{2}^{* }$ and ${\rho }_{3}={\rho }_{4}^{* }$, then the solution u given by (3.18) describes an interaction solution between two lumps and a bell-shaped soliton. To exhibit the behaviors, we choose ρ1=1+2i, ${\rho }_{3}=\tfrac{1}{2}+2{\rm{i}}$, ${\kappa }_{5}=\tfrac{1}{3}$ and ${\rho }_{5}=\tfrac{1}{3}$, the interactions of two lumps and a bell-shaped soliton are shown in figure 12.
Figure 12.
New window|Download| PPT slide Figure 12.The interactions of the solution consisting of two lumps and a bell-shaped soliton with γ=−1. The parameters are chosen as: ρ1=1+2i, ${\rho }_{3}=\tfrac{1}{2}+2{\rm{i}}$, ${\kappa }_{5}=\tfrac{1}{3}$, ${\rho }_{5}=\tfrac{1}{3}$.
4. Conclusion
The (2+1)-dimensional BLMP equation is an important integrable model, which has been extensively used in the areas of fluid, plasmas, traffic flow and engineering phenomena. In this paper, we introduce a special transformation to the BLMP equation and thereby construct its N-soliton solutions via the Hirota bilinear method. Then by taking the long wave limit and imposing the conjugation conditions to the related N-soliton solutions, we obtain various types of localized wave solutions, including interaction solutions between one bell-shaped soliton and one breather/lump (see figures 1-2), two breathers (see figure 3), two lumps (see figure 7), interactions between two bell-shaped solitons and one breather/lump (see figures 4 and 6), one lump and one breather (see figure 5), three bell-shaped solitons and one breather/lump (see figures 9 and 11), one bell-shaped soliton and two breathers/lumps (see figures 8 and 12), as well as interactions between one breather, one lump and one bell-shaped soliton (see figure 10). The dynamical behaviors of these solutions are analyzed numerically, which shows that the choices of the parameters can affect the types of the solutions and their propagation properties. However, for the more complicated interaction solutions when N>5, due to the complexity in selecting the parameters, it becomes rather difficult to study their interactions, but they are worthy of deep investigation. Hence, in the future, we shall focus on studying how the parameters involved in the N-soliton solutions affect the behaviors of the higher-order localized interaction solutions. The method proposed can be used to construct localized interaction solutions of other nonlinear evolution equations. The results obtained provide a possible tool to study the propagation phenomena of localized waves.
Acknowledgments
We would like to express sincere thanks to the reviewers for the kind comments and valuable suggestions. This work is supported by the National Natural Science Foundation of China under grant No. 11775116 and Jiangsu Qinglan high-level talent Project.
AblowitzM JClarksonP A1991Soliton, Nonlinear Evolution Equations and Inverse Scatting New York Cambridge University Press [Cited within: 2]
BlumanG WKumeiS1989Symmetries and Differential Equations New York Springer [Cited within: 1]
RogersCSchiefW2002Bäcklund and Darboux Transformations Geometry and Modern Applications in Soliton Theory New York Cambridge University Press [Cited within: 1]
MatveevV ASalleM A1991Darboux Transformations and Solitons Berlin Springer
,∗College of Sciences, 
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