# New Interaction Solutions for a (2 + 1)-Dimensional Vakhnenko Equation

Authors: Qing Meng; Bin He
Journal: Complexity
 (2020)

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## Abstract

In this paper, we focus on the interaction solutions of a (2 + 1)-dimensional Vakhnenko equation. By using Hirota’s transformation combined with the three-wave method and with symbolic computation, some interaction solutions which include interaction solutions between exponential and trigonometric functions and interaction solutions between exponential and trigonometric and hyperbolic functions are presented.


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## Body

## 1. Introduction

It is well known that nonlinear partial differential equations (NPDEs) and their solutions play a significant role in interpreting many important phenomena in nonlinear sciences. A variety of powerful methods are developed for finding the exact solutions of NPDEs, such as Hirota’s method [1, 2], simplified Hirota’s method [3, 4], the Lie symmetry analysis method [5, 6], the simplest equation method [5, 6], the invariant subspace method [7], and the nonlinear steepest descent method [8]. Very recently, the lump and interaction solutions [9–11] have attracted the attention of many scholars because of lump’s applications in nonlinear optics, physics, oceanography, etc, and the interaction solutions are valuable in analyzing the nonlinear dynamics of waves in shallow water and can be used for forecasting the appearance of rogue waves [12, 13].In order to describe high-frequent wave propagations in a relaxing medium, the Vakhnenko equation [14], the generalized Vakhnenko equation [15], and the modified generalized Vakhnenko equation [16] were presented. Many different kinds of valuable results have been obtained [17–23]. Vakhnenko and Parkes [17] obtained the two-loop soliton solution for the Vakhnenko equation using Hirota’s bilinear method. Vakhnenko et al. [18] derived a Bäcklund transformation both in the bilinear and in ordinary form for the generalized Vakhnenko equation and found the exact N-soliton solution via the inverse scattering method. Wazwaz [19] derived multiple soliton solutions and multiple singular soliton solutions for the Vakhnenko equation, the generalized Vakhnenko equation, and the modified generalized Vakhnenko equation by the simplified form of the bilinear method. Wang and Chen [20] investigated the integrability of the modified generalized Vakhnenko equation and presented the quasiperiodic solution by applying Hirota direct method and Riemann theta function. Brunelli and Sakovich [21] obtained a bi-Hamiltonian formulation for the Vakhnenko equation via the Miura-type transformations. The dynamical behaviours and exact traveling wave solutions of the modified generalized Vakhnenko equation were studied in [22]. Hashemi et al. [23] determined the Lie symmetry group, the corresponding symmetry reductions, and invariant solutions of the modified generalized Vakhnenko equation by the Li group analysis method, and so on.In 2008, Victor et al. [24] initially derived a (2 + 1)-dimensional Vakhnenko equation(1)∂∂x+∂∂y∂∂t+u∂∂x+∂∂yu+u=0,which is to model high-frequent wave perturbations in relaxing high-rate active barothropic media and involves two spatial variables x,y and a temporal variable t. With the aid of symbolic computation and Hirota’s method, Victor et al. [24] unearthed some typical solitary wave solutions to equation (1) and depicted single- and multivalued solutions depending on the dissipative parameter. Morrison and Parkes [15, 16] showed that ux,y,t=UT1,T2,X under the following transformation:(2)x=T1+∫−∞XUT1,T2,X′dX′+x0,y=T2+∫−∞XUT1,T2,X′dX′+y0,t=X,where x0 and y0 are two constants and T1,T2, and X are three independent variables. Li et al. [25] introduced a new function W defined by(3)WT1,T2,X=∫−∞XUT1,T2,X′dX′.Then,Wx=U and equation (1) becomes(4)WXXT1+WXXT2+WXWT1+WXWT2+WX=0.Moreover, Li et al. [25] indicated that if W=WT1,T2,X is an explicit solution of equation (4), then(5)ux,y,t=WXT1,T2,X,x=T1+WT1,T2,X+x0,y=T2+WT1,T2,X+y0,t=X,is an implicit solution of equation (1). Some 1-loop, 2-loop, and 3-loop soliton solutions were presented applying the improved Hirota method, and the traveling and interaction processes for the N-loop soliton solutions are explored in [25].In this paper, we investigate interaction solutions of equation (1) via Hirota’s transformation [26] and three-wave methods [27–30].


