# Resonant Soliton and Soliton-Cnoidal Wave Solutions for a (3+1)-Dimensional Korteweg-de Vries-Like Equation

Authors: Zheng-Yi Ma; Jin-Xi Fei; Jun-Chao Chen; Quan-Yong Zhu
Journal: Complexity
 (2019)

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## Abstract

The residual symmetry of a (3+1)-dimensional Korteweg-de Vries (KdV)-like equation is constructed using the truncated Painlevé expansion. Such residual symmetry can be localized and the (3+1)-dimensional KdV-like equation is extended into an enlarged system by introducing some new variables. By using Lie’s first theorem, the finite transformation is obtained for this localized residual symmetry. Further, the linear superposition of multiple residual symmetries is localized and then-th Bäcklund transformation in the form of the determinants is constructed for this equation. For illustration more detail, the first three multiple wave solutions-the collisions of resonant solitons are depicted. Finally, with the aid of the link between the consistent tanh expansion (CTE) method and the truncated Painlevé expansion, the explicit soliton-cnoidal wave interaction solution containing three kinds of Jacobian elliptic functions for this equation is derived.


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## Body

## 1. Introduction

In scientific and engineering fields, nonlinear evolution equations have been studied in wide applications, such as in the nonlinear optics [1–7], plasma physics [8, 9], fluid mechanics [10, 11], textile engineering [12], and wave propagation phenomena [13–16]. Explicitly, for finding solutions, which including solitons, cnoidal waves, Painlevé waves, Airy waves, Bessel waves, etc., people often take the symmetry reduction approach with nonlocal symmetries with the aid of Darboux transformation, Bäklund transformation, and residual symmetry [17–21]. Methodology, for finding nonlinear evolution equations having infinitely many symmetries or flows, Olver proposed a general method which preserve them and it was employed to the KdV, modified Korteweg-de Vries (mKdV), Burgers’, and sine-Gordon equations [22, 23]. Integrable nonlinear evolution equations possessing a remarkably rich algebraic structures, which including infinitely many symmetries and conserved quantities, existence of a bi-Hamiltonian formulation through a recursion operator Φ were reviewed [24]. After that, Lou concluded that the residues of the truncated Painlevé expansions were all nonlocal symmetries of the original system for any Painlevé integrable systems [25]. This criterion was applied some well-known concrete examples, such as the KdV equation, the fifth order KdV equation, the Sawada-Kotera (SK) system, the Kaup-Kupershmidt (KK) equation, the Boussinesq equation, and the Kadomtsev-Petvishvili (KP) system. At the same time, a consistent Riccati expansion (CRE) method, which can be considered as an extension of the usual Riccati equation method and the tanh function expansion method was proposed for some integrable systems [26]. This method was systematic applied to many CRE solvable systems, such as the KdV equation and the (2+1)-dimensional asymmetric Nizhnik-Novikov-Veselov (ANNV) model, and the derived exact solution shared a similar determining equation which describing the interaction between a soliton and a cnoidal wave.A (3+1)-dimensional KdV-like equation can be described as(1)ut+α6uux+uxxx+β4uuy+2ux∫uydx+uxxy+γ4uuz+2ux∫uzdx+uxxz=0.Its soliton solutions were constructed by means of the simplified Hirota’s method [27]. The bilinear form, Bäklund transformations, Lax pairs and infinite conservation laws of Eq. (1) by means of the Bell polynomial method were constructed [28]. Its N-soliton solutions and periodic wave solutions were also presented with the aid of the bilinear formula and Riemann theta function.This paper is organized as follows. In Section2, with the aid of the truncated Painlevé expansion, the residual symmetry of the (3+1)-dimensional KdV-like equation is derived, and this nonlocal symmetry is localized by introducing eight auxiliary variables. Subsequently, we can obtain the finite symmetry transformation by solving the initial value problem. In Section 3, through localizing the linear superposition of multiple residual symmetries and construct the infinite transformation for this equation, the multiple residual symmetries and the n-th Bäcklund transformation are obtained. A direct result shows that one can derive special soliton solutions-the collisions of resonant solitons. In Section 4, using the CTE method, the explicit soliton-cnoidal wave interaction solution of the variables u,v and w with the Jacobian elliptic functions for this equation is obtained. A brief summary is given in Section 5.


