COMPLEXITY Complexity 1099-0526 1076-2787 Hindawi 10.1155/2019/8034731 8034731 Research Article Resonant Soliton and Soliton-Cnoidal Wave Solutions for a (3+1)-Dimensional Korteweg-de Vries-Like Equation https://orcid.org/0000-0002-5389-2052 Ma Zheng-Yi 1 2 Fei Jin-Xi 3 https://orcid.org/0000-0001-8911-6031 Chen Jun-Chao 1 https://orcid.org/0000-0001-6592-7905 Zhu Quan-Yong 1 Boutayeb Mohamed 1 Institute of Nonlinear Analysis and Department of Mathematics Lishui University Lishui 323000 China lsu.edu.cn 2 Department of Mathematics Zhejiang Sci-Tech University Hangzhou 310018 China zstu.edu.cn 3 Department of Photoelectric Engineering Lishui University Lishui 323000 China lsu.edu.cn 2019 2872019 2019 04 05 2019 12 07 2019 15 07 2019 2872019 2019 Copyright © 2019 Zheng-Yi Ma et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 the n-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.

National Natural Science Foundation of China 11705077 11775104 11447017 Natural Science Foundation of Zhejiang Province LY14A010005 Scientific Research Foundation of the First-Class Discipline of Zhejiang Province (B) 201601
1. Introduction

In scientific and engineering fields, nonlinear evolution equations have been studied in wide applications, such as in the nonlinear optics , plasma physics [8, 9], fluid mechanics [10, 11], textile engineering , and wave propagation phenomena . 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 . 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 . 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 . 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 . 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+2uxuydx+uxxy+γ4uuz+2uxuzdx+uxxz=0.Its soliton solutions were constructed by means of the simplified Hirota’s method . The bilinear form, Bäklund transformations, Lax pairs and infinite conservation laws of Eq. (1) by means of the Bell polynomial method were constructed . 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 Section 2, 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 ff(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)Cftfx,Kfyfx,Qfzfx,Sfxxxfx-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 faf+b/cf+d,acbd .

It is all know that the set of solutions u0,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 .

For the above purpose, the eight auxiliary variables gg(x,y,z,t), hh(x,y,z,t),pp(x,y,z,t), qq(x,y,z,t), ψψ(x,y,z,t), ϕϕ(x,y,z,t), rr(x,y,z,t), and ss(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. <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M48"><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:math></inline-formula>-th Bäcklund Transformation Related to Multiple Residual Symmetries

In fact, one can further study the n-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=2i=1ncifi,xx,σv=2i=1ncifi,xy,σw=2i=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 λiλj(ij), 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 variables gigi(x,y,z,t), hihi(x,y,z,t), pipi(x,y,z,t), qiqi(x,y,z,t), ψiψi(x,y,z,t), ϕiϕi(x,y,z,t), riri(x,y,z,t), and sisi(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=-2k=1nckhk,σv=-2k=1nckpk,σw=-2k=1nckqk,σfj=14kjckgkhj-gjhk2gkgjλk-λj2,σgj=-kjckgkhj-gjhkλk-λj,σhj=2kjckgkgj+12kjckgj2hk2-gk2hj2gkgjλk-λj,σpj=kjckgjwk-gkwj+pjhk-pkhjλk-λj,σqj=kjckgjrk-gkrj+qjhk-qkhjλk-λj,σψj=kjckgkpj-gjpkgj2hk2-gk2hj2+2gkgjgjhk-gkhjgjwk-gkwj4gk2gj2λk-λj2,σϕj=kjckgjqk-gkqjgk2hj2-gj2hk2+2gkgjgkrj-gjrkgkhj-gjhk4gk2gj2λk-λj2,σrj=2kjckgjpk+gkpj+12kjckgkpj-gjpkgk2hj2+gj2hk2+2gkgjgj2hkrk-gk2hjrjgk2gj2λk-λj,σsj=2kjckgjqk+gkqj-12kjckgkqj-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+2lnΔxx,Vε=v+2lnΔxy,Wε=v+2lnΔxz,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+1c1εμ1,2c1εμ1,i-1c1εμ1,ic1εμ1,i+1c1εμ1,nc2εμ1,2c2εf2+1c2εμ2,i-1c2εμ2,ic2εμ2,i+1c2εμ2,nci-1εμ1,i-1ci-1εμ2,i-1ci-1εfi-1+1ci-1εμi-1,ici-1εμi-1,i+1ci-1εμi-1,nμ1,iμ2,iμi,i-1fiμi,i+1μi,nci+1εμ1,i+1ci+1εμ2,i+1ci+1εμi-1,i+1ci+1εμi,i+1ci+1εfi+1+1ci+1εμi+1,ncnεμ1,ncnεμ2,ncnεμi-1,ncnεμi,ncnεμi+1,ncnεfn+1.and(26)D=c1εf1+1c1εμ1,2c1εμ1,jc1εμ1,nc2εμ1,2c2εf2+1c2εμ2,jc2εμ2,ncjεμ1,jcjεμ2,jcjεfj+1cjεμj,ncnεμ1,ncnεμ2,ncnεμj,ncnεfn+1,μi,j=gihj-gjhi2gigjλi-λj.

