# A Study on Lump Solutions to a Generalized Hirota-Satsuma-Ito Equation in (2+1)-Dimensions

Authors: Wen-Xiu Ma; Jie Li; Chaudry Masood Khalique
Journal: Complexity
 (2018)

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## Abstract

The Hirota-Satsuma-Ito equation in (2+1)-dimensions passes the three-soliton test. This paper aims to generalize this equation to a new one which still has abundant interesting solution structures. Based on the Hirota bilinear formulation, a symbolic computation with a new class of Hirota-Satsuma-Ito type equations involving general second-order derivative terms is conducted to require having lump solutions. Explicit expressions for lump solutions are successfully presented in terms of coefficients in a generalized Hirota-Satsuma-Ito equation. Three-dimensional plots and contour plots of a special presented lump solution are made to shed light on the characteristic of the resulting lump solutions.


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## Body

## 1. Introduction

In the classical theory of differential equations, the main question is to study the existence of solutions to given equations, including many nonlinear equations describing real-world problems. Cauchy problems are to deal with the existence, uniqueness, and stability of solutions satisfying initial data. Laplace’s method is developed for solving Cauchy problems for linear ordinary differential equations and the Fourier transform method for linear partial differential equations. In modern soliton theory, the isomonodromic transform method and the inverse scattering transform method have been designed to solve Cauchy problems for nonlinear ordinary and partial differential equations [1–3]. Explicitly solvable differential equations include various constant-coefficient and linear differential equations, but it is extremely difficult to compute exact solutions to variable-coefficient or nonlinear equations.However, the Hirota bilinear method provides us with a working approach to soliton solutions, historically found for nonlinear integrable equations [4, 5]. Soliton solutions are analytic ones exponentially localized in all directions in space and time. Let a polynomial P determine a Hirota bilinear differential equation:(1)PDx,Dy,Dtf·f=0,in (2+1)-dimensions, where Dx,Dy, and Dt are Hirota’s bilinear derivatives. The corresponding partial differential equation with a dependent variable u is determined usually by one of the logarithmic transformations: u=2(ln⁡f)x and u=2(ln⁡f)xx. Within the Hirota bilinear formulation, soliton solutions are expressed through(2)f=∑μ=0,1exp⁡∑i=1Nμiξi+∑i<jμiμjaij,where ∑μ=0,1 means the sum over all possibilities for μ1,μ2,…,μN taking either 0 or 1, and the wave variables and the phase shifts are defined by(3)ξi=kix+liy-ωit+ξi,0,1≤i≤N,and(4)eaij=-Pki-kj,li-lj,ωj-ωiPki+kj,li+lj,ωj+ωi,1≤i<j≤N,in which ki,li, and ωi, 1≤i≤N, satisfy the corresponding dispersion relation and ξi,0, 1≤i≤N, are arbitrary phase shifts.Lump solutions are a class of analytic rational solutions which are localized in all directions in space, originated from solving integrable equations in (2+1)-dimensions (see, e.g., [6–8]). Taking long wave limits of N-soliton solutions can generate special lumps [9]. Many integrable equations in (2+1)-dimensions exhibit the remarkable richness of lump solutions (see, e.g., [6, 7]). Such equations contain the KPI equation [10], whose special lump solutions have been derived from N-soliton solutions [11], the three-dimensional three-wave resonant interaction [12], the BKP equation [13, 14], the Davey-Stewartson equation II [9], the Ishimori-I equation [15], and the KP equation with a self-consistent source [16]. An important step in the process of getting lumps is to determine positive quadratic function solutions to bilinear equations [6]. Then, through the mentioned logarithmic transformations, we present lump solutions to nonlinear equations (see, e.g., [6] for the case of Hirota bilinear equations and [7] for the case of generalized bilinear equations).In this paper, we would like to generalize the Hirota-Satsuma-Ito (HSI) equation in (2+1)-dimensions to a new one which still has abundant interesting solution structures. Hirota bilinear forms are the starting point for our discussion (see, e.g., [6, 7, 17, 18] for other equations). We will consider a general class of HSI type equations while keeping the existence of lump solutions. A general such generalized HSI equation in (2+1)-dimensions and its lump solutions will be determined through symbolic computations with Maple. For a special presented lump solution, three-dimensional plots and contour plots will be made via the Maple plot tool, to shed light on the characteristic of the presented lump solutions. A few concluding remarks will be given in the last section.