## 2. Residual Symmetry and Finite Transformation of the (3+1)-Dimensional KdV-Like Equation

The (3+1)-dimensional KdV-like equation is integrability and its equivalent form has(2)ut+α6uux+uxxx+β4uuy+2uxv+uxxy+γ4uuz+2uxw+uxxz=0,(3)vx-uy=0,uz-wx=0,three parameters α,β,γ are real constants.The truncated Painlevé expansion of (2) and (3) reads (4)u=u2f2+u1f+u0,v=v2f2+v1f+v0,w=w2f2+w1f+w0,where ui,vj,wk(i,j,k=0,1,2) are functions of f and f≡f(x,y,z,t) is a real function. After substituting (4) into (2) and (3), one can derive the following results through vanishing the coefficients of powers f and solving an over-determined equations (λ is a arbitrary parameter) (5)u0=-fxxx2fx+fxx24fx2+λ,u1=2fxx,u2=-2fx2,(6)v0=-fxxy+2λfyfx-αfxxx+ft2βfx+3αfxx2+4βfxxfxy4βfx2,v1=2fxy,v2=-2fxfy,(7)w0=-fxxz+2λfzfx+fxxfxzfx2-3αλγ,w1=2fxz,w2=-2fxfz, which should hold on two consistent conditions (8)Cx+βKxxx+αSx+2βKxS+βKSx+4λβKx=0,(9)Qxxx+2SQx+QSx+4λQx=0, where the four variable quantities (10)C≡ftfx,K≡fyfx,Q≡fzfx,S≡fxxxfx-3fxx22fx2. Indeed, (8) and (9) possess Schwarzian structure for the (3+1)-dimensional KdV-like equation (1) and have invariant form under the Möbious transformation f→af+b/cf+d,ac≠bd [29–31].It is all know that the set of solutionsu0,v0 and w0 of (5)-(7) establish a link between the (3+1)-dimensional KdV-like equation (1) or (2) and (3) and their consistent conditions (8) and (9). That is to say, if the function f satisfies the consistent conditions (8) and (9), then (4) with (5)-(7) is an auto-Bäcklund transformation between the solutions u,v,w and u0,v0,w0 of (2) and (3). According to the residual symmetry theorem, the residuals u1,v1 and w1 are just the nonlocal symmetry with respect to the solutions u0,v0 and w0 of (5)-(7) and one should transform them to a local Lie point symmetry for studying this nonlocal symmetry [25].For the above purpose, the eight auxiliary variablesg≡g(x,y,z,t), h≡h(x,y,z,t),p≡p(x,y,z,t), q≡q(x,y,z,t), ψ≡ψ(x,y,z,t), ϕ≡ϕ(x,y,z,t), r≡r(x,y,z,t), and s≡s(x,y,t) are introduced, which should obey the rule (11)fx=g,gx=h,gy=p,gz=q,fy=ψ,fz=ϕ,hy=r,hz=s.Therefore, the local Lie point symmetry of the prolonged system of (2), (3), (8), (9), and (11) for the residual symmetry {σu,σv,σw}={2fxx,2fxy,2fxz} becomes(12)σu=2h,σv=2p,σw=2q,σf=-f2,σg=-2fg,σh=-2g2-2fh,σp=-2gψ-2fp,σq=-2gϕ-2fq,σψ=-2fψ,σϕ=-2fϕ,σr=-4gp-2hψ-2fr,σs=-4gq-2ϕh-2fs.Correspondingly, by solving the initial value problem(13)dUεdε=2Hε,dVεdε=2Pε,dWεdε=2Qε,dFεdε=-F2ε,dGεdε=-2FεGε,dHεdε=-2G2ε+FεHε,dPεdε=-2GεΨε+FεPε,dQεdε=-2GεΦε+FεQε,dΨεdε=-2FεΨε,dΦεdε=-2FεΦε,dRεdε=-22GεPε+HεΨε+FεRε,dSεdε=-22GεQε+HεΨε+FεSε,and(14)U0=u,V0=v,W0=w,F0=f,G0=g,H0=h,P0=p,Q0=q,Ψ0=ψ,Φ0=ϕ,R0=r,S0=s,with the infinitesimal parameter ε, one can write down the following finite transformation theorem.Theorem 1.
If{u,v,w,f,g,h,p,q,ψ,ϕ,r,s} is a solution of the extended system (2), (3), (8), (9), and (11), so are {U(ε),V(ε),W(ε), F(ε),G(ε),H(ε), P(ε),Q(ε),Ψ(ε), and  Φ(ε),R(ε),S(ε)}, where(15)Uε=u+2εh-g2ε+fhεfε+12,Vε=v+2εp-gψε+fpεfε+12,Wε=w+2εq-gϕε-fqεfε+12,Fε=ffε+1,Gε=gfε+12,Hε=h-2g2ε+fhεfε+13,Pε=p-2gψε+fpεfε+13,Qε=q-2gϕε+fqεfε+13,Ψε=ψfε+12,Φε=ϕfε+12,Rε=rfε+12-2εhψ+2gpfε+13+6g2ψε2fε+14,Sε=sfε+12-2εhϕ+2gqfε+13+6g2ϕε2fε+14.


