# Different Wave Structures for the (2+1)-Dimensional Korteweg-de Vries Equation

Authors: Chun-Rong Qin; Jian-Guo Liu; Wen-Hui Zhu; Guo-Ping Ai; M. Hafiz Uddin
Journal: Advances in Mathematical Physics
 (2022)

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## Abstract

In this article, a (2+1)-dimensional Korteweg-de Vries equation is investigated. Abundant periodic wave solutions are obtained based on the Hirota’s bilinear form and a direct test function. Meanwhile, the interaction solutions between lump and periodic waves are presented. What is more, we derive the interaction solutions among lump, periodic, and solitary waves. Based on the extended homoclinic test technique, some new double periodic-soliton solutions are presented. Finally, some 3D and density plots are simulated and displayed to respond the dynamic behavior of these obtained solutions.


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## Body

## 1. Introduction

Korteweg-de Vries (KdV) equation [1–11]
(1)ut+6uux+uxxx=0,has been used to depict the shallow-water waves, stratified internal waves, lattice dynamics, and so on, where u=ux,t. Its extensions, namely, the KdV-type models, have been presented in fields such as fluid flows, plasma physics, and solid-state physics [12–15]. Solitary wave solutions have wide applications in many fields of natural science such as plasmas, hydrodynamics, nonlinear optics, fiber optics, and solid state physics, and that the interaction of solitons plays an important role [16, 17], which can keep their velocities and shapes after the elastic collisions [18–21]. Periodic waves, as solitary waves, have amusing applications in nature. For the ultrashort pulse-train generation from the beating of two-mode signals, for instance, one must research periodic wave solutions of nonlinear equations governing the fiber system [22]. However, the interaction properties between periodic waves are rarely discussed because the mathematics is more involved.In this paper, based on symbolic computation [23–30], we will investigate the following (2+1)-dimensional KdV equation for nonlinear waves such as the shallow-water waves and surface and internal waves [31]
(2)ut+3uvx+uxxx=0,ux=vy,where u=ux,y,t, v=vx,y,t. Equation (1) was obtained by Boiti et al. in Ref. [31] by using the weak Lax pair, also named as Boiti-Leon-Manna-Pempinelli equation [32] and read as the ubiquitous KdV equation when v=u and y=x [33]. The rich dromion structures and localized structures [34, 35], exact periodic solitary wave and Jacobi elliptic function double periodic solutions [33], periodic type of three-wave solutions [36], lump solutions [37], a new Bäklund transformation and new representation of the N-soliton solution [38], invariant solutions [39], M-lump solutions [40], and breathers and interaction solutions [41, 42] for Equation (1) have been studied. Ma [43] obtained N-soliton solutions and given the Hirota N-soliton conditions of Equation (1) by using the Hirota bilinear formulation. However, the interaction solutions between lump and periodic waves and interaction solutions among lump, periodic, and solitary waves have not been seen in literature, which will become our main work.The organization of this article is as follows. In Section2, abundant periodic wave solutions are obtained based on the Hirota’s bilinear form and a direct test function. In Section 3, the interaction solutions between lump and periodic waves are obtained. In Section 4, we present the interaction solutions among lump, periodic, and solitary waves. Dynamic behavior is analyzed by some 3D and density plots. In Section 5, we present new double periodic-soliton solutions for the (2+1)-dimensional KdV equation by using the extended homoclinic. In Section 6, the conclusions are made.


