# Lump Solutions and Interaction Solutions for the Dimensionally Reduced Nonlinear Evolution Equation

Authors: Baoyong Guo; Huanhe Dong; Yong Fang
Journal: Complexity
 (2019)

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## Abstract

In this paper, by means of the Hirota bilinear method, a dimensionally reduced nonlinear evolution equation is investigated. Through its bilinear form, lump solutions are obtained. We construct interaction solutions between lump solutions and one soliton solution by choosing quadratic functions and exponential function. Interaction solutions with the combinations of exponential functions and sine function are also given. Meanwhile, the figures of these solutions are plotted. The dynamical characteristics and properties of obtained solutions are discussed, respectively. The results show that the corresponding physical quantities and properties of nonlinear waves are associated with the values of the parameters.


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## Body

## 1. Introduction

Nonlinear evolution equations (NLEEs) are becoming more and more important in modern science. People have paid much more attention than ever before on the study of NLEEs. They have significant applications in many subject fields, especially in nonlinear science, for instance, mathematical physics, nonlinear mechanics, particle physics, marine science, atmospheric science, and automation. This trend stems from the fact that NLEEs can explain a lot of natural phenomena; for example, in mathematical physics, many physical quantities of nonlinear waves can be described by the parameters of equations and the solutions of equations can also well interpret the propagation of water waves. In order to obtain the solutions of NLEEs, researchers have put forward many methods, including the Hirota direct method [1], Painlevé analysis method [2], inverse scattering transformation (IST) [3, 4], Riemann–Hilbert method [5–7], Lie symmetry method, and so on [8–12]. Among these methods, the Hirota direct method is so prompt and effective. Based on this approach, researchers have obtained many different kinds of solutions, including lump solutions [13–15], breather solutions [16–18], rogue wave solutions [19–21], interaction solutions, and so on [22–26]. With the help of the Riemann–Hilbert method, people also acquire soliton solutions of integrable hierarchies and coupled systems [27–31]. By taking the long wave limit of soliton solutions, rational solutions of NLEEs are presented [32]. Meanwhile, some difference equations also possess lump solutions and interaction solutions, such as the Toda lattice equation [33]. In recent years, researchers generalize the existing NLEEs to new ones and obtain the corresponding lump solutions [34, 35]. These results are good supplements to the theory of exact solutions for NLEEs.In this paper, we focus on the (3 + 1)-dimensional nonlinear evolution equation [36]; its form is(1)3uxz−2ut−2uux+2uxxxy+2ux∫xuyx=0,where u=ux,y,z,t,∫x is the integral with respect to x.Letz=y; we derive dimensionally reduced situation of equation (1) as follows:(2)3uxy−2ut−2uux+2uxxxy+2ux∫xuyx=0.Equation (2) has wide applications in different areas, for example, mathematical physics, ocean science, engineering, and others. It could describe propagation of shallow water wave in nonlinear dispersive channel. So, it is very important to find the exact solution for this dimensionally reduced nonlinear evolution equation.The structure of this paper is as follows. In Section2, we present the Hirota bilinear form and lump solutions of equation (2). In Section 3, we obtain interaction solutions with the combination of lump solutions and one soliton solution. In Section 4, we acquire interaction solutions with the combination of two exponential functions and one sine function. In Section 5, some conclusions are given.


## 2. Hirota Bilinear Form and Lump Solutions

### 2.1. Hirota Bilinear Form of the Dimensionally Reduced Nonlinear Evolution Equation

By means of variable transformationu=−6lnFxx, we transform equation (2) into the Hirota bilinear form [37, 38]:(3)2DyDt+2Dx3Dy−3DxDyF⋅F=0,where D is the Hirota bilinear differential operator which is defined as follows:(4)DxmDynDtlF⋅G=∂∂x−∂∂x′m∂∂y−∂∂y′n∂∂t−∂∂t′lFx,y,tGx′,y′,t′x′=x,y′=y,t′=t.


