# A New Nonlinear Equation with Lump-Soliton, Lump-Periodic, and Lump-Periodic-Soliton Solutions

Authors: Bo Ren; Ji Lin; Zhi-Mei Lou
Journal: Complexity
 (2019)

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## Abstract

An extended (2+1)-dimensional Calogero-Bogoyavlenskii-Schiff-like equation is proposed by using the generalized bilinear operators based on a prime numberp=3. By combining multiexponential functions with a quadratic function, the interaction between lumps and multikink soliton is generated. In the meanwhile, the interaction of lump with periodic waves and the interaction among lumps, periodic waves, and multikink soliton can be obtained by introducing the ansätz forms. The dynamics of these interaction solutions are analyzed graphically by selecting appropriate parameters.


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## Body

## 1. Introduction

Finding exact solutions of nonlinear partial differential equations (NPDEs) plays an important topic in study of various fields of sciences, especially in physics [1–7]. During the past decades, many powerful methods have been presented for finding exact solutions of the NPDEs in mathematical physics [8–12]. Among these exact solutions, lump wave which was first discovered by Manakov et al. decays in all directions of the spaces [13]. The study of lump waves has a widespread application in many fields, such as the oceanographic engineering [14], capillary flow [15], atmosphere [16], nonlinear optics [17], and finance [18]. Many high-dimensional NPDEs admit the lump solutions such as the Ishimori-I equation [19], the Kadomtsev-Petviashvili (KP) equation [20], the generalized KP equation [21], the variable-coefficient nonlinear Schrödinger equation [22], the KP-Boussinesq equation [23], and the Sawada-Kotera equation [24]. In order to describe complex situations, some interaction solutions are constructed by the localization procedure related with the nonlocal symmetry [25–27] and the homoclinic breather limit approach [28, 29]. By mixing a positive quadratic function with the exponential/trigonometric functions, interactive lump-kink [30–35], lump-soliton [36–38], and lump-periodic waves [39] of the NPDEs have been presented. Besides, new NPDEs can be constructed by extending Hirota’s bilinear form, which involve different prime numbers and possess potential applications [40]. It is interesting to study lump solutions of these kinds of the NPDEs associated with the generalized bilinear operators [41–43]. In this paper, the main purpose is to construct an NPDE by using generalized bilinear operators which possesses lump solutions. In the meanwhile, interaction between a lump and multikink soliton, and interaction of lump with periodic waves, and interaction among a lump, periodic waves, and multikink soliton are obtained by introducing the ansätz forms. The interaction among a lump, periodic waves, and multikink soliton to the NPDEs has not been reported in other studies.We want to construct an NPDE associated with the generalized bilinear operators by an extended (2+1)-dimensional Calogero-Bogoyavlenskii-Schiff (eCBS) equation. The eCBS equation reads(1)uxt+uxxxy+6uxxuy+6uxuxy+δ1uxy+δ2uxx+δ3uxxxx+12uxuxx=0,where δ1,δ2, and δ3 are arbitrary constants. By the dependent variable transformation(2)u=ln⁡ϕxx,the eCBS equation (1) becomes the following bilinear form:(3)DxDt+Dx3Dy+δ1DxDy+δ2Dx2+δ3Dx4=0,where Dx,Dy, and Dt represent the Hirota bilinear operators [44]:(4)DxlDynDtmf·g=∂∂x-∂∂x′l∂∂y-∂∂y′n∂∂t-∂∂t′mfx,y,t·gx′,y′,t′x=x′,y=y′,t=t′.The eCBS equation (1) will be transformed to a (2+1)-dimensional CBS equation with δ1=δ2=δ3=0. The CBS equation describes the (2+1)-dimensional interaction of Riemann wave propagation with the long-wave propagation [45].Based on the generalized bilinear operators [40], a generalized bilinear form on a prime number p=3 has(5)D3,xD3,t+D3,x3D3,y+δ1D3,xD3,y+δ2D3,x2+δ3D3,x4=2ϕϕxt-2ϕtϕx+6ϕxxϕxy+2δ1ϕϕxy-ϕxϕy+2δ2ϕϕxx-ϕx2+6δ3ϕxx2=0,with the D-operators in (5) defined by(6)Dp,xaDp,ybDp,tcf·g=∂∂x+αp∂∂x′a∂∂y+αp∂∂y′b∂∂t+αp∂∂t′cfx,y,t·gx′,y′,t′x=x′,y=y′,t=t′=∑i=0a∑j=0b∑k=0caibjckαpiαpjαpk∂a-i∂xa-i∂i∂x′i∂b-j∂yb-j∂j∂y′j∂c-k∂tc-k∂k∂t′kx=x′,y=y′,t=t′,while(7)αps=-1rps,s=rpsmod⁡p.Employing a dependent variable transformation (8)(8)u=ϕxϕ,w=ϕyϕ,the generalized bilinear form of (5) can be cast as an eCBS like (eCBSL) equation in the following form:(9)ut+3uxuy+3u2uy+3uuxw+3u3w+δ1uy+δ2ux+3δ3u2+2ux2=0,uy-wx=0.The purpose of the present paper is to study some novel interaction solutions of the above eCBSL equation (9).This paper is organized as follows. In Section2, the interaction between lump waves and multikink soliton is obtained by adding additional exponential functions to a quadratic function. In Section 3, the interacted lumps with periodic waves and interaction among lumps, periodic waves, and multikink soliton are derived by using the ansätz forms. Some concluding remarks will be devoted to the last section.