## 2. Periodic Wave Solutions

Under two logarithmic transformations [36]
(3)u=2lnfxy,v=2lnfxx,Equation (2) has the following bilinear form:
(4)DyDt+Dx3Dyf·f=fytf−fyft+ffxxxy+3fxyfxx−3fxfxxy−fyfxxx=0,where f=fx,y,t. To study the periodic solitary wave solutions of Equation (1), suppose that
(5)f=k1eθ1+e−θ1+k2tanθ2+k3tanhθ3,where θi=αix+βiy+δit+σii=1,2,3 and αi, βi, δi, and σi are undetermined constants. Substituting Equation (5) into Equation (4) and equating all the coefficients of different powers of eθ1, e−θ1, tanθ2, and tanhθ3 and constant term to zero, we haveCase 1.
(6)k3=β1=α2=δ2=0,δ1=−α13.
Substituting Equation (6) into Equation (5), we have
(7)f=e−tα13+xα1+σ1k1+etα13−xα1−σ1+k2tanyβ2+σ2.
Thus, the first new periodic wave solution is obtained as(8)u1=−2sec2yβ2+σ2k2e−tα13+xα1+σ1k1α1−etα13−xα1−σ1α1β2e−tα13+xα1+σ1k1+etα13−xα1−σ1+k2tanyβ2+σ22,(9)v1=2etα13−xα1−σ1α12+e−tα13+xα1+σ1k1α12e−tα13+xα1+σ1k1+etα13−xα1−σ1+k2tanyβ2+σ2−2e−tα13+xα1+σ1k1α1−etα13−xα1−σ1α12e−tα13+xα1+σ1k1+etα13−xα1−σ1+k2tanyβ2+σ22.
Dynamic behavior of Equation (8) is shown in Figure 1 in x−y.Case 2.
(10)k3=k1=α2=δ2=0,δ1=−α13.
Substituting Equation (10) into Equation (5), we have
(11)f=k2tanyβ2+σ2+etα13−xα1−yβ1−σ1.
Thus, we derive the second new periodic wave solution as follows:(12)u2=2etα13−xα1−yβ1−σ1α1β1k2tanyβ2+σ2+etα13−xα1−yβ1−σ1+2etα13−xα1−yβ1−σ1α1sec2yβ2+σ2k2β2−etα13−xα1−yβ1−σ1β1k2tanyβ2+σ2+etα13−xα1−yβ1−σ12,(13)v2=2etα13−xα1−yβ1−σ1α12k2tanyβ2+σ2+etα13−xα1−yβ1−σ1−2e2tα13−2xα1−2yβ1−2σ1α12k2tanyβ2+σ2+etα13−xα1−yβ1−σ12.
Dynamic behavior of Equation (12) is shown in Figure 2 in x−y.Case 3.
(14)k2=k1=α3=δ3=0,δ1=−α13.
Substituting Equation (14) into Equation (5), we have
(15)f=k3tanhyβ3+σ3+etα13−xα1−yβ1−σ1.
Thus, the third new periodic wave solution is(16)u3=2etα13−xα1−yβ1−σ1α1β1k3tanhyβ3+σ3+etα13−xα1−yβ1−σ1+2etα13−xα1−yβ1−σ1α1sech2yβ3+σ3k3β3−etα13−xα1−yβ1−σ1β1k3tanhyβ3+σ3+etα13−xα1−yβ1−σ12,(17)v3=2etα13−xα1−yβ1−σ1α12k3tanhyβ3+σ3+etα13−xα1−yβ1−σ1−2e2tα13−2xα1−2yβ1−2σ1α12k3tanhyβ3+σ3+etα13−xα1−yβ1−σ12.
Dynamic behavior of Equation (16) is shown in Figure 3 in x−y.Case 4.
(18)k2=β1=α3=δ3=0,δ1=−α13.
Substituting Equation (20) into Equation (5), we have
(19)f=e−tα13+xα1+σ1k1+etα13−xα1−σ1+k3tanhyβ3+σ3.
Then, the fourth new periodic wave solution is presented as follows:(20)u4=−2sech2yβ3+σ3k3e−tα13+xα1+σ1k1α1−etα13−xα1−σ1α1β3e−tα13+xα1+σ1k1+etα13−xα1−σ1+k3tanhyβ3+σ32,(21)v4=2etα13−xα1−σ1α12+e−tα13+xα1+σ1k1α12e−tα13+xα1+σ1k1+etα13−xα1−σ1+k3tanhyβ3+σ3−2e−tα13+xα1+σ1k1α1−etα13−xα1−σ1α12e−tα13+xα1+σ1k1+etα13−xα1−σ1+k3tanhyβ3+σ32.
Dynamic behavior of Equation (20) is shown in Figure 4 in x−y.Figure 1
α1=σ2=−1, k1=5, k2=σ1=−2, β2=1, and t=0. (a) 3D plot and (b) density plot.
(a)(b)Figure 2
α1=σ2=−1, β1=−1, k2=2, σ1=1, β2=5, and t=0. (a) 3D plot and (b) density plot.
(a)(b)Figure 3
α1=β1=1, σ1=−1, k3=−2, σ3=β3=5, and t=0. (a) 3D plot and (b) density plot.
(a)(b)Figure 4
α1=β3=σ1=1, k3=−2, δ1=−5, k1=σ3=5, and t=0. (a) 3D plot and (b) density plot.
(a)(b)