## 2. Lump Solutions

It is known that the Hirota-Satsuma shallow water wave equation [4], (5)ut=uxxt+3uut-3uxvt-ux,vx=-u,has a bilinear form, (6)DtDx3-DtDx-Dx2f·f=0,under the logarithmic transformations u=2(ln⁡f)xx and v=2(ln⁡f)x. An integrable (2+1)-dimensional extension of the Hirota-Satsuma equation reads (7)3uxutx+uxxxt+uyt+uxx=0,which passes the Hirota three-soliton test [19], and has a bilinear form under the logarithmic transformation u=2(ln⁡f)x:(8)Dx3Dt+DyDt+Dx2f·f=0.Equation (7) is called the Hirota-Satsuma-Ito (HSI) equation in (2+1)-dimensions [19]. We would like to add three terms to generalize the abovementioned HSI equation to a new one which still possesses abundant interesting solution structures: (9)Pu=3uxutx+uxxxt+δ1uyt+δ2uxx+δ3uxy+δ4uxt+δ5uyy=0.This generalized HSI equation has a bilinear form under the logarithmic transformation u=2(lnf)x:(10)Bf=Dx3Dt+δ1DyDt+δ2Dx2+δ3DxDy+δ4DxDt+δ5Dy2f·f=0.Precisely, under u=2(ln⁡f)x, we have the relation P(u)=(B(f)/f2)x. In what follows, we would like to determine lump solutions to the gHSI equation in (2+1)-dimensions (9), through symbolic computations with Maple.We start to search for positive quadratic solutions to the gHSI bilinear equation (10) to generate lump solutions to the gHSI equation (9): (11)f=a1x+a2y+a3t+a42+a5x+a6y+a7t+a82+a9.Plugging this function into the gHSI bilinear equation (10) generates a system of nonlinear algebraic equations on the parameters ai, 1≤i≤9. Conducting direct symbolic computation to solve this system gives a set of solutions for the parameters where(12)a3=-b1a1δ4+a2δ12+a5δ4+a6δ12,a7=-b2a1δ4+a2δ12+a5δ4+a6δ12,a9=3a12+a52b3a1a6-a2a52δ12δ2-δ1δ3δ4+δ42δ5,and all other ai’s are arbitrary. The involved three constants are defined as follows:(13)b1=a12a2+2a1a5a6-a2a52δ2+a1a22+a62δ3+a2a22+a62δ5δ1+a1a12+a52δ2+a2a12+a52δ3+a1a22+2a2a5a6-a1a62δ5δ4,b2=-a12a6+2a1a2a5+a52a6δ2+a5a22+a62δ3+a6a22+a62δ5δ1+a5a12+a52δ2+a6a12+a52δ3+-a22a5+2a1a2a6+a5a62δ5δ4,b3=a12+a52a1a2+a5a6δ1δ2+δ3δ4+a12+a52a22+a62δ1δ3+a12+a522δ2δ4+a22+a62a1a2+a5a6δ1δ5+a1a2+a5a62-a1a6-a2a52δ4δ5.Those formulas in (12) and (13) were obtained under a simplification process with Maple.From (12), we can easily see that it is sufficient to guarantee f>0 if we require (14)δ12δ2-δ1δ3δ4+δ42δ5b3>0,and, thus, the function f defined by (12) and (13) under the abovementioned condition and (15)a1a6-a2a5≠0leads to lump solutions (16)u=2ln⁡fx=2fxfto the gHSI equation in (2+1)-dimensions (9).When one takes(17)δ1=1,δ2=1,δ3=δ4=δ5=0,one obtains the original HSI equation in (2+1)-dimensions (7), and the function f by (12) and (13) presents a class of lump solutions to the HSI equation (7):(18)u=2ln⁡fx,f=a1x+a2y+a3t+a42+a5x+a6y+a7t+a82+a9,where(19)a3=-a12a2+2a1a5a6-a2a52a22+a62,a7=a12a6-2a1a2a5-a52a6a22+a62,a9=3a12+a522a1a2+a5a6a1a6-a2a52,and all other ai’s are arbitrary. Solving the abovementioned parameter solutions on a3 and a7 for a2 and a6 and substituting the resulting expressions for a2 and a6 into the formula for a9 in (19), we get(20)a2=-a12a3+2a1a5a7-a3a52a32+a72,a6=a12a7-2a1a3a5-a52a7a32+a72,a9=-3a12+a52a32+a72a1a3+a5a7a1a7-a3a52.It is easy to see that(21)a1a6-a2a5=a12+a52a1a7-a3a5a32+a72,and, thus, the conditions of (22)a1a3+a5a7<0,a1a7-a3a5≠0guarantee that (16) with (11) and (20) will present lump solutions to the HSI equation (7) [20].Particularly taking(23)δ1=1,δ2=1,δ3=-1,δ4=1,δ5=-1,we obtain a special gHSI equation as follows: (24)uxxxt+3uxutx+uyt+uxx-uxy+uxt-uyy=0,which has a Hirota bilinear form (25)Dx3Dt+DyDt+Dx2-DxDy+DxDt-Dy2f·f=0,under the logarithmic transformation (16). Associated with (26)a1=1,a2=-2,a4=2,a5=2,a6=-3,a8=-5,(16) with (11) and (22) present the lump solution to the special gHSI equation (24): (27)u=210x-16y-t-16x-2y-3/2t+22+2x-3y+1/2t-52+15.Three three-dimensional plots and contour plots of this lump solution are made via Maple plot tools, to shed light on the characteristic of the presented lump solutions, in Figure 1.Figure 1
Profiles ofu when t=0,3,6: 3D plots (top) and contour plots (bottom).All the exact solutions generated above add valuable insights into the existing theories on soliton solutions and dromion-type solutions, developed through various powerful solution techniques including the Hirota perturbation approach, the Riemann-Hilbert approach, the Wronskian technique, symmetry reductions, and symmetry constraints (see, e.g., [21–31]).