## 2. Interaction Solutions of Equation (1)

Under the transformationW=6lnfX, equation (4) becomes the Hirota bilinear equation:(6)DX3DT1+DX3DT2+DX2f⋅f=0,where f=fT1,T2,X is a real function, and the Hirota bilinear differential operator DxmDtn was defined by [26](7)DxmDtnf⋅g=∂∂x−∂∂x′m∂∂t−∂∂t′nfx,tgx′,t′x,t=x′,t′.In fact, equation (6) is equivalent to one special case of a generalized Bogoyavlensky–Konopelchenko equation [31] upon combining T1 and T2 as t.Consider equation (6) as well as the following novel test function:(8)f=b1eξ1+b2e−ξ1+b3sinξ2+b4cosξ3+b5sinhξ4,which is a combination of f=k1eξ1+k2e−ξ1+k3sinξ2+k4cosξ3 [28] and f=k1eξ1+k2e−ξ1+k3sinξ2+k4sinhξ3 [29], where ξi=ωiX+ciT1+kiT2,ωi,ci,ki,bj1≤i≤4,1≤j≤5 are unknown constants to be determined later.Substituting (8) into (6), we can obtain an algebraic system of ωi,ci,ki,bj1≤i≤4,1≤j≤5.Case 1.
Choosingb5=0 and with the aid of Maple, we present some solutions of the algebraic system as follows:(9)b1=0,b2=b2,b3=b3,b4=b4,b5=0,ω1=3ω2,c1=3−4ω2k14ω2,k1=k1,ω2=ω2,c2=1−4ω2k24ω2,k2=k2,ω3=ω2,c3=1−4ω2k34ω2,k3=k3,where ω2≠0.(10)b1=b42ω32ω12−3ω324b2ω12ω32−3ω12,b2=b2,b3=0,b4=b4,b5=0,ω1=ω1,c1=−ω1+k1ω12+ω32ω12+ω32,k1=k1,ω2=ω2,c2=c2,k2=k2,ω3=ω3,c3=ω3−k3ω12+ω32ω12+ω32,k3=k3,where b2≠0,ω1≠0,andω3≠±3ω1.(11)b1=b1,b2=0,b3=b3,b4=b4,b5=0,ω1=3ω2,c1=3+4ω2k14ω2,k1=k1,ω2=ω2,c2=1−4ω2k24ω2,k2=k2,ω3=−ω2,c3=−1+4ω2k34ω2,k3=k3,where ω2≠0.
The parameters in set (9) generate the following class of interaction solutions to equation (6) as(12)f1=b2e−3ω2X+3−4ω2k1/4ω2T1+k1T2+b3sinω2X+1−4ω2k24ω2T1+k2T2+b4cosω2X+1−4ω2k34ω2T1+k3T2,which further leads to furnish a class of interaction solutions to equation (1) as follows:(13)ux,y,t=6G1f1−H12f12,x=T1+6H1f1+x0,y=T2+6H1f1+y0,t=X,where b2,b3,b4,ω2≠0,k1,k2,k3,x0,y0 are arbitrary constants and(14)G1=ω223b2e−3ω2X+3−4ω2k1/4ω2T1+k1T2−b3sinω2X+1−4ω2k24ω2T1+k2T2−b4cosω2X+1−4ω2k34ω2T1+k3T2,H1=ω2−b23e−3ω2X+3−4ω2k1/4ω2T1+k1T2+b3cosω2X+1−4ω2k24ω2T1+k2T2−b4sinω2X+1−4ω2k34ω2T1+k3T2.
The parameters in set (10) generate the following class of interaction solutions to equation (6) as(15)f2=b42ω32ω12−3ω324b2ω12ω32−3ω12eω1X−ω1+k1ω12+ω32/ω12+ω32T1+k1T2+b2e−ω1X−ω1+k1ω12+ω32/ω12+ω32T1+k1T2+b4cosω3X+ω3−k3ω12+ω32ω12+ω32T1+k3T2,which further leads to furnish a class of interaction solutions to equation (1) as