## 3. Lump-Periodic Waves

Under the transformations [41]
(22)u=2lnfxy,v=γ+2lnfxx,Equation (2) has the more general bilinear form
(23)DyDt+Dx3Dy+3γDxDyf·f=fytf−fyft+3γfyxf−fyfx+ffxxxy+3fxyfxx−3fxfxxy−fyfxxx=0.To discuss the interaction between lump and periodic waves, assume(24)f=α9+k1sinα14+α13t+α11x+α12y+α8+α7t+α5x+α6y2+α4+α3t+α1x+α2y2+k2cosα24+α23t+α21x+α22y,where αii=1,⋯,9, αj1, αj2, αj3, and αj4j=1,2 are undetermined constants. Substituting Equation (24) into Equation (23), we have
(25)α13=α113−3α11γ,α23=α213−3α21γ,α12=α22=0,α6=−α1α2α5,α7=−3α5γ,α3=−3α1γ.Equation (24) will become
(26)f=α9+k1sinα14+tα113−3α11γ+α11x+α4−3α1γt+α1x+α2y2+α8−3α5γt+α5x−α1α2yα52+k2cosα24+tα213−3α21γ+α21x.Combining Equations (22) and (26), we can obtain the following interaction solution:
(27)u=2lnfxy,v=γ+2lnfxx.Dynamic behavior of Equation (27) is shown in Figure 5. Lump and periodic waves can be seen in Figure 5. With the change of y value, the amplitude of wave changes correspondingly and reaches the maximum at a certain moment.Figure 5
α1=α4=α11=α9=k1=1, k2=α2=α21=3, γ=α8=α14=2, α24=4, α5=−2. (a) y=−3, (b) y=0, and (c) y=2.
(a)(b)(c)In addition, we can also derive another three sets of solutions for the parametersαii=1,⋯,9, αj1, αj2, αj3, and αj4j=1,2.
(28)α13=α113−3α11γ,α23=α12=α21=0,α6=−α1α2α5,α7=−3α5γ,α3=−3α1γ,α13=α11=α22=0,α23=α213−3α21γ,α6=−α1α2α5,α7=−3α5γ,α3=−3α1γ,α13=α23=α11=α21=0,α6=−α1α2α5,α7=−3α5γ,α3=−3α1γ.Substituting these sets of solutions for the parameters into Equations (22) and (24), the corresponding interaction solutions can be obtained.