From the above Theorem 2, 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=2lnΔ1xx,v2=2lnΔ1xy,w2=a+2lnΔ1xz,Δ1=c1c2ε2f1f2-μ1,22+εc1f1+c2f2+1,and(30)u3=2lnΔ2xx,v3=2lnΔ2xy,w3=a+2lnΔ2xz,Δ2=c1εf1+1c1εμ1,2c1εμ1,3c2εμ1,2c2εf2+1c2εμ2,3c3εμ1,3c3εμ2,3c3εf3+1.

For illustration of more detail, 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. 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.

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.

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.

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 . While the CTE method is a special form of CRE and this approach is a generalization of the traditional tanh function expansion method . For the (3+1)-dimensional KdV-like equation (2) and (3), through introducing the tangent transformation (31)f=abtanhϕ+c,the truncated Painlevé expansion of (2) and (3) can be reduced(32)u=-2ϕx2tanh2ϕ+2ϕxxtanhϕ+ϕxx24ϕx2-ϕxxx2ϕx+ϕx2+λ,(33)v=-2ϕxϕytanh2ϕ+2ϕxytanhϕ+3αϕxx24βϕx2+ϕxxϕxyϕx2-αϕxxx+ϕt2βϕx-ϕxxy+2λϕyϕx+αϕx2β+2ϕxϕy,(34)w=-2ϕxϕztanh2ϕ+2ϕxztanhϕ+ϕxxϕxzϕx2-ϕxxz+2λϕzϕx+2ϕxϕz-3αλγ, where a,b,c are three constants and ϕϕ(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+ω2tk1x+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+12arctanhmsnX,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+k22mm2-1SNtanhϕ-k2CN1+m2SN22Ξ+k22mm2-1SN24Ξ2+λ,(43)v=1DN2Ξml2CN+2l1DN2cosh2ϕ+k2l2mm2-1SNtanhϕ-k22mm2-1CN1+m2SN2Ξl2+k2αβ+k23m2m2-12SN2Ξ2l2+3k2α4β-1Ξ2λl2mCN+2l1DN+mω2CN+2ω1DN2β+αΞ24βDN2,(44)w=1DN2Ξmm2CN+2m1DN2cosh2ϕ+k2mm2m2-1SNtanhϕ-k22mm2m2-1CN1+m2SN2Ξ+k23m2m2m2-12SN2Ξ2-2λmm2CN+2m1DNΞ-3αλγ.with SNsn(X,m),CNcn(X,m),DNdn(X,m),Ξk2mCN+2k1DN, and Xk2x+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 .

The u,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 , rogue waves, bright-dark solitons, and traveling wave solutions of the (3+1)-dimensional generalized KP equation  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.

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant Nos. 11705077, 11775104, and 11447017), the Natural Science Foundation of Zhejiang Province (Grant No. LY14A010005), and the Scientific Research Foundation of the First-Class Discipline of Zhejiang Province (B) (No. 201601).

