# Nonlocal Symmetry and Bäcklund Transformation of a Negative-Order Korteweg–de Vries Equation

Authors: Jinxi Fei; Weiping Cao; Zhengyi Ma
Journal: Complexity
 (2019)

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## Abstract

The residual symmetry of a negative-order Korteweg–de Vries (nKdV) equation is derived through its Lax pair. Such residual symmetry can be localized, and the original nKdV equation is extended into an enlarged system by introducing four new variables. By using Lie’s first theorem, we obtain the finite transformation for the localized residual symmetry. Furthermore, we localize the linear superposition of multiple residual symmetries and constructn-th Bäcklund transformation for this nKdV equation in the form of the determinants.


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## Body

## 1. Introduction

It is well known that infinitely many symmetries or flows appear for finding evolution equations, and a general method was proposed by Olver, which was applied to the Korteweg–de Vries (KdV), modified Korteweg–de Vries (mKdV), Burgers, and sine-Gordon equations [1, 2]. Then, Olver’s concept of a recursion operator for symmetries of an evolution equation was extended to negative powers of the operator, and some negative-order Korteweg–de Vries (nKdV) equations were derived, including the following nKdV equation [3–12]:(1)ux−vt=0,uxxx+4uxv+2uvx=0.Several years later, Lou reobtained the negative KdV hierarchy from the conformal invariance of its Schwartz form [13]. This new method used for the KdV equation can be extended to get more symmetries from known ones for arbitrary partial differential equations without using a recursion operator. Also, the theory can be used as a new criterion to verify whether a model is integrable or not. Based on the regular KdV system, Qiao originally studied nKdV equations, particularly their Hamiltonian structures, Lax pairs, conservation laws, and explicit multisoliton and multikink wave solutions through bilinear Bäcklund transformations [14–17]. By using the simplified form of Hirota’s direct method, Wazwaz developed the nKdV equation and negative-order Kadomtsev–Petvishvili (nKP) equation in 2 + 1 dimensions and found multiple soliton solutions with free coefficients [18]. However, Kudryashov showed that existence of multisoliton solutions for the nonlinear evolution equation is the consequence of complete integrability [19, 20]. According to Theorem 1 in Reference [14], the nKdV equation (1) admits a Lax pair with the parameter λ as follows:(2)Lψ≡ψxx+vψ=λψ,ψt=−12λuψx+14λuxψ,and also possesses a Lax pair without the parameter as follows:(3)Lψ=∂x2+vψ=0,Mψ=4∂x2∂t+4v∂t+2u∂x+3uxψ=0.This paper is organized as follows: In Section2, with the aid of the Lax pair, the residual symmetry of the nKdV equation (1) is derived, and this nonlocal symmetry is localized by introducing four 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 constructing the infinite transformation for the nKdV