## 2. Interaction between Lumps and Multisoliton

### 2.1. Interaction between a Lump and One-Kink Soliton

In order to obtain interaction between lump waves and one soliton of (9), we assume that an interaction solution is determined by a sum of a quadratic function and an exponential function:(10)ϕ=g2+h2+a9+b1exp⁡k1x+p1y+q1t,g=a1x+a2y+a3t+a4,h=a5x+a6y+a7t+a8,f1=k1x+p1y+q1t,with ai(i=1,2,⋯,9),b1,k1,p1, and q1 being thirteen undetermined real parameters. By substituting (10) into (5) and vanishing the different powers of x,y,t, and exp, we present two families of lump solutions for the parameters.Case I.
(11) a 1 = 0 , a 3 = - δ 1 a 2 , a 6 = - δ 3 a 5 , a 7 = - δ 1 a 6 - δ 2 a 5 , p 1 = - δ 3 k 1 , q 1 = - δ 1 p 1 - δ 2 k 1 .The solution of u and w will be localized in all directions of spaces, while the parameters satisfy the conditions(12)a9>0,b1>0.By substituting (10) into (8) and combining the parameters relations (11), the first class of the interacted lump with one-kink soliton of (9) can be obtained:(13)u=2a5h+k1p1exp⁡fϕ,w=2a2g+2a6h+k1q1exp⁡fϕ,where(14)ϕ=g2+h2+a9+k1exp⁡f,g=a2y-δ1a2t+a4,h=a5x-δ3a5y-δ1a6+δ2a5t+a8,f1=k1x-δ3k1y-δ1p1+δ2k1t.Case II.
(15) a 2 = - δ 3 a 1 - a 5 a 1 a 6 + δ 3 a 5 , a 3 = - δ 1 a 2 - δ 2 a 1 , a 7 = - δ 1 a 6 - δ 2 a 5 , p 1 = - δ 3 k 1 , q 1 = k 1 δ 1 δ 3 - δ 2 ,with(16)a1≠0,a9>0,b1>0,the solution of u and w will be localized in all directions of spaces. By substituting (10) into (8) and combining the relations (15), we get the second class of interacted lump with one-kink soliton of (9):(17)u=2a1g+2a5h+b1k1exp⁡f1ϕ,w=2a2g+2a6h+b1p1exp⁡f1ϕ,where(18)ϕ=g2+h2+a9+b1exp⁡f1,g=a1x-δ3a1+a5a1a6+δ3a5y-δ1a2+δ2a1t+a4,h=a5x+a6y-δ1a6+δ2a5t+a8,f1=k1x-δ3k1y+k1δ1δ3-δ2t.Remark. The interaction between lump waves and one-kink soliton will become lump waves with b1=0.