## 4. Lump-Periodic-Solitary Waves

In order to investigate the interaction among lump, periodic, and solitary waves, suppose(29)f=α9+k1expα14+α13t+α11x+α12y+α8+α7t+α5x+α6y2+α4+α3t+α1x+α2y2+k2cosα24+α23t+α21x+α22y.Substituting Equation (29) into Equation (23), we have
(30)α13=−α113−3α11γ,α23=α213−3α21γ,α12=α22=0,α6=−α1α2α5,α7=−3α5γ,α3=−3α1γ.Equation (29) will become
(31)f=α9+k1eα14+t−3α11γ−α113+α11x+α4−3α1γt+α1x+α2y2+α8−3α5γt+α5x−α1α2yα52+k2cosα24+tα213−3α21γ+α21x.Combining Equations (22) and (31), we can obtain the following interaction solution:
(32)u=2lnfxy,v=γ+2lnfxx.Dynamic behavior of Equation (32) is shown in Figure 6. Lump, periodic, and solitary waves can be found in Figure 6.Figure 6
α1=α4=α11=α9=k1=1, k2=α2=α21=3, γ=α8=α14=2, α24=4, α5=−2. (a) t=−1, (b) t=0, and (c) t=1.
(a)(b)(c)In addition, we can also derive another three sets of solutions for the parametersαii=1,⋯,9, αj1, αj2, αj3, and αj4j=1,2.
(33)α13=−α113−3α11γ,α23=α12=α21=0,α6=−α1α2α5,α7=−3α5γ,α3=−3α1γ,α13=α11=α22=0,α23=α213−3α21γ,α6=−α1α2α5,α7=−3α5γ,α3=−3α1γ,α13=α23=α11=α21=0,α6=−α1α2α5,α7=−3α5γ,α3=−3α1γ.Substituting these sets of solutions for the parameters into Equations (22) and (29), the corresponding interaction solutions can be obtained.