Jia J. Lin J. Solitons in nonlocal nonlinear kerr media with exponential response function Optics Express 2012 20 7 7469 7479 Lin J. Chen W. W. Jia J. Abundant soliton solutions of general nonlocal nonlinear Schrödinger system with external field Journal of the Optical Society of America A 2014 31 1 188 195 Dai C.-Q. Wang Y. Liu J. Spatiotemporal Hermite-Gaussian solitons of a (3+1)-dimensional partially nonlocal nonlinear Schrodinger equation Nonlinear Dynamics 2016 84 3 1157 1161 10.1007/s11071-015-2560-9 MR3486561 Dai C.-Q. Fan Y. Zhou G.-Q. Zheng J. Chen L. Vector spatiotemporal localized structures in (3+1)-dimensional strongly nonlocal nonlinear media Nonlinear Dynamics 2016 86 2 999 1005 10.1007/s11071-016-2941-8 MR3546555 Kong L. Q. Liu J. Jin D. Q. Ding D. J. Dai C. Q. Soliton dynamics in the three-spine -helical protein with inhomogeneous effect Nonlinear Dynamics 2017 87 83 92 Dai C.-Q. Zhou G.-Q. Chen R.-P. Lai X.-J. Zheng J. Vector multipole and vortex solitons in two-dimensional Kerr media Nonlinear Dynamics 2017 88 4 2629 2635 10.1007/s11071-017-3399-z MR3656543 Wang Y.-Y. Chen L. Dai C.-Q. Zheng J. Fan Y. Exact vector multipole and vortex solitons in the media with spatially modulated cubic-quintic nonlinearity Nonlinear Dynamics 2017 90 2 1269 1275 10.1007/s11071-017-3725-5 MR3716663 Wang J. Y. Cheng X. P. Tang X. Y. Yang J. R. Ren B. Oblique propagation of ion acoustic soliton-cnoidal waves in a magnetized electron-positron-ion plasma with superthermal electrons Physics of Plasmas 2014 21 3 032111 Wang J. Y. Cheng X. P. Zeng Y. Zhang Y. X. Ge N. Y. Quasi-soliton solution of Korteweg-de Vries equation and its application in ion acoustic waves Acta Physica Sinica 2018 67 110201 Farmer D. M. Smith J. D. Tidal interaction of stratified flow with a sill in knight inlet Deep Sea Research 1980 27A 239 254 Akylas T. R. Grimshaw R. H. Solitary internal waves with oscillatory tails Journal of Fluid Mechanics 1992 242 279 298 10.1017/S0022112092002374 MR1184524 Zbl0754.76014 Ding D. J. Jin D. Q. Dai C. Q. Analytical solutions of differential-difference sine-Gordon equation Thermal Science 2017 21 4 1701 1705 10.2298/TSCI160809056D Williams G. Kourakis I. Nonlinear dynamics of multidimensional electrostatic excitations in nonthermal plasmas Plasma Physics and Controlled Fusion 2013 55 5 055005 Singh S. V. Devanandhan S. Lakhina G. S. Bharuthram R. Effect of ion temperature on ion-acoustic solitary waves in a magnetized plasma in presence of superthermal electrons Physics of Plasmas 2013 20 012306 Wang J. Y. Tang X. Y. Lou S. Y. Gao X. N. Jia M. Nanopteron solution of the Korteweg-de Vries equation with an application in ion-acoustic waves Europhysics Letters 2014 108 20005 Wang Y.-Y. Zhang Y.-P. Dai C.-Q. Re-study on localized structures based on variable separation solutions from the modified tanh-function method Nonlinear Dynamics 2016 83 3 1331 1339 10.1007/s11071-015-2406-5 MR3449473 Gu C. Hu H. Zhou Z. Darboux Transformations in Integrable Systems Theory and Their Applications to Geometry: Mathematical Physics Studies 2005 Dordrecht, Netherlands Springer-Verlag Press MR2174988 Lou S. Y. Hu X. B. Non-local symmetries via Darboux transformations Journal of Physics A: Mathematical and Theoretical 1997 30 5 L95 L100 