### 2.2. Interaction between a Lump and Two-Kink Soliton

For interaction between lump waves and two-kink soliton, we use the form of a quadratic function with two exponential functions:(19)ϕ=g2+h2+a9+b1exp⁡f1+b2exp⁡f2,g=a1x+a2y+a3t+a4,h=a5x+a6y+a7t+a8,fi=kix+piy+qit,i=1,2.By inserting (19) into (5) and gathering the coefficients of x,y,t, and exp, the parameters are yielded by solving the algebraic equations(20)a3=-δ1a2-δ2a1,a6=-δ3a5-a1a5a2+δ3a1,a7=a5δ1δ3-δ2+δ1a1a5a2+δ3a1,pi=-δ3ki,qi=kiδ1δ3-δ2.The parameters need to satisfy the conditions(21)a5≠0,a9>0,b1>0,b2>0,then, the solution of u and w is localized in all directions of spaces. By substituting (19) into (8) and combining the parameters relations (20), we get the following interaction solution of (9):(22)u=2a1g+2a5h+b1k1exp⁡f1+b2k2exp⁡f2ϕ,w=2a2g+2a6h+b1p1exp⁡f1+b2p2exp⁡f2ϕ,where(23)ϕ=g2+h2+a9+b1exp⁡f1+b2exp⁡f2,g=a1x+a2y-δ1a2+δ2a1t+a4,h=a5x-δ3a5+a1a5a2+δ3a1y+a5δ1δ3-δ2+δ1a1a5a2+δ3a1t+a8,fi=kix-δ3kiy+kiδ1δ3-δ2t.The ansätz form of (19) is a combination of a quadratic function and two exponential functions. This kind of interaction solution is thus called a special rogue wave [46]. Figure 1 shows this kind of a special rogue wave by selecting the parameters as a1=-1, a2=-3, a4=-1/2, a5=1,a8=1, a9=1; b1=1, b2=1; k1=1/2, k2=-1/3; and δ1=2, δ2=1, δ3=1. The spatial structure of a lump wave caught in two-kink soliton is described in Figure 1(a) at t=0. Figure 1(b) displays the corresponding density plot of the lump-kink wave. Figure 1(c) shows the wave along x-axis with y=0 and different values t=-15,t=0,t=15.Figure 1
Profile of the interaction between a lump wave and two-kink soliton in (22). (a) 3-dimensional plot with t=0, (b) the corresponding density plot, and (c) the curve plot with various values of time.
(a) (b) (c)


## 2.1. Interaction between a Lump and One-Kink Soliton

In order to obtain interaction between lump waves and one soliton of (9), we assume that an interaction solution is determined by a sum of a quadratic function and an exponential function:(10)ϕ=g2+h2+a9+b1exp⁡k1x+p1y+q1t,g=a1x+a2y+a3t+a4,h=a5x+a6y+a7t+a8,f1=k1x+p1y+q1t,with ai(i=1,2,⋯,9),b1,k1,p1, and q1 being thirteen undetermined real parameters. By substituting (10) into (5) and vanishing the different powers of x,y,t, and exp, we present two families of lump solutions for the parameters.Case I.
(11) a 1 = 0 , a 3 = - δ 1 a 2 , a 6 = - δ 3 a 5 , a 7 = - δ 1 a 6 - δ 2 a 5 , p 1 = - δ 3 k 1 , q 1 = - δ 1 p 1 - δ 2 k 1 .The solution of u and w will be localized in all directions of spaces, while the parameters satisfy the conditions(12)a9>0,b1>0.By substituting (10) into (8) and combining the parameters relations (11), the first class of the interacted lump with one-kink soliton of (9) can be obtained:(13)u=2a5h+k1p1exp⁡fϕ,w=2a2g+2a6h+k1q1exp⁡fϕ,where(14)ϕ=g2+h2+a9+k1exp⁡f,g=a2y-δ1a2t+a4,h=a5x-δ3a5y-δ1a6+δ2a5t+a8,f1=k1x-δ3k1y-δ1p1+δ2k1t.Case II.
(15) a 2 = - δ 3 a 1 - a 5 a 1 a 6 + δ 3 a 5 , a 3 = - δ 1 a 2 - δ 2 a 1 , a 7 = - δ 1 a 6 - δ 2 a 5 , p 1 = - δ 3 k 1 , q 1 = k 1 δ 1 δ 3 - δ 2 ,with(16)a1≠0,a9>0,b1>0,the solution of u and w will be localized in all directions of spaces. By substituting (10) into (8) and combining the relations (15), we get the second class of interacted lump with one-kink soliton of (9):(17)u=2a1g+2a5h+b1k1exp⁡f1ϕ,w=2a2g+2a6h+b1p1exp⁡f1ϕ,where(18)ϕ=g2+h2+a9+b1exp⁡f1,g=a1x-δ3a1+a5a1a6+δ3a5y-δ1a2+δ2a1t+a4,h=a5x+a6y-δ1a6+δ2a5t+a8,f1=k1x-δ3k1y+k1δ1δ3-δ2t.Remark. The interaction between lump waves and one-kink soliton will become lump waves with b1=0.