## 5. Double Periodic-Soliton Solutions

Supposing the functionf in Equation (4) has the following double-periodic soliton structures:
(34)f=eθ1γ1cosθ2+γ2sinθ2+k1e2θ1+eθ3γ3cosθ4+γ4sinθ4+k2eθ4,where θi=αix+βiy+δit,i=1,2,3,4 and αi, βi, and δi are constants to be determined later. Substituting Equation (34) into Equation (4), we can obtain a set of algebraic equations for αi, βi, and δi yields a set of algebraic equations. Solving these algebraic equations with the aid of symbolic computation, we obtain the following:Case 5.
(35)k1=β2=β4=0,β3=β1,δ2=α2α22−3α1−2α42,(36)δ1=−α13+3α22α1−14α43−12α1−2α3α42+6α12−α22−α32α4,(37)δ3=−α33+15α42α3−20α43,δ4=α4−3α32+12α4α3−11α42.Case 6.
(38)k1=β2=δ4=α4=0,δ1=−α13+3α3α12+3α22−α32α1−3α22α3,γ4=−β4γ32β4−β3,γ2=α22+α1α3−α1γ1α22α1−α3,δ3=−α33,β1=2β4,δ2=α2α22−3α1−α32.Case 7.
(39)k1=α3=α4=β2=δ3=δ4=0,γ4=−β4γ3β1−β3,γ2=2α1α2γ1α12−α22,δ1=3α1α22−α13,δ2=α23−3α12α2.Case 8.
(40)k1=α2=β4=δ2=0,δ1=−2α43α32−12α4α3+11α42,δ3=−α33+15α42α3−20α43,δ4=−α43α32−12α4α3+11α42,γ4=−α3−3α4α3−α4γ32α3−2α4α4,α1=2α4.Case 9.
(41)k1=α2=α4=δ2=δ4=0,α1=α3,γ4=−β4γ1γ3β1−β3γ1+β2γ2,δ1=δ3=−α33.Case 10.
(42)k2=α2=β4=β2=δ2=0,β3=2β1,δ1=−α13,γ3=−α32+2α1α3+α42γ42α1−α3α4,δ3=3α3−α1α42−α33α12−3α3α1+α32,δ4=α43−3α1−α32α4.Case 11.
(43)k2=α4=β2=δ4=0,γ4=β1−β3γ3β4,α3=2α1,δ1=3α1α22−α13,δ2=α23−3α12α2,δ3=6α1α22−2α13.Case 12.
(44)k2=α2=δ2=α4=δ4=0,α3=2α1,δ3=−2α13,δ1=−α13,γ4=−β2γ1+β3−β1γ2γ3β4γ2.Case 13.
(45)α1=α2=δ1=δ2=α4=δ4=0,β4=β1,δ3=−α33,γ4=−β2γ1+β3−β1γ2γ3β1γ2.Case 14.
(46)α4=β2=δ4=0,β4=β1,α3=2iϵα2,α1=iϵα2,δ3=8iϵα23,(47)δ2=4α23,δ1=4iϵα23,γ4=1−β3β1γ3.Case 15.
(48)δ1=α1=α4=δ4=β2=0,δ3=−α33,α2=iϵα3,γ2=−iϵγ1,γ3=−β1−β3γ4β4,δ2=−iϵα33,where ε=±1. Substituting Equations (35)–(47) into Equations (3) and (34), respectively, we can obtain abundant double-periodic soliton solutions of Equation (1). As an example, substituting Equation (47) into Equation (34), we have
(49)f=e2xα4+t−3α32+12α4α3−11α42α4k2+cosxα2+tα22−3α1−2α42α2γ1+sinxα2+tα22−3α1−2α42α2γ2Expxα1+t−α13+3α22α1−14α43−12α1−2α3α42+6α12−α22−α32α4+yβ1+exα3+t−α33+15α42α3−20α43+yβ1cosxα4+t−3α32+12α4α3−11α42α4γ3+sinxα4+t−3α32+12α4α3−11α42α4γ4.
Therefore, the corresponding double-periodic soliton solutions can be presented as follows:(50)u1=2exα1+yβ1+tδ1α1β1cosxα2+tδ2γ1+sinxα2+tδ2γ2+exα1+yβ1+tδ1β1cosxα2+tδ2α2γ2−sinxα2+tδ2α2γ1+exα3+yβ1+tδ3α3β1cosxα4+tδ4γ3+sinxα4+tδ4γ4+exα3+yβ1+tδ3β1cosxα4+tδ4α4γ4−sinxα4+tδ4α4γ3/e2xα4+tδ4k2+exα1+yβ1+tδ1cosxα2+tδ2γ1+sinxα2+tδ2γ2+exα3+yβ1+tδ3cosxα4+tδ4γ3+sinxα4+tδ4γ4−2exα1+yβ1+tδ1β1cosxα2+tδ2γ1+sinxα2+tδ2γ2+exα3+yβ1+tδ3β1cosxα4+tδ4γ3+sinxα4+tδ4γ42e2xα4+tδ4k2α4+exα1+yβ1+tδ1α1cosxα2+tδ2γ1+sinxα2+tδ2γ2+exα1+yβ1+tδ1cosxα2+tδ2α2γ2−sinxα2+tδ2α2γ1+exα3+yβ1+tδ3α3cosxα4+tδ4γ3+sinxα4+tδ4γ4+exα3+yβ1+tδ3cosxα4+tδ4α4γ4−sinxα4+tδ4α4γ3/e2xα4+tδ4k2+exα1+yβ1+tδ1cosxα2+tδ2γ1+sinxα2+tδ2γ2+exα3+yβ1+tδ3cosxα4+tδ4γ3+sinxα4+tδ4γ42.
Dynamic behavior of expression (49) is shown in Figure 7.Figure 7
Evolution of the periodic wave for solution (49) at α1=α3=γ2=−1, α2=α4=γ3=1, β1=γ1=5, k2=−2, and γ4=2. (a) t=−5, (b) t=0, and (c) t=5.
(a)(b)(c)


## 6. Conclusion

In this paper, we study a (2+1)-dimensional KdV equation. Abundant periodic wave solutions are obtained based on the Hirota’s bilinear form and a direct test function. Corresponding dynamic behavior is shown in Figures1–4. Meanwhile, the interaction solutions between lump and periodic waves are obtained. Corresponding dynamic behavior is seen in Figure 5. From Figure 5, we can observe the interaction between lump and periodic waves. With the change of y value, the amplitude of wave changes correspondingly and reaches the maximum at a certain moment. We present the interaction solutions among lump, periodic, and solitary waves. Corresponding dynamic behavior is seen in Figure 6. From Figure 6, we can observe the lump wave, periodic wave, and solitary wave at the same time. Finally, with the aid of the extended homoclinic test technique and an ansätz functions, double periodic-soliton solutions of the (2+1)-dimensional Korteweg-de Vries equation are obtained. Corresponding dynamic behavior is shown in Figure 7.



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