## 3.n-th Bäcklund Transformation Related to Multiple Residual Symmetries

In fact, one can further study then-th Bäcklund transformation related to multiple residual symmetries of (2) and (3). The reason is that these residual symmetries just depend on the solution of the Schwarzian structure (8) and (9). The infinite residual symmetries of the fields u,v and w have (16)σu=2∑i=1ncifi,xx,σv=2∑i=1ncifi,xy,σw=2∑i=1ncifi,xz,where n is an arbitrary positive integer, ci(i=1,⋯,n) are arbitrary real constants, and fi(i=1,⋯,n) are different solutions of the Schwarzian Equations (8) and (9) with different parameters λi≠λj(i≠j), which need to satisfy(17)ftfxx+βfyfxxxx+αfxxxfx-3fxx22fx2x+2βfyfxxfxxxfx-3fxx22fx2+βfyfxfxxxfx-3fxx22fx2x+4λiβfyfxx=0,(18)fzfxxxx+2fxxxfx-3fxx22fx2fzfxx+fzfxfxxxfx-3fxx22fx2x+4λifzfxx=0,and three variable quantities (19)u=-fi,xxx2fi,x+fi,xx24fi,x2+λi,(20)v=-fi,xxy+2λifi,yfi,x-αfi,xxx+fi,t2βfi,x+3αfi,xx2+4βfi,xxfi,xy4βfi,x2,(21)w=-fi,xxz+2λifi,zfi,x+fi,xxfi,xzfi,x2-3αλiγ.Correspondingly, the eight auxiliary variablesgi≡gi(x,y,z,t), hi≡hi(x,y,z,t), pi≡pi(x,y,z,t), qi≡qi(x,y,z,t), ψi≡ψi(x,y,z,t), ϕi≡ϕi(x,y,z,t), ri≡ri(x,y,z,t), and si≡si(x,y,t) are renew introduced, which have the relation (22)fi,x=gi,gi,x=hi,gi,y=pi,gi,z=qi,fi,y=ψi,fi,z=ϕi,hi,y=ri,hi,z=si,i=1,2,⋯,nin order to localize the residual symmetry σu,σv and σw in (16).After writing the linearized form of some enlarged system, the symmetry (16) is localized to a Lie point symmetry(23)σu=-2∑k=1nckhk,σv=-2∑k=1nckpk,σw=-2∑k=1nckqk,σfj=14∑k≠jckgkhj-gjhk2gkgjλk-λj2,σgj=-∑k≠jckgkhj-gjhkλk-λj,σhj=2∑k≠jckgkgj+12∑k≠jckgj2hk2-gk2hj2gkgjλk-λj,σpj=∑k≠jckgjwk-gkwj+pjhk-pkhjλk-λj,σqj=∑k≠jckgjrk-gkrj+qjhk-qkhjλk-λj,σψj=∑k≠jckgkpj-gjpkgj2hk2-gk2hj2+2gkgjgjhk-gkhjgjwk-gkwj4gk2gj2λk-λj2,σϕj=∑k≠jckgjqk-gkqjgk2hj2-gj2hk2+2gkgjgkrj-gjrkgkhj-gjhk4gk2gj2λk-λj2,σrj=2∑k≠jckgjpk+gkpj+12∑k≠jckgkpj-gjpkgk2hj2+gj2hk2+2gkgjgj2hkrk-gk2hjrjgk2gj2λk-λj,σsj=2∑k≠jckgjqk+gkqj-12∑k≠jckgkqj-gjqkgk2hj2+gj2hk2+2gkgjgj2hksk-gk2hjsjgk2gj2λk-λj.For the Lie point symmetry (23), the following n-th Bäcklund transformation theorem can be summarized according to Lie’s first principle with the aid of its initial value problem.Theorem 2.