## 2.2. Interaction between a Lump and Two-Kink Soliton

For interaction between lump waves and two-kink soliton, we use the form of a quadratic function with two exponential functions:(19)ϕ=g2+h2+a9+b1exp⁡f1+b2exp⁡f2,g=a1x+a2y+a3t+a4,h=a5x+a6y+a7t+a8,fi=kix+piy+qit,i=1,2.By inserting (19) into (5) and gathering the coefficients of x,y,t, and exp, the parameters are yielded by solving the algebraic equations(20)a3=-δ1a2-δ2a1,a6=-δ3a5-a1a5a2+δ3a1,a7=a5δ1δ3-δ2+δ1a1a5a2+δ3a1,pi=-δ3ki,qi=kiδ1δ3-δ2.The parameters need to satisfy the conditions(21)a5≠0,a9>0,b1>0,b2>0,then, the solution of u and w is localized in all directions of spaces. By substituting (19) into (8) and combining the parameters relations (20), we get the following interaction solution of (9):(22)u=2a1g+2a5h+b1k1exp⁡f1+b2k2exp⁡f2ϕ,w=2a2g+2a6h+b1p1exp⁡f1+b2p2exp⁡f2ϕ,where(23)ϕ=g2+h2+a9+b1exp⁡f1+b2exp⁡f2,g=a1x+a2y-δ1a2+δ2a1t+a4,h=a5x-δ3a5+a1a5a2+δ3a1y+a5δ1δ3-δ2+δ1a1a5a2+δ3a1t+a8,fi=kix-δ3kiy+kiδ1δ3-δ2t.The ansätz form of (19) is a combination of a quadratic function and two exponential functions. This kind of interaction solution is thus called a special rogue wave [46]. Figure 1 shows this kind of a special rogue wave by selecting the parameters as a1=-1, a2=-3, a4=-1/2, a5=1,a8=1, a9=1; b1=1, b2=1; k1=1/2, k2=-1/3; and δ1=2, δ2=1, δ3=1. The spatial structure of a lump wave caught in two-kink soliton is described in Figure 1(a) at t=0. Figure 1(b) displays the corresponding density plot of the lump-kink wave. Figure 1(c) shows the wave along x-axis with y=0 and different values t=-15,t=0,t=15.Figure 1
Profile of the interaction between a lump wave and two-kink soliton in (22). (a) 3-dimensional plot with t=0, (b) the corresponding density plot, and (c) the curve plot with various values of time.
(a) (b) (c)


## 3. Interaction among Lumps, Periodic Waves, and Multisoliton

### 3.1. Interaction between a Lump and Periodic Waves

Similar to the lump-soliton solutions, the interaction between lump waves and triangular periodic waves for (5) can be assumed as follows:(24)ϕ=g2+h2+a9+b1cos⁡f1,g=a1x+a2y+a3t+a4,h=a5x+a6y+a7t+a8,f1=k1x+p1y+q1t.By substituting (24) into (5) and collecting the coefficients of x,y,t,sin, and cos, the parameters are determined:(25)a3=-δ1a2-δ2a1,a7=-δ1a6-δ2a5,p1=-δ3k1,q1=k1δ1δ3-δ2,δ3=-a1a2+a5a6a12+a52.The parameters satisfy(26)a1a5≠0,a9>b1,then, the solution of u and w is localized in all directions of spaces. By substituting (24) into (8) and combining the parameters relations (25), we get the following interaction solution of (9):(27)u=2a1g+2a5h-b1k1sin⁡f1ϕ,w=2a2g+2a6h-b2p1sin⁡f1ϕ,where(28)ϕ=g2+h2+a9+b1cos⁡f1,g=a1x+a2y-δ1a2+δ2a1t+a4,h=a5x+a6y-δ1a6+δ2a5t+a8,f1=k1x-δ3k1y+k1δ1δ3-δ2t.The 3D plot, density plot, and curve plot for the interaction between a lump and triangular periodic waves of (27) are depicted in Figure 2. The specific parameters are chosen as a1=1/2, a2=-1, a4=1/2, a5=-1/4, a6=-3, a8=1, a9=20; b1=15; k1=1/3; and δ1=1, δ2=2. The interaction between a lump and periodic waves of u is presented in Figure 2(a) at t=0. Figure 2(b) displays the corresponding density plot of the lump-periodic wave. Figure 2(c) represents the curve plot with y=0 and various values of time t=-3,t=0,t=3.Figure 2
Profile of the interaction between a lump wave and triangular periodic waves in (27). (a) 3-dimensional plot with t=0, (b) the corresponding density plot, and (c) the curve plot with various values of time.
(a) (b) (c)


