# Rational Solutions for the (2+1)-Dimensional Modified KdV-CBS Equation

Authors: Yan Li; Temuer Chaolu; Yuexing Bai
Journal: Advances in Mathematical Physics
 (2019)

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## Abstract

In this paper, with the help of symbolic computation, three types of rational solutions for the (2+1)-dimensional modified KdV-Calogero-Bogoyavlenkskii-Schiff equation are derived. By means of the truncated Painlevé expansion, we show that the (2+1)-dimensional modified KdV-Calogero-Bogoyavlenkskii-Schiff equation can be written as a trilinear-linear equation, from which we get explicit representation for rational solutions of the (2+1)-dimensional modified KdV-Calogero-Bogoyavlenkskii-Schiff equation.


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## Body

## 1. Introduction

It is an important problem to seek exact solutions in the study of the nonlinear evolution equations (NLEEs). With regard to the methods for solving NLEEs, various techniques and approaches have been proposed, for instance, the Darboux transformation [1], the Lie group method [2–4], and the Hirota bilinear method [5]. Recently, based on the Hirota bilinear method, a method for constructing a lump solution by taking the function f in its bilinear form as a positive quadratic function, proposed by Ma [6], has been used to solve many NLEEs. By using this method, the phenomenon of the rogue wave was found in Refs. [7–9]. And then, more and more integrable soliton equations are found to have lump solutions and mixed kink-lump solutions [10–23].In this paper, we focus on the following (2+1)-dimensional modified KdV-Calogero-Bogoyavlenkskii-Schiff (KdV-CBS) equation
(1)ut−4u2uy−2ux∂x−1u2y+2uxxy−6u2ux+uxxx=0,which can be derived from the (2+1)-dimensional KdV-CBS equation [24, 25]
(2)qt+4qqy−2qx∂x−1qy+qxxy−6qqx+qxxx=0,by means of the Miura transformation q=u2−ux [26]. Here, the subscripts x and y denote the partial derivatives with respect to x and y, respectively. In order to treat the integral appearing in equation (1), so that equation (1) can be better analyzed and solved, let vx=2uuy; equation (1) is transformed into the following system:
(3)ut−4u2uy−2uxv+uxxy−6u2ux+uxxx=0,2uuy=vx.The soliton-cnoidal wave solutions of equation (3) were obtained in Ref. [27]. Interesting studies on the family of KdV-type equations have been carried out in the following papers [28–31]. The aim of this paper is to construct the rational solutions of system (3) by solving a trilinear-linear equation related to its truncated Painlevé expansion.The outline of this paper is as follows. In Section2, using the truncated Painlevé expansion, we convert the original modified KdV-CBS equation to a trilinear-linear equation. In Section 3, the first class of rational solutions for the modified KdV-CBS equation is obtained by taking function f as a positive quadratic function. In Section 4, the second class of rational solutions is obtained by taking function f as a positive quadratic and an exponential function. In Section 5, the third class of rational solutions is obtained by taking function f as a positive quadratic and two exponential functions. The conclusion will be given in Section 6.


## 2. Trilinear-Linear Equation

Based on the Painlevé analysis proposed in Ref. [32], equation (3) possesses a truncated Painlevé expansion as follows:
(4)u=u0+u1f,v=v0+v1f+v2f2,with u0, u1, v0, v1, v2, and f being the function of x, y, and t. We substitute (4) into (3); the results can be obtained as follows:
(5)v1=−fxy,v2=fxfy,u0=−fxx2fx,u1=fx,v0=12ftfx−12fxxfxyfx2+12fxxyfx+12fxxxfx−34fxx2fx2,with f satisfying the following trilinear-linear equation:
(6)3fxxfyxxfx−fyxxxfx2−fxtfx2−fxxxxfx2+fxxftfx+4fxxxfxxfx+fyfxxxfx−3fxx3−3fxx2fyx=0.By solving the trilinear-linear equation (6), we can get
(7a)u=−12fxxfx+fxf,(7b)v=12ftfx−12fxxfxyfx2+12fyxxfx+12fxxxfx−34fxx2fx2−fyx2f+fxfyf2.Based on the idea in Refs. [6, 7], we take f as
(8)f=a1x+a2y+a3t+a42+a5x+a6y+a7t+a82+a9+αexpk1x+k2y+k3t+k4+βexp−k1x+k2y+k3t+k4,where ai, k, kj, 1≤i≤9, and 1≤j≤4 are real parameters to be determined. Substituting (8) into trilinear-linear equation (6), we get a set of solutions. In the following, we will give three types of rational solutions of equation (3) via (8).