## 3. Concluding Remarks

We have studied a generalized (2+1)-dimensional Hirota-Satsuma-Ito (HSI) equation to explore different equations which possess lump solutions, through symbolic computations with Maple. The results enrich the theory of lumps and solitons, providing a new example of (2+1)-dimensional nonlinear equations, which possess beautiful lump structures. Three-dimensional plots and contour plots of a specially chosen lump solution were made by using the plot tool in Maple.Many nonlinear equations possess lump solutions, which include (2+1)-dimensional generalized KP, BKP, KP-Boussinesq, Sawada-Kotera, and Bogoyavlensky-Konopelchenko equations [32–36]. Some recent studies also demonstrate the strikingly high richness of lump solutions to linear partial differential equations [37] and nonlinear partial differential equations in (2+1)-dimensions (see, e.g., [38–41]) and (3+1)-dimensions (see, e.g., [42–48]). Diversity of lump solutions supplements exact solutions generated from different kinds of combinations (see, e.g., [49–52]) and can yield various Lie-Bäcklund symmetries, which can be used to determine conservation laws by symmetries and adjoint symmetries [53–55]. Moreover, diverse interaction solutions [35] have been exhibited for many integrable equations in (2+1)-dimensions, including lump-soliton interaction solutions (see, e.g., [56–58]) and lump-kink interaction solutions (see, e.g., [59–62]).We finally remark that we could add one more term to the gHSI equation (9) to formulate a more generalized HSI bilinear equation,(28)Dx3Dt+δ1DyDt+δ2Dx2+δ3DxDy+δ4DxDt+δ5Dy2+δ6Dt2f·f=0,where δi, 1≤i≤6, are all constants, but we failed to drive any lump solution to the corresponding nonlinear equation on u=2(ln⁡f)x. The first term in the abovementioned bilinear equation is crucial in determining lump solutions but the last term brings the difficulty to work out lump solutions. There is no hint on how to solve any big system of resulting nonlinear algebraic equations. Nevertheless, some general considerations on the existence of lumps have been made for the Hirota bilinear case [6] and the generalized bilinear cases [7].



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