### 3.2. Interaction among a Lump, Periodic Waves, and One-Kink Soliton

The interaction among lump waves, triangular periodic waves, and one-kink soliton for (5) can be assumed as follows:(29)ϕ=g2+h2+a9+b1cos⁡f1+b2exp⁡f2,g=a1x+a2y+a3t+a4,h=a5x+a6y+a7t+a8,fi=kix+piy+qit,i=1,2.Substituting (29) into (5) leads to(30)pi=-δ3ki,qi=kiδ1δ3-δ2,δ1=-a1a7-a3a5a1a6-a2a5,δ2=-δ1δ3-B,δ3=-a1a2+a5a6a12+a52B=a1a3+a5a7a12+a52.The parameters satisfy(31)a1a5≠0,a1a6-a2a5≠0,a9>b1,b2>0,then, the solution of u and w is localized in all directions of spaces. By substituting (29) into (8) and combining the parameters relations (30), we get the following interaction solution of (9):(32)u=2a1g+2a5h-b1k1sin⁡f1+b2p2exp⁡f2ϕ,w=2a2g+2a6h-b2p1sin⁡f1+b2q2exp⁡f2ϕ,where(33)ϕ=g2+h2+a9+b1cos⁡f1+b2exp⁡f2,g=a1x+a2y+a3t+a4,h=a5x+a6y+a7t+a8,fi=kix-δ3kiy+kiδ1δ3-δ2t.The parameters are selected as a1=1/2, a2=-1, a3=-1, a4=1/2, a5=-1/4, a6=-3, a7=1, a8=1, a9=30; b1=25, b2=1/4; and k1=1/3, k2=-1/4. The interaction among a lump, triangular periodic waves, and one-kink soliton of u is presented in Figure 3(a) at t=0. Figure 3(b) displays the corresponding density plot of this interaction solution. Figure 3(c) represents the wave along x-axis with y=0 and different time t=-13,t=0,t=13.Figure 3
Profile of the interaction among a lump wave, triangular periodic waves, and one-kink soliton in (32). (a) 3-dimensional plot with t=0, (b) the corresponding density plot, and (c) the curve plot at various values of time.
(a) (b) (c)


### 3.3. Interaction among a Lump, Periodic Waves, and Two-Kink Soliton

The interaction among lump waves, triangular periodic waves, and two-kink soliton for (5) can be assumed as follows:(34)ϕ=g2+h2+a9+b1cos⁡f1+b2exp⁡f2+b3exp⁡f3,g=a1x+a2y+a3t+a4,h=a5x+a6y+a7t+a8,fi=kix+piy+qit,i=1,2,3.By substituting (34) into (5) and collecting the coefficients of x,y,t,sin,cos, and exp, the parameters are stated by solving the algebraic equations(35)pi=-δ3ki,qi=kiδ1δ3-δ2,δ1=-a1a7-a3a5a1a6-a2a5,δ2=-δ1δ3-B,δ3=-a1a2+a5a6a12+a52B=a1a3+a5a7a12+a52.The parameters satisfy(36)a1a5≠0,a1a6-a2a5≠0,a9>b1,b2>0,b3>0,then, the solution of u and w is localized in all directions of spaces. Substituting (34) into (8) and combining the parameters relations (35), we get the following interaction solution of (9):(37)u=2a1g+2a5h-b1k1sin⁡f1+b2p2exp⁡f2+b3p3exp⁡f3ϕ,w=2a2g+2a6h-b2p1sin⁡f1+b2q2exp⁡f2+b2q3exp⁡f3ϕ,where(38)ϕ=g2+h2+a9+b1cos⁡f1+b2exp⁡f2+b3exp⁡f3,g=a1x+a2y+a3t+a4,h=a5x+a6y+a7t+a8,fi=kix-δ3kiy+kiδ1δ3-δ2t.The parameters are selected as a1=1/2, a2=-2, a3=3, a4=1/2, a5=-1/4, a6=-3, a7=1, a8=1, a9=55; b1=52, b2=1/2, b3=5; and k1=-1/3, k2=1/4, k3=-1/5. The interaction among a lump wave, triangular periodic waves, and two-kink soliton of u is presented in Figure 4(a) at t=0. Figure 4(b) displays the corresponding density plot of this interaction solution. Figure 4(c) represents the wave along x-axis with y=0 and different time t=-8,t=0,t=10. This kind of interaction solution has not been given in other studies.Figure 4
Profile of the interaction among a lump, triangular periodic waves, and two-kink soliton in (37). (a) 3-dimensional plot with t=0, (b) the corresponding density plot, and (c) the curve plot at various values of time.
(a) (b) (c)