## 3. Rational Solutions from the Quadratic Function

Substitution equation (8) into trilinear-linear equation (6), we will get the following set of constraining equations for the parameters
(9)a1=a1,a2=−a12+a52+a5a6a1,a3=a3,a4=a4,a5=a5,a6=a6,a7=a3a5a1,a8=a8,a9=a9,α=0,β=0,k1=k1,k2=k2,k3=k3,k4=k4,where
(10)a1≠0,a9>0,to guarantee that the corresponding f is positive. By substituting (9) into (8), the function f can be obtained as follows:
(11)f=a1x−a12+a52+a5a6a1y+a3t+a42+a5x+a6y+a3a5ta1+a82+a9.By means of equation (7a) and equation (7b), we get the solutions of (3):
(12a)u=4a1g+a5h2−2a12+2a52f2ga1+2ha5f,(12b)v=a3a1g−a5h2a1a1g+a5h−a12+a5224a1g+a5h2+2fa12+a522+Nf2,where
(13)f=g2+h2+a9,g=a1x−a12+a52+a5a6a1y+a3t+a4,h=a5x+a6y+a3a5a1t+a8,N=−2ga12+a52+a5a6a1+2ha6a1g+a5h.The solutions ofu via (12a) are singular which can be seen in Figure 1(a); Figure 1(b) is the corresponding density plot. The solutions of v via (12b) can be seen in Figure 1(c) and Figure 1(d) is the corresponding density plot.Figure 1
Profiles of the solutionu via (12a) and profiles of the solution v via (12b) with a1=1, a3=1.5, a4=0, a5=2, a6=4, a8=0, and a9=1. (a) and (c) are the corresponding 3-dimensional plots; (b) and (d) are the corresponding density plots.
(a)
t=0
(b)
t=0
(c)
t=0
(d)
t=0


## 4. Rational Solutions Obtained by Adding an Exponential Term to the Quadratic Function

Two types of solutions are obtained in the following.Case I.
(14)a1=0,a2=a2,a3=0,a4=a4,a5=a5,a6=−a5,a7=−k3a5k2,a8=a8,a9=a9,α=0,β=β,k1=−k2,k2=k2,k3=k3,k4=k4,which should satisfy the constraint conditions
(15)k2≠0,a9>0,β>0,to ensure that the corresponding f is positive and well defined. By substituting (14) into (8), the function f reads
(16)f=a2y+a42+a5x−a6y−a3a5tk2+a82+βexpk1x+k2y+k3t+k4+a9.
Using equation (7a) and equation (7b), the solutions for (3) can be obtained:
(17a)u=2a5h+k2βexpξ2−2a52+k22βexpξf2k2βexpξ+2ha5f,(17b)v=−k3k2βexpξ+2a5hk2k2βexpξ+4ha5−B24A2+A2ga2−A+Bff2,where
(18)f=g2+h2+βexpξ+a9,g=a2y+a4,h=a5x−a5y+k3a5k2t+a8,ξ=k2x−k2y−k3t−k4,A=k2βexpξ+2ha5,B=k22βexpξ+2a52.Case II.
(19)a1=0,a2=a2,a3=0,a4=a4,a5=a5,a6=−a5,a7=−k3a5k2,a8=a8,a9=a9,α=α,β=0,k1=k1,k2=k2,k3=k3,k4=k4,a1=a3=0,a6=−a5,a7=−k3a5k2,β=0,which should satisfy the constraint conditions
(20)k2≠0,a9>0,α>0,to guarantee that the corresponding f is positive and well defined. Substituting (19) into (8), we then have the function f:
(21)f=a2y+a42+a5x−a6y−a3a5tk2+a82+αexpk1x+k2y+k3t+k4+a9.
The solutions of this case are similar to the solutions of equation (17a) and equation (17b).