If{u,v,w,fi,gi,hi,pi,qi,ψi,ϕi,ri,si} is a solution of the prolonged (3+1)-dimensional KdV-like system (2), (3) and (17)-(21), so are {U(ε),V(ε),Wi(ε),Fi(ε),Gi(ε),Hi(ε),Pi(ε),Qi(ε), and Ψi(ε),Φi(ε),Ri(ε),Si(ε)}, here(24)Uε=u+2ln⁡Δxx,Vε=v+2ln⁡Δxy,Wε=v+2ln⁡Δxz,Fiε=-ΔiΔ,Giε=Fi,xε,Hiε=Fi,xxε,Piε=Fi,xyε,Qiε=Fi,xzε,Ψiε=Fi,yε,Ψiε=Fi,zε,Riε=Fi,xxyε,Siε=Fi,xxzε,i=1,2,⋯,n.where Δi and Δ are two determinants of the matrices Di and D, which satisfy(25)Di=c1εf1+1c1εμ1,2⋯c1εμ1,i-1c1εμ1,ic1εμ1,i+1⋯c1εμ1,nc2εμ1,2c2εf2+1⋯c2εμ2,i-1c2εμ2,ic2εμ2,i+1⋯c2εμ2,n⋯⋯⋯⋯⋯⋯⋯⋯ci-1εμ1,i-1ci-1εμ2,i-1⋯ci-1εfi-1+1ci-1εμi-1,ici-1εμi-1,i+1⋯ci-1εμi-1,nμ1,iμ2,i⋯μi,i-1fiμi,i+1⋯μi,nci+1εμ1,i+1ci+1εμ2,i+1⋯ci+1εμi-1,i+1ci+1εμi,i+1ci+1εfi+1+1⋯ci+1εμi+1,n⋯⋯⋯⋯⋯⋯⋯⋯cnεμ1,ncnεμ2,n⋯cnεμi-1,ncnεμi,ncnεμi+1,n⋯cnεfn+1.and(26)D=c1εf1+1c1εμ1,2⋯c1εμ1,j⋯c1εμ1,nc2εμ1,2c2εf2+1⋯c2εμ2,j⋯c2εμ2,n⋯⋯⋯⋯⋯⋯cjεμ1,jcjεμ2,j⋯cjεfj+1⋯cjεμj,n⋯⋯⋯⋯⋯⋯cnεμ1,ncnεμ2,n⋯cnεμj,n⋯cnεfn+1,μi,j=gihj-gjhi2gigjλi-λj.From the above Theorem2, one can derive an infinite number of explicit solutions from a suitable seed solution of (2) and (3) under some special circumstances. Especially, we have the recursive soliton solutions from the known one for this system. For example, when the seed solution takes u=0,v=0,w=a for (2) and (3), it is not difficult to verify that the prolonged (3+1)-dimensional KdV-like system (2), (3), and (17)-(21) possesses the following soliton function fi:(27)fi=1+ekix+liy+miz+nit,mi=-3αki2+4aγ2γki,ni=ki2αki-2βli2,λi=ki24.The corresponding first three multiple wave solutions of (2) and (3) are explicitly written(28)u1=2c1εf1,xxc1εf1+1-2c12ε2f1,x2c1εf1+12,v1=2c1εf1,xyc1εf1+1-2c12ε2f1,xf1,yc1εf1+12,w1=a+2c1εf1,xzc1εf1+1-2c12ε2f1,xf1,zc1εf1+12,(29)u2=2ln⁡Δ1xx,v2=2ln⁡Δ1xy,w2=a+2ln⁡Δ1xz,Δ1=c1c2ε2f1f2-μ1,22+εc1f1+c2f2+1,and(30)u3=2ln⁡Δ2xx,v3=2ln⁡Δ2xy,w3=a+2ln⁡Δ2xz,Δ2=c1εf1+1c1εμ1,2c1εμ1,3c2εμ1,2c2εf2+1c2εμ2,3c3εμ1,3c3εμ2,3c3εf3+1.For illustration of more detail, the parametersα,β,γ,ε,a,c1,c2,c3,k1 are all taken 1, but l1=-l3=1/2,k2=-l2=3/2,k3=2. Figure 1 displays the bell-like bright and dark solitons for the above condition of (28). Figures 1(a) and 1(b) are two line solitons for u1 and v1 with the amplitude 0.5 and 0.25, respectively, while Figure 1(c) is a dark one for w1 with the amplitude 1.75. Figures 2 and 3 are the collisions of two-resonant solitons and three-resonant solitons of (29) and (30), respectively.Figure 1
The plots of three line solitons expressed by (28) for the solution of (2) and (3) with the parameters α,β,γ,ε,a,c1,k1 are all taken 1, but l1=1/2.Figure 2
Three interactions of two-resonant solitons expressed by (29) for the solution of (2) and (3) with the parameters α,β,γ,ε,a,c1,c2,k1 are all taken 1, but l1=1/2,k2=-l2=3/2.Figure 3
Three interactions of three-resonant solitons expressed by (30) for the solution of (2) and (3) with the parameters α,β,γ,ε,a,c1,c2,c3,k1 are all taken 1, but l1=-l3=1/2,k2=-l2=3/2,k3=2.