## 3.1. Interaction between a Lump and Periodic Waves

Similar to the lump-soliton solutions, the interaction between lump waves and triangular periodic waves for (5) can be assumed as follows:(24)ϕ=g2+h2+a9+b1cos⁡f1,g=a1x+a2y+a3t+a4,h=a5x+a6y+a7t+a8,f1=k1x+p1y+q1t.By substituting (24) into (5) and collecting the coefficients of x,y,t,sin, and cos, the parameters are determined:(25)a3=-δ1a2-δ2a1,a7=-δ1a6-δ2a5,p1=-δ3k1,q1=k1δ1δ3-δ2,δ3=-a1a2+a5a6a12+a52.The parameters satisfy(26)a1a5≠0,a9>b1,then, the solution of u and w is localized in all directions of spaces. By substituting (24) into (8) and combining the parameters relations (25), we get the following interaction solution of (9):(27)u=2a1g+2a5h-b1k1sin⁡f1ϕ,w=2a2g+2a6h-b2p1sin⁡f1ϕ,where(28)ϕ=g2+h2+a9+b1cos⁡f1,g=a1x+a2y-δ1a2+δ2a1t+a4,h=a5x+a6y-δ1a6+δ2a5t+a8,f1=k1x-δ3k1y+k1δ1δ3-δ2t.The 3D plot, density plot, and curve plot for the interaction between a lump and triangular periodic waves of (27) are depicted in Figure 2. The specific parameters are chosen as a1=1/2, a2=-1, a4=1/2, a5=-1/4, a6=-3, a8=1, a9=20; b1=15; k1=1/3; and δ1=1, δ2=2. The interaction between a lump and periodic waves of u is presented in Figure 2(a) at t=0. Figure 2(b) displays the corresponding density plot of the lump-periodic wave. Figure 2(c) represents the curve plot with y=0 and various values of time t=-3,t=0,t=3.Figure 2
Profile of the interaction between a lump wave and triangular periodic waves in (27). (a) 3-dimensional plot with t=0, (b) the corresponding density plot, and (c) the curve plot with various values of time.
(a) (b) (c)


## 3.2. Interaction among a Lump, Periodic Waves, and One-Kink Soliton

The interaction among lump waves, triangular periodic waves, and one-kink soliton for (5) can be assumed as follows:(29)ϕ=g2+h2+a9+b1cos⁡f1+b2exp⁡f2,g=a1x+a2y+a3t+a4,h=a5x+a6y+a7t+a8,fi=kix+piy+qit,i=1,2.Substituting (29) into (5) leads to(30)pi=-δ3ki,qi=kiδ1δ3-δ2,δ1=-a1a7-a3a5a1a6-a2a5,δ2=-δ1δ3-B,δ3=-a1a2+a5a6a12+a52B=a1a3+a5a7a12+a52.The parameters satisfy(31)a1a5≠0,a1a6-a2a5≠0,a9>b1,b2>0,then, the solution of u and w is localized in all directions of spaces. By substituting (29) into (8) and combining the parameters relations (30), we get the following interaction solution of (9):(32)u=2a1g+2a5h-b1k1sin⁡f1+b2p2exp⁡f2ϕ,w=2a2g+2a6h-b2p1sin⁡f1+b2q2exp⁡f2ϕ,where(33)ϕ=g2+h2+a9+b1cos⁡f1+b2exp⁡f2,g=a1x+a2y+a3t+a4,h=a5x+a6y+a7t+a8,fi=kix-δ3kiy+kiδ1δ3-δ2t.The parameters are selected as a1=1/2, a2=-1, a3=-1, a4=1/2, a5=-1/4, a6=-3, a7=1, a8=1, a9=30; b1=25, b2=1/4; and k1=1/3, k2=-1/4. The interaction among a lump, triangular periodic waves, and one-kink soliton of u is presented in Figure 3(a) at t=0. Figure 3(b) displays the corresponding density plot of this interaction solution. Figure 3(c) represents the wave along x-axis with y=0 and different time t=-13,t=0,t=13.Figure 3
Profile of the interaction among a lump wave, triangular periodic waves, and one-kink soliton in (32). (a) 3-dimensional plot with t=0, (b) the corresponding density plot, and (c) the curve plot at various values of time.
(a) (b) (c)