follows:(16)ux,y,t=6G2f2−H22f22,x=T1+6H2f2+x0,y=T2+6H2f2+y0,t=X,where b2≠0,b4,ω1≠0,ω3≠±3ω1,k1,k3,x0,y0 are arbitrary constants and(17)G2=b42ω32ω12−3ω324b2ω32−3ω12eω1X−ω1+k1ω12+ω32/ω12+ω32T1+k1T2+b2ω12e−ω1X−ω1+k1ω12+ω32ω12+ω32T1+k1T2−b4ω32cosω3X+ω3−k3ω12+ω32ω12+ω32T1+k3T2,H2=b42ω32ω12−3ω324b2ω1ω32−3ω12eω1X−ω1+k1ω12+ω32/ω12+ω32T1+k1T2−b2ω1e−ω1X−ω1+k1ω12+ω32/ω12+ω32T1+k1T2−b4ω3sinω3X+ω3−k3ω12+ω32ω12+ω32T1+k3T2.
Settingb2=−b4ω3/2ω1ω12−3ω32/3ω12−ω32, solution (15) becomes(18)f¯2=b4ω3ω1ω12−3ω323ω12−ω32sinhω1X−ω1+k1ω12+ω32ω12+ω32T1+k1T2+b4cosω3X+ω3−k3ω12+ω32ω12+ω32T1+k3T2,which further leads to furnish a class of interaction solutions to equation (1) as follows:(19)ux,y,t=6G¯2f¯2−H¯22f¯22,x=T1+6H¯2f¯2+x0,y=T2+6H¯2f¯2+y0,t=X,where b4,ω1≠0,ω3ω12−3ω323ω12−ω32>0,k1,k3,x0,y0 are arbitrary constants and(20)G¯2=b4ω3ω1ω12−3ω323ω12−ω32sinhω1X−ω1+k1ω12+ω32ω12+ω32T1+k1T2−ω3cosω3X+ω3−k3ω12+ω32ω12+ω32T1+k3T2,H¯2=b4ω3ω12−3ω323ω12−ω32coshω1X−ω1+k1ω12+ω32ω12+ω32T1+k1T2−sinω3X+ω3−k3ω12+ω32ω12+ω32T1+k3T2.
The parameters in set (11) generate the following class of interaction solutions to equation (6) as(21)f3=b1e3ω2X+3+4ω2k1/4ω2T1+k1T2++b3sinω2X+1−4ω2k24ω2T1+k2T2+b4cosω2X+1+4ω2k34ω2T1−k3T2,which further leads to furnish a class of interaction solutions to equation (1) as follows:(22)ux,y,t=6G3f3−H32f32,x=T1+6H3f3+x0,y=T2+6H3f3+y0,t=X,where b1,b3,b4,ω2≠0,k1,k2,k3,x0,y0 are arbitrary constants and(23)G3=ω223b1e3ω2X+3+4ω2k1/4ω2T1+k1T2−b3sinω2X+1−4ω2k24ω2T1+k2T2−b4cosω2X+1+4ω2k34ω2T1−k3T2,H3=ω2b13e3ω2X+3+4ω2k1/4ω2T1+k1T2+b3cosω2X+1−4ω2k24ω2T1+k2T2−b4sinω2X+1+4ω2k34ω2T1−k3T2.Case 2.
. Choosingb4=0 and with the aid of Maple, we present some solutions of the algebraic system as follows:(24)b1=0,b2=0,b3=b5ω4ω2ω22−3ω423ω22−ω42,b4=0,b5=b5,ω1=ω1,c1=c1,k1=k1,ω2=ω2,k2=ω2−c2ω22+ω42ω22+ω42,c2=c2,ω4=ω4,c4=−ω4+k4ω22+ω42ω22+ω42,k4=k4,where ω2≠0,ω22−3ω423ω22−ω42>0.(25)b1=b1,b2=b2,b3=0,b4=0,b5=b5,ω1=ω1,c1=−1+4ω1k14ω1,k1=k1,ω2=ω2,c2=c2,k2=k2,ω4=−ω1,c4=1−4ω1k44ω1,k4=k4,where ω1≠0.
The parameters in set (24) generate the following class of interaction solutions to equation (6) as(26)f4=b5ω4ω2ω22−3ω423ω22−ω42sinω2X+c2T1+ω2−c2ω22+ω42ω22+ω42T2+b5sinhω4X−ω4+k4ω22+ω42ω22+ω42T1+k4T2,which further leads to furnish a class of interaction solutions to equation (1) as follows:(27)ux,y,t=6G4f4−H42f42,x=T1+6H4f4+x0,y=T2+6H4f4+y0,t=X,where b5,ω2≠0,ω4ω22−3ω423ω22−ω42>0,c2,k4,x0,y0 are arbitrary constants and(28)G4=b5ω4−ω2ω22−3ω423ω22−ω42sinω2X+c2T1+ω2−c2ω22+ω42ω22+ω42T2+ω4sinhω4X−ω4+k4ω22+ω42ω22+ω42T1+k4T2,H4=b5ω4ω22−3ω423ω22−ω42cosω2X+c2T1+ω2−c2ω22+ω42ω22+ω42T2+coshω4X−ω4+k4ω22+ω42ω22+ω42T1+k4T2.