10.1088/0305-4470/30/5/004 Zbl1001.35501 Hu X. R. Lou S. Y. Chen Y. Explicit solutions from eigenfunction symmetry of the Korteweg-de Vries equation Physical Review E 2012 85 056607 Rogers C. Schief W. K. Bäcklund and Darboux Transformations, Geometry and Modern Applications in Soliton Theory 2002 Cambridge, UK Cambridge University Press Cambridge Texts in Applied Mathematics 10.1017/CBO9780511606359 MR1908706 Zbl1019.53002 Gao X. N. Lou S. Y. Tang X. Y. Bosonization, singularity analysis, nonlocal symmetry reductions and exact solutions of supersymmetric KdV equation Journal of High Energy Physics 2013 2013, article 29 10.1007/JHEP05(2013)029 Olver P. J. Evolution equations possessing infinitely many symmetries Journal of Mathematical Physics 1977 18 6 1212 1215 10.1063/1.523393 MR521611 Zbl0348.35024 Olver P. J. Applications of Lie Groups to Differential Equations 1993 2nd New York, NY, USA Springer-Verlag Press Fokas A. S. Symmetries and integrability Studies in Applied Mathematics 1987 77 3 253 299 MR1002293 10.1002/sapm1987773253 Zbl0639.35075 2-s2.0-0023542072 Lou S. Y. Residual symmetries and Bäcklund transformations 2013, https://arxiv.org/abs/1308.1140 Lou S. Y. Consistent Riccati expansion for integrable systems Studies in Applied Mathematics 2015 134 3 372 402 10.1111/sapm.12072 MR3322699 Zbl1314.35145 2-s2.0-84925012331 Wazwaz A.-M. Compacton solutions of higher order nonlinear dispersive KdV-like equations Applied Mathematics and Computation 2004 147 2 449 460 10.1016/S0096-3003(02)00738-5 MR2012585 Zbl1045.35071 Wang X.-B. Tian S.-F. Xua M.-J. Zhang T.-T. On integrability and quasi-periodic wave solutions to a (3+1)-dimensional generalized KdV-like model equation Applied Mathematics and Computation 2016 283 216 233 10.1016/j.amc.2016.02.028 MR3478598 Weiss J. Tabor M. Carnevale G. The Painlevé property for partial differential equations Journal of Mathematical Physics 1983 24 3 522 526 10.1063/1.525721 MR692140 2-s2.0-36749112076 Lou S. Y. KdV extensions with Painleve property Journal of Mathematical Physics 1998 39 4 2112 2121 10.1063/1.532298 MR1614770 Zhang S.-L. Tang X.-Y. Lou S.-Y. High-dimensional Schwarzian derivatives and Painleve integrable models Communications in Theoretical Physics 2002 38 5 513 516 10.1088/0253-6102/38/5/513 MR1963698 Chen C. L. Lou S. Y. CTE solvability and exact solution to the broer-kaup system Chinese Physics Letters 2013 30 11 110202 Tian S.-F. Initial-boundary value problems of the coupled modified Korteweg--de Vries equation on the half-line via the Fokas method Journal of Physics A: Mathematical and General 2017 50 395204 MR3708079 Tian S. F. Initial-boundary value problems for the coupled modified Korteweg-de Vries equation on the interval Communications on Pure and Applied Analysis 2018 17 3 923 957 10.3934/cpaa.2018046 Peng W.-Q. Tian S.-F. Zhang T.-T. Analysis on lump, lumpoff and rogue waves with predictability to the (2+1)-dimensional B-type Kadomtsev-Petviashvili equation Physics Letters A 2018 382 38 2701 2708 10.1016/j.physleta.2018.08.002 MR3842626 Qin C. Y. Tian S. F. Wang X. B. Zhang T. T. Li J. Rogue waves, bright-dark solitons and traveling wave solutions of the (3+1)-dimensional generalized Kadomtsev-Petviashvili equation Computers & Mathematics with Applications 2018 75 4221 4231