## 3.3. Interaction among a Lump, Periodic Waves, and Two-Kink Soliton

The interaction among lump waves, triangular periodic waves, and two-kink soliton for (5) can be assumed as follows:(34)ϕ=g2+h2+a9+b1cos⁡f1+b2exp⁡f2+b3exp⁡f3,g=a1x+a2y+a3t+a4,h=a5x+a6y+a7t+a8,fi=kix+piy+qit,i=1,2,3.By substituting (34) into (5) and collecting the coefficients of x,y,t,sin,cos, and exp, the parameters are stated by solving the algebraic equations(35)pi=-δ3ki,qi=kiδ1δ3-δ2,δ1=-a1a7-a3a5a1a6-a2a5,δ2=-δ1δ3-B,δ3=-a1a2+a5a6a12+a52B=a1a3+a5a7a12+a52.The parameters satisfy(36)a1a5≠0,a1a6-a2a5≠0,a9>b1,b2>0,b3>0,then, the solution of u and w is localized in all directions of spaces. Substituting (34) into (8) and combining the parameters relations (35), we get the following interaction solution of (9):(37)u=2a1g+2a5h-b1k1sin⁡f1+b2p2exp⁡f2+b3p3exp⁡f3ϕ,w=2a2g+2a6h-b2p1sin⁡f1+b2q2exp⁡f2+b2q3exp⁡f3ϕ,where(38)ϕ=g2+h2+a9+b1cos⁡f1+b2exp⁡f2+b3exp⁡f3,g=a1x+a2y+a3t+a4,h=a5x+a6y+a7t+a8,fi=kix-δ3kiy+kiδ1δ3-δ2t.The parameters are selected as a1=1/2, a2=-2, a3=3, a4=1/2, a5=-1/4, a6=-3, a7=1, a8=1, a9=55; b1=52, b2=1/2, b3=5; and k1=-1/3, k2=1/4, k3=-1/5. The interaction among a lump wave, triangular periodic waves, and two-kink soliton of u is presented in Figure 4(a) at t=0. Figure 4(b) displays the corresponding density plot of this interaction solution. Figure 4(c) represents the wave along x-axis with y=0 and different time t=-8,t=0,t=10. This kind of interaction solution has not been given in other studies.Figure 4
Profile of the interaction among a lump, triangular periodic waves, and two-kink soliton in (37). (a) 3-dimensional plot with t=0, (b) the corresponding density plot, and (c) the curve plot at various values of time.
(a) (b) (c)


## 4. Conclusion

In summary, some novel interaction solutions of a (2+1)-dimensional eCBSL equation (9) were considered by using the ansätz forms. By combining a quadratic function with multiexponential functions, interaction between lumps and multikink soliton were found. The interacted lumps with periodic waves and interaction among lumps, periodic waves, and multikink soliton were derived by introducing additional trigonometric and exponential functions. The phenomena of interaction between a lump and two-kink soliton, interaction of lump with periodic waves, and interaction among a lump, periodic waves, and multikink soliton for the eCBSL equation were generated as illustrative examples. In particular, the behavior of interaction among a lump, periodic waves, and two-kink soliton has not been found in other studies. The results presented in this paper can be applied to other high-dimensional NPDEs. It may be helpful for the theories of the associated Hirota bilinear equation. In the meanwhile, the multisoliton, one-breather, and rogue wave can be obtained by the Darboux transformation [47]. These kinds of solutions for (9) are worthy of further study.



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