The parameters in set (25) generate the following class of interaction solutions to equation (6) as(29)f5=b1eω1X−1+4ω1k1/4ω1T1+k1T2+b2e−ω1X−1+4ω1k1/4ω1T1+k1T2−b5sinhω1X−1−4ω1k44ω1T1−k4T2,which further leads to furnish a class of interaction solutions to equation (1) as follows:(30)ux,y,t=6G5f5−H52f52,x=T1+6H5f5+x0,y=T2+6H5f5+y0,t=X,where b1,b2,b5,ω1≠0,k1,k4,x0,y0 are arbitrary constants and(31)G5=ω12b1eω1X−1+4ω1k1/4ω1T1+k1T2+b2e−ω1X−1+4ω1k1/4ω1T1+k1T2−b5sinhω1X−1−4ω1k44ω1T1−k4T2,H5=ω1b1eω1X−1+4ω1k1/4ω1T1+k1T2−b2e−ω1X−1+4ω1k1/4ω1T1+k1T2−b5coshω1X−1−4ω1k44ω1T1−k4T2.
Settingb2=−b1, solution (29) becomes(32)f¯5=2b1sinhω1X−1+4ω1k14ω1T1+k1T2−b5sinhω1X−1−4ω1k44ω1T1−k4T2,which further leads to furnish a class of interaction solutions to equation (1) as follows:(33)ux,y,t=6G¯5f¯5−H¯52f¯52,x=T1+6H¯5f¯5+x0,y=T2+6H¯5f¯5+y0,t=X,where b1,b5,ω1≠0,k1,k4,x0,y0 are arbitrary constants and(34)G¯5=ω122b1sinhω1X−1+4ω1k14ω1T1+k1T2−b5sinhω1X−1−4ω1k44ω1T1−k4T2,H¯5=ω12b1coshω1X−1+4ω1k14ω1T1+k1T2−b5coshω1X−1−4ω1k44ω1T1−k4T2.
Settingb2=b1, solution (29) becomes(35)f˜5=2b1coshω1X−1+4ω1k14ω1T1+k1T2−b5sinhω1X−1−4ω1k44ω1T1−k4T2,which further leads to furnish a class of interaction solutions to equation (1) as follows:(36)ux,y,t=6G˜5f˜5−H˜52f˜52,x=T1+6H˜5f˜5+x0,y=T2+6H˜5f˜5+y0,t=X,where b1,b5,ω1≠0,k1,k4,x0,y0 are arbitrary constants and(37)G˜5=ω122b1coshω1X−1+4ω1k14ω1T1+k1T2−b5sinhω1X−1−4ω1k44ω1T1−k4T2,H˜5=ω12b1sinhω1X−1+4ω1k14ω1T1+k1T2−b5coshω1X−1−4ω1k44ω1T1−k4T2.


## 3. Conclusions

In this paper, we explore a (2 + 1)-dimensional Vakhnenko equation by Hirota’s transformation combined with the three-wave method. With symbolic computation, five types of interaction solutions are obtained. It should be pointed out that some similar types of interaction solutions as that of the solution presented in this paper are not shown here for brevity. Future investigation may be applied to the method in this paper to search for lump solutions and interaction solutions between lump and other functions for the equation.



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