### 2.2. Lump Solutions Consisting of Two Quadratic Functions

In order to obtain lump solutions of equation (2), we take function F in the following form:(5)F=f12+f22+α9,f1=α1x+α2y+α3t+α4,f2=α5x+α6y+α7t+α8,where αi1≤i≤9 are real parameters to be determined.Substituting equation (5) into equation (3) and considering the coefficients of all powers of the variables to be 0, with the help of Maple, we have the following relations:(6)α3=32α1,α6=−α1α2α5,α7=32α5,where α1,α2,α4,α5,α8,andα9 are free parameters.For convenience, we let the parameters to be(7)α1=1,α2=1,α4=0,α5=1,α8=0,α9=1.Consequently, the lump solutions of equation (2) can be written as(8)u=−6lnFxx=−62α12+α52α1x+α2y+3/2α1t+α42+α5x−α1α2/α5y+3/2α5t+α82+α9−2α1α1x+α2y+3/2α1t+α4+α5α5x−α1α2/α5y+3/2α5t+α82⋅α1x+α2y+3/2α1t+α42+α5x−α1α2/α5y+3/2α5t+α82+α92=−64x+y+3/2t2+4x−y+3/2t2−4x+6t2+4⋅x+y+3/2t2+x−y+2/3t2+12−1.From equations (5) and (6), we can find that the lump solution is a kind of rational solution. Based on parameter relations (6), we know F is analytical if and only if α5≠0 and α9>0. Let us make a simple dynamical characteristic analysis. Figure 1(a) shows the lump solutions of equation (2). When α4=0,α8=0, and t=0, the lump wave will center at the origin; here we take α1=1,α2=1,α4=0,α5=1,α8=0,α9=1,andt=1. The lump wave is located in arbitrary directions in the space, so it is a kind of localized wave actually. From equation (8), we can find that if x⟶±∞ or y⟶±∞, then f1 and f2⟶∞, u⟶0. The lump wave has one peak and two valleys, and valleys are symmetrically distributed on both sides of the peak. According to the extreme value principle of multivariate functions, by the calculation, we obtain that the lump wave’s minimum point (extreme value point) is −3α12+α52t+2α1α4+α5α8/2α12+α52,α5α1α8−α4α5/α2α12+α52=−3/2,0 and the corresponding amplitude is −12α12+α52/α9=−24. It propagates along the line y=0 with the velocity of −3/2. This mode of motion is uniform linear motion in physics.Figure 1
(a) Plots of lump solutions (8) with α1=1,α2=1,α4=0,α5=1,α8=0,α9=1,andt=1; (b) contour map in the (x, y) plane.
(a)
(b)


### 2.3. Lump Solutions Consisting of Three Quadratic Functions

We will seek for the lump solutions consisting of three quadratic functions. This situation has rarely been seen in existing literatures [13–15, 35]. In order to do it, we take function F in the following form:(9)F=f12+f22+f32+α13,f1=α1x+α2y+α3t+α4,f2=α5x+α6y+α7t+α8,f3=α9x+α10y+α11t+α12,where αi1≤i≤13 are real parameters to be determined.Substituting equation (9) into equation (3) and considering the coefficients of all powers of the variables to be 0, with the help of Maple, we have the following relations:(10)α1=−132α6α7+3α9α10α2,α3=−122α6α7+3α9α10α2,α5=23α7,α11=32α9,where α2,α4,α6,α7,α8,α9,α10,α12,andα13 are free parameters.For convenience, we let the parameters to be(11)α2=1,α4=0,α6=1,α7=1,α8=0,α9=1,α10=1,α12=0,α13=1.Consequently, the lump solutions consisting of three quadratic functions of equation (2) can be written as(12)u=−6lnFxx=−676/9−5/3x+y−5/2t2+76/92/3x+y+t2+76/9x+y+3/2t2−76/9x+38/3t2+76/9−5/3x+y−5/2t2+2/3x+y+t2+x+y+3/2t2+12.Based on parameter relations (10), we know F is analytical if and only if α2≠0 and α13>0. Figure 2(a) shows the lump solutions consisting of three quadratic functions of equation (2). Similar to the previous dynamical characteristic analysis, when α4=0,α8=0,α12=0, and t=0, the lump wave will center at the origin; here we take α2=1,α4=0,α6=1,α7=1,α8=0,α9=1,α10=1,α12=0,α13=1,andt=1. From equation (12), we can find that if x⟶±∞ or y⟶±∞, then f1, f2, and f3⟶∞, u⟶0. The lump wave consisting of three quadratic functions also has one peak and two valleys, and valleys are symmetrically distributed on both sides of the peak. Similarly, we obtain that the lump wave’s minimum point (extreme value point) is −34α72α22+α62+9α92α22+α102+12α6α7α9α10t+4α2α7α2α8−α4α6+6α2α9α2α12−α4α10/24α72α22+α62+9α92α22+α102+12α6α7α9α10,−α2α4+α6α8+α10α12/α22+α62+α102=−3/2,0 and the corresponding amplitude is −42α6α7+3α9α102/3α22+16/3α72+12α92/α13=−152/3. It propagates along the line y=0 with the velocity of −3/2. This mode of motion is also uniform linear motion in physics.Figure 2
(a) Plots of lump solutions (12) with α2=1,α4=0,α6=1,α7=1,α8=0,α9=1,α10=1,α12=0,α13=1,andt=1; (b) contour map in the (x, y) plane.
(a)
(b)Compared with the previous results (Section2.2), when α4=0,α8=0,andα12=0, we realize that lump solutions (8) and lump solutions (12) have the same minimum point (extreme value point); however, they have different extreme values (amplitudes) at the same minimum point. These two kinds of lump wave have the same mode of motion.