## 5. Rational Solutions Obtained by Adding Two Exponential Terms to the Quadratic Function

In this section, in order to find interaction solutions of equation (3), we further add two exponential terms to the quadratic function. Three sets of solutions are obtained as follows.Case I.
(22)a1=a1,a2=−a12+a52+a5a6a1,a3=a3,a4=a4,a5=a5,a6=a6,a7=a3a5a1,a8=a8,a9=a9,α=α,β=β,k1=0,k2=k2,k3=0,k4=k4,where
(23)a1≠0,a9>0,α>0,β>0,to ensure that the corresponding f is positive and well defined. By substituting (22) into (8), we can get
(24)f=a1x−a12+a52+a5a6a1y+a3t+a42+a5x+a6y+a3a5ta1a82+a9,where ai, kj, 1≤i≤9, and 1≤j≤4 are all real parameters to be determined. Then, the solutions of (3) will be obtained:
(25a)u=−4a1h+a5g2−a12+a52f2fa1h+a5g,(25b)v=a1a3h+a3a5g2a1a1h+a5g−a1+a528a1a1h+a5g+Df2,with
(26)f=h2+g2+a9+αexpξ+βexp−ξ,h=a1x−a12+a52+a5a6a1y+a3t+a4,g=a5x+a6y+a3a5ta1+a8,ξ=k2y+k4,
D=2a6g−2ha12+a52+a5a6/a1+k2αexpξ−βexp−ξ+2a1h+a5g+2fa12+a52.The solution for (25a) can be seen in Figure 2(a); Figures 2(b)–2(d) are the corresponding density plots with different times.Case II.
(27)a1=a1,a2=−a12+a52+a5a6a1,a3=a1k3k1,a4=a4,a5=a5,a6=a6,a7=a5k3k1,a8=a8,a9=a9,α=α,β=β,k1=k1,k2=−k1,k3=k3,k4=k4,where
(28)a1≠0,k1≠0,a9>0,α>0,β>0,to guarantee that the corresponding f is positive and well defined. We substitute (27) into (8); hence, we can reinstall function f as the following formula:
(29)f=a1x−a12+a52+a5a6a1y+a1k3tk1+a42+a5x+a6y+a5k3ta1+a82+a9,where ai, kj, 1≤i≤9, and 1≤j≤4 are all real parameters to be determined. Then, by substituting equation (27) into (29), and with equation (3), we have
(30a)u=−2B2−Af2Bf,(30b)v=k32k1+A24B2+B2ha6−2ha12+a52+a5a6/a1+k1αexpξ−βexp−ξ−Aff2,where
(31)f=h2+g2+αexpξ+βexp−ξ+a9,h=a1x−a12+a52+a5a6a1y+a1k3tk1+a4,g=a5x+a6y+a5k3ta1+a82+a9,ξ=k1x−k1y+k3t+k4,A=2a12h+a52g+k12αexpξ+βexp−ξ,B=2a1h+a5+k1αexpξ−βexp−ξ.Figure 2
Profiles of the solutionu via (25a) with a1=1, a3=1.5, a4=0, a5=2, a6=4, a8=0, a9=1, α=1, β=1, k2=1, and k4=0. (a) 3-dimensional plot; (b–d) the corresponding density plots with different times.
(a)
t=0
(b)
t=−20
(c)
t=0
(d)
t=20The solution for (30a) as a rational solution can be seen in Figure 3(a) and in Figures 3(b)–3(d) the corresponding density plots with different times. The solution for (30b) as a rational solution which is singular can be seen in Figure 4(a) and in Figures 4(b)–4(d) the corresponding density plots with different times.Case III.
(32)a1=a1,a2=−a12k32+a6a7k1k3+a72k12a1k32,a3=a1k3k1,a4=a4,a5=a7k1k3,a6=a6,a7=a7,a8=a4a7k1k3a1,a9=a9,α=α,β=β,k1=k1,k2=−k1,k3=k3,k4=k4,with
(33)k3a1≠0,a9>0,α>0,β>0,to guarantee that the corresponding f is positive and well defined. We substitute (32) into (8), then function f reads