## 4. CTE and Soliton-Cnoidal Wave Interaction Solution

With the help of a Riccati equation, Lou proposed the CRE for solving nonlinear systems [26]. While the CTE method is a special form of CRE and this approach is a generalization of the traditional tanh function expansion method [32]. For the (3+1)-dimensional KdV-like equation (2) and (3), through introducing the tangent transformation (31)f=abtanh⁡ϕ+c,the truncated Painlevé expansion of (2) and (3) can be reduced(32)u=-2ϕx2tanh2⁡ϕ+2ϕxxtanh⁡ϕ+ϕxx24ϕx2-ϕxxx2ϕx+ϕx2+λ,(33)v=-2ϕxϕytanh2⁡ϕ+2ϕxytanh⁡ϕ+3αϕxx24βϕx2+ϕxxϕxyϕx2-αϕxxx+ϕt2βϕx-ϕxxy+2λϕyϕx+αϕx2β+2ϕxϕy,(34)w=-2ϕxϕztanh2⁡ϕ+2ϕxztanh⁡ϕ+ϕxxϕxzϕx2-ϕxxz+2λϕzϕx+2ϕxϕz-3αλγ, where a,b,c are three constants and ϕ≡ϕ(x,y,z,t) is a real function.At the same time, the Schwarzian structure (8) and (9) owns the following consistent condition:(35)ϕxt=-3ϕxx2αϕxx+βϕxyϕx2+ϕxxx4αϕxx+βϕxyϕx+ϕxx4αϕx3+3βϕxxy+4βλϕy+ϕtϕx+4βϕxyϕx2-λ-αϕxxxx-βϕxxxy,(36)ϕxxxz=-3ϕxx2ϕxzϕx2+ϕxzϕxxx+4ϕx3-4λϕx+ϕxx3ϕxxz+4λϕzϕx.In order to derive the soliton-cnoidal wave solution, the solution of (35) and (36) should assume as the following form: (37)ϕ=k1x+l1y+m1z+ω1t+Gk2x+l2y+m2z+ω2t≡k1x+l1y+m1z+ω1t+GX.After substituting (37) into the consistent condition (35) and (36), the elliptic equation is satisfied (38)G1X2X=4G14X+a1G13X+a2G12X+a3G1X+a4,G1X=GXX,where the coefficients ai(i=1,2,3,4) are(39)a1=-2C2k22+8k1k2,a2=2C1k2-6C2k1k2+4k12k22,a3=4C1k1-6C2k12+4λk2m1-k1m2k22m2=4C1k1-6C2k12+4λβk2l1-k1l2+k2ω1-k1ω2k23αk2+βl2,a4=2k12C1-C2k1k2+4λk1k2m1-k1m2k24m2=2k12C1-C2k1k2+4λβk1k2l1-k1l2+k1k2ω1-k1ω2k24αk2+βl2,C1 and C2 are two arbitrary constants.One concrete solution of (37) is (40)ϕ=k1x+l1y+m1z+ω1t+12arctanh⁡msnX,m,with sn(X,m) is a Jacobian elliptic sine function with modulus m. For this time, the relations of the coefficients are (41)m1=m22k12m2-1+λ2λ,ω1=8αk13+4βk12l2m2-1+2βλl2-2l1+12ω2,k2=2k1.As the result, the soliton-cnoidal wave interaction solution of (2) and (3) is (42)u=1DN2Ξ2212-tanh2ϕ+k22mm2-1SNtanh⁡ϕ-k2CN1+m2SN22Ξ+k22mm2-1SN24Ξ2+λ,(43)v=1DN2Ξml2CN+2l1DN2cosh2⁡ϕ+k2l2mm2-1SNtanh⁡ϕ-k22mm2-1CN1+m2SN2Ξl2+k2αβ+k23m2m2-12SN2Ξ2l2+3k2α4β-1Ξ2λl2mCN+2l1DN+mω2CN+2ω1DN2β+αΞ24βDN2,(44)w=1DN2Ξmm2CN+2m1DN2cosh2ϕ+k2mm2m2-1SNtanh⁡ϕ-k22mm2m2-1CN1+m2SN2Ξ+k23m2m2m2-12SN2Ξ2-2λmm2CN+2m1DNΞ-3αλγ.with