## 2.1. Hirota Bilinear Form of the Dimensionally Reduced Nonlinear Evolution Equation

By means of variable transformationu=−6lnFxx, we transform equation (2) into the Hirota bilinear form [37, 38]:(3)2DyDt+2Dx3Dy−3DxDyF⋅F=0,where D is the Hirota bilinear differential operator which is defined as follows:(4)DxmDynDtlF⋅G=∂∂x−∂∂x′m∂∂y−∂∂y′n∂∂t−∂∂t′lFx,y,tGx′,y′,t′x′=x,y′=y,t′=t.


## 2.2. Lump Solutions Consisting of Two Quadratic Functions

In order to obtain lump solutions of equation (2), we take function F in the following form:(5)F=f12+f22+α9,f1=α1x+α2y+α3t+α4,f2=α5x+α6y+α7t+α8,where αi1≤i≤9 are real parameters to be determined.Substituting equation (5) into equation (3) and considering the coefficients of all powers of the variables to be 0, with the help of Maple, we have the following relations:(6)α3=32α1,α6=−α1α2α5,α7=32α5,where α1,α2,α4,α5,α8,andα9 are free parameters.For convenience, we let the parameters to be(7)α1=1,α2=1,α4=0,α5=1,α8=0,α9=1.Consequently, the lump solutions of equation (2) can be written as(8)u=−6lnFxx=−62α12+α52α1x+α2y+3/2α1t+α42+α5x−α1α2/α5y+3/2α5t+α82+α9−2α1α1x+α2y+3/2α1t+α4+α5α5x−α1α2/α5y+3/2α5t+α82⋅α1x+α2y+3/2α1t+α42+α5x−α1α2/α5y+3/2α5t+α82+α92=−64x+y+3/2t2+4x−y+3/2t2−4x+6t2+4⋅x+y+3/2t2+x−y+2/3t2+12−1.From equations (5) and (6), we can find that the lump solution is a kind of rational solution. Based on parameter relations (6), we know F is analytical if and only if α5≠0 and α9>0. Let us make a simple dynamical characteristic analysis. Figure 1(a) shows the lump solutions of equation (2). When α4=0,α8=0, and t=0, the lump wave will center at the origin; here we take α1=1,α2=1,α4=0,α5=1,α8=0,α9=1,andt=1. The lump wave is located in arbitrary directions in the space, so it is a kind of localized wave actually. From equation (8), we can find that if x⟶±∞ or y⟶±∞, then f1 and f2⟶∞, u⟶0. The lump wave has one peak and two valleys, and valleys are symmetrically distributed on both sides of the peak. According to the extreme value principle of multivariate functions, by the calculation, we obtain that the lump wave’s minimum point (extreme value point) is −3α12+α52t+2α1α4+α5α8/2α12+α52,α5α1α8−α4α5/α2α12+α52=−3/2,0 and the corresponding amplitude is −12α12+α52/α9=−24. It propagates along the line y=0 with the velocity of −3/2. This mode of motion is uniform linear motion in physics.Figure 1
(a) Plots of lump solutions (8) with α1=1,α2=1,α4=0,α5=1,α8=0,α9=1,andt=1; (b) contour map in the (x, y) plane.
(a)
(b)