(34)f=a1x−a12k32+a6a7k1k3+a72k12a1k32y+a1k3tk1+a42+a7k1k3x+a6y+a7t+a4a7k1k3a12+a9+αexpk1x−k1y+k3t+k4+βexp−k1x+k1y−k3t−k4,where ai, kj, 1≤i≤9, and 1≤j≤4 are all real parameters to be determined. Then, the rational solution of system (3) can be obtained again:
(35a)u=−Jf−2D22Df,(35b)v=2a1k3h/k1+2a7g+N2D+J24D2+DMh+2a7g+N+2a1M−2a7k1/k3+k12αexpξ+βexp−ξf2,where
(36)f=h2+g2+αexpξ+βexp−ξ+a9,h=a1x−a12+k32+a6a7k1k3+a72k12a1k32y+a1k3tk1+a4,g=a7k1k3x+a6y+a7t+a4a7k1k3a1,ξ=k1x−k1y+k3t+k4,D=2a1h+2a7k1k3g+k1αexpξ−βexp−ξ,J=2a12+2a7k1k32+k12αexpξ+βexp−ξ,M=2a12k32+a6a7k1k3+a72k12a1k32,N=k3αexpξ−βexp−ξ.Figure 3
Profiles of the solutionu via (30a) with a1=1, a4=0, a5=2, a6=4, a8=0, a9=1, α=1, β=1, k1=1, k2=1.5, k3=2, and k4=0. (a) 3-dimensional plot with the time t=0; (b–d) the corresponding density plots with different times.
(a)
t=0
(b)
t=−20
(c)
t=0
(d)
t=20Figure 4
Profiles of the solutionv via (30b) with a1=1, a4=0, a5=2, a6=4, a8=0, a9=1, α=1, β=1, k1=1, k2=1.5, k3=2, and k4=0. (a) 3-dimensional plot with the time t=0; (b–d) the corresponding density plots with different times.
(a)
t=0
(b)
t=−10
(c)
t=0
(d)
t=10The solution for (35a) is rational and can be seen in Figure 5(a) and Figure 5(b) is the corresponding density plot; the solution for (35b) is rational and can be seen in Figure 5(c) and Figure 5(d) is the corresponding density plot.Figure 5
Profiles of the solutionu via (35a) and profiles of the solution v via (35b) with a1=1, a4=0, a6=2, a7=1, a8=0, a9=1, α=1, β=1, k1=1, k3=2, and k4=0. (a) and (c) are the 3-dimensional plots; (b) and (d) are the corresponding density plots.
(a)
t=0
(b)
t=0
(c)
t=0
(d)
t=0


## 6. Conclusions

Construction of rational solutions for NLEEs is an important part in nonlinear science. In this study, based on the trilinear-linear equation (6), we obtain three classes of rational solutions ((12a), (12b), (17a), (17b), (25a), (25b), (30a), (30b), (35a), and (35b)) for the (2+1)-dimensional modified KdV-CBS equation (3). Three-dimensional plots and the corresponding density plots of the three classes of rational solutions are given, respectively, in Figures 1–7 in this paper.Figure 6
Profiles of the solutionu via (17a) with a2=1, a4=0, a5=1, a8=0, a9=1, beta=1, k2=2, k3=1, and k4=0. (a) 3-dimensional plot with the time t=0; (b–d) the corresponding density plots with different times.
(a)
t=0
(b)
t=−20
(c)
t=0
(d)
t=20Figure 7
Profiles of the solutionv via (17b) with a2=1, a4=0, a5=1, a8=0, a9=1, beta=1, k2=2, k3=1, and k4=0. (a) 3-dimensional plot with the time t=0; (b–d) the corresponding density plots with different times.
(a)
t=0
(b)
t=−20
(c)
t=0
(d)
t=20



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