SN≡sn(X,m),CN≡cn(X,m),DN≡dn(X,m),Ξ≡k2mCN+2k1DN, and X≡k2x+l2y+m2z+ω2t, while sn(X,m),cn(X,m), and dn(X,m) are three kinds of Jacobian elliptic functions with modulus m. Figure 4 shows that one bright/dark soliton resides on a cnoidal wave instead of constant background for the above solution u,v, and w of (2) and (3) when the coefficients are taken α=k2=ω2=2,β=γ=λ=k1=l1=m2=1,m1=-0.25,ω1=-14,l2=-1,m=0.5. For this soliton-cnoidal wave, the influence of the electron superthermality, positron concentration, and magnetic field obliqueness was investigated in detail [8].Figure 4
Theu,v, and w plots of the soliton-cnoidal wave interaction solution (42), (43), and (44), respectively.


## 5. Summary

The KdV equation is well known and it higher dimensional extension is important. For example, initial-boundary value problems of the coupled mKdV equation on the half-line through the Fokas method [33, 34], analysis on lump, lumpoff and rogue waves with predictability to the (2+1)-dimensional B-type KP equation [35], rogue waves, bright-dark solitons, and traveling wave solutions of the (3+1)-dimensional generalized KP equation [36] were studied in detail. In this paper, the nonlocal symmetry of the (3+1)-dimensional KdV-like equation is obtained with the aid of its the truncated Painlevé expansion. After introducing the auxiliary variables g,h,p,q,ψ,ϕ,r and s, an enlarged system which possesses a Lie point symmetry for the nonlocal symmetry is taken. By applying Lie’s first theorem for the localized point symmetries, we obtain the corresponding finite transformation. Further, we can localize the linear superposition of multiple residual symmetries and construct the infinite transformation for this equation. From Theorem 2, the n-th Bäcklund transformation can be expressed in a compact way of determinants. According to this conclusion, one can derive special soliton solutions-the collisions of resonant solitons from some seed solutions. At the same time, the explicit soliton-cnoidal wave interaction solution of the variables u,v, and w with three Jacobian elliptic functions for this (3+1)-dimensional KdV-like equation is also derived using the CTE method.



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