## 2.3. Lump Solutions Consisting of Three Quadratic Functions

We will seek for the lump solutions consisting of three quadratic functions. This situation has rarely been seen in existing literatures [13–15, 35]. In order to do it, we take function F in the following form:(9)F=f12+f22+f32+α13,f1=α1x+α2y+α3t+α4,f2=α5x+α6y+α7t+α8,f3=α9x+α10y+α11t+α12,where αi1≤i≤13 are real parameters to be determined.Substituting equation (9) into equation (3) and considering the coefficients of all powers of the variables to be 0, with the help of Maple, we have the following relations:(10)α1=−132α6α7+3α9α10α2,α3=−122α6α7+3α9α10α2,α5=23α7,α11=32α9,where α2,α4,α6,α7,α8,α9,α10,α12,andα13 are free parameters.For convenience, we let the parameters to be(11)α2=1,α4=0,α6=1,α7=1,α8=0,α9=1,α10=1,α12=0,α13=1.Consequently, the lump solutions consisting of three quadratic functions of equation (2) can be written as(12)u=−6lnFxx=−676/9−5/3x+y−5/2t2+76/92/3x+y+t2+76/9x+y+3/2t2−76/9x+38/3t2+76/9−5/3x+y−5/2t2+2/3x+y+t2+x+y+3/2t2+12.Based on parameter relations (10), we know F is analytical if and only if α2≠0 and α13>0. Figure 2(a) shows the lump solutions consisting of three quadratic functions of equation (2). Similar to the previous dynamical characteristic analysis, when α4=0,α8=0,α12=0, and t=0, the lump wave will center at the origin; here we take α2=1,α4=0,α6=1,α7=1,α8=0,α9=1,α10=1,α12=0,α13=1,andt=1. From equation (12), we can find that if x⟶±∞ or y⟶±∞, then f1, f2, and f3⟶∞, u⟶0. The lump wave consisting of three quadratic functions also has one peak and two valleys, and valleys are symmetrically distributed on both sides of the peak. Similarly, we obtain that the lump wave’s minimum point (extreme value point) is −34α72α22+α62+9α92α22+α102+12α6α7α9α10t+4α2α7α2α8−α4α6+6α2α9α2α12−α4α10/24α72α22+α62+9α92α22+α102+12α6α7α9α10,−α2α4+α6α8+α10α12/α22+α62+α102=−3/2,0 and the corresponding amplitude is −42α6α7+3α9α102/3α22+16/3α72+12α92/α13=−152/3. It propagates along the line y=0 with the velocity of −3/2. This mode of motion is also uniform linear motion in physics.Figure 2
(a) Plots of lump solutions (12) with α2=1,α4=0,α6=1,α7=1,α8=0,α9=1,α10=1,α12=0,α13=1,andt=1; (b) contour map in the (x, y) plane.
(a)
(b)Compared with the previous results (Section2.2), when α4=0,α8=0,andα12=0, we realize that lump solutions (8) and lump solutions (12) have the same minimum point (extreme value point); however, they have different extreme values (amplitudes) at the same minimum point. These two kinds of lump wave have the same mode of motion.


## 3. Interaction Solutions Consisting of Lump Solutions and One Soliton Solution

We will seek for the interaction solutions between lump solutions and one soliton solution. In order to do it, suppose thatF has the following form:(13)F=f12+f22+leλ+α9,f1=α1x+α2y+α3t+α4,f2=α5x+α6y+α7t+α8,λ=ξ1x+ξ2y+ξ3t,where αi1≤i≤9,l, and ξi1≤i≤3 are real parameters to be determined.Substituting equation (13) into equation (3) and considering the coefficients of all powers of the variables to be 0, with the help of Maple, we have the following relations:(14)α3=32α1,α6=−α1α2α5,α7=32α5,ξ2=0,ξ3=−ξ1−32+ξ12,where α1,α2,α4,α5,α8,α9,l,andξ1 are free parameters.Thus, the interaction solutions between lump solutions and one soliton solution can be written as(15)u=−62α12+α52+lξ12eξ1x−ξ1−3/2+ξ12tα1x+α2y+3/2α1t+α42+α5x−α1α2/α5y+3/2α5t+α82+α9+leξ1x−ξ1−3/2+ξ12t−2α1α1x+α2y+3/2α1t+α4+α5α5x−α1α2/α5y+3/2α5t+α8+lξ1eξ1x−ξ1−3/2+ξ12t2⋅α1x+α2y+3/2α1t+α42+α5x−α1α2/α5y+3/2α5t+α82+α9+leξ1x−ξ1−3/2+ξ12t2−1.Figure3 shows the interaction solutions between lump solutions and one soliton solution. Similar to previous section, we know F is analytical if and only if α5≠0,α9>0, and l>0. In order to facilitate dynamic analysis, we take α1=1,α2=1,α4=1,α5=1,α8=1,α9=1,ξ1=1,andl=1. When t<−3, there is only a solitary wave; at about t=−2, a special phenomenon occurs and the solitary wave starts to split into two parts: one is the lump wave and the other is the solitary wave. In this process, the amplitude of the lump wave has changed. When t>−2, the solitary wave still moves in the same direction as before, but the lump wave moves in the opposite direction, and they are farther and farther. This is a fission phenomenon. Figure 4 shows the opposite state of motion (ξ1=−1), where a lump wave and a solitary wave merge into a solitary wave. It is a fusion phenomenon. No matter fission or fusion, during the interaction, the solitary wave keeps its same amplitude and shape, and it is elastic. On the contrary, the lump wave’s amplitude has changed, and it is inelastic.Figure 3
Fission phenomenon: plots of interaction solutions (15) with α1=1,α2=1,α4=1,α5=1,α8=1,α9=1,ξ1=1,andl=1. (a) t=−10; (b) t=−3; (c) t=−2; (d) t=0; (e) t=6.
(a)
(b)
(c)
(d)
(e)Figure 4
Fusion phenomenon: plots of interaction solutions (15) with α1=1,α2=1,α4=1,α5=1,α8=1,α9=1,ξ1=−1,andl=1. (a) t=−10; (b) t=−2; (c) t=0; (d) t=2; (e) t=6.
(a)
(b)
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(e)


## 4. Interaction Solutions Consisting of Two Exponential Functions and One Sine Function

In this section, we will seek for the interaction solutions consisting of two exponential functions and one sine function. In order to do it, suppose thatF has the following form:(16)F=e−a1x+b1y+c1t+l1ea1x+b1y+c1t+l2sina2x+b2y+c2t,where a1,a2,b1,b2,c1,c2,l1,andl2 are parameters to be determined.Substituting equation (16) into equation (3), with the help of Maple, we have the following relations:(17)a1=64,a2=−64,b1=b2,c1=c2,l1=l224,where b2,c2,andl2 are free parameters.For convenience, we let the parameters to be(18)b2=3,c2=1,l2=1.Consequently, the interaction solutions consisting of two exponential functions and one sine function of equation (2) can be written as(19)u=−6lnFxx=−63/8e−6/4x+3y+t+3/8isinh6/4x+3y+t⋅e−6/4x+3y+t+1/4e6/4+isinh6/4x+3y+t−−6/4e−6/4x+3y+t+6/4icosh6/4x+3y+t2⋅e−6/4x+3y+t+1/4e6/4+isinh6/4x+3y+t2.Figure5 shows the interaction solutions consisting of two exponential functions and one sine function. Because u is a complex solution, we plot its real part and imaginary part, respectively. From Figure 4(a), we find that the real part of u is very similar to soliton solution. Its peaks are sharp while the soliton solution’s peak is smooth. All of its peaks have the same amplitude. From Figure 4(c), we can see the imaginary part of u has two rows of peaks. All of its all peaks have the same amplitude, and this is similar to the real part. When x or y is given, the amplitude and shape of the real part of u remain the same when t changes, and this is similar to the imaginary part.Figure 5
Plots of interaction solutions (19) with b2=3,c2=1,l2=1,andx=0: (a) the real part of u; (b) contour map of the real part; (c) the imaginary part of u; (d) contour map of the imaginary part.
(a)
(b)
(c)
(d)The obtained solution’s real part has one row of sharp peaks, but its imaginary part has two rows of sharp peaks. As far as we know, such solutions with sharp peaks in both real and imaginary parts have rarely been seen in existing literatures.


## 5. Conclusion

As a summary, we investigate different kinds of solutions for a dimensionally reduced nonlinear evolution equation, including lump solutions and two kinds of interaction solutions. Dynamical characteristics and properties of obtained solutions are discussed, respectively. For these solutions, we discover many special physical phenomena, for example, the fission and fusion phenomena from the first kind of interaction solutions. The results show that the Hirota bilinear method is so prompt and effective to obtain solutions for nonlinear evolution equations. Based on this point, many other kinds of solutions for equations especially nonlinear mathematical physics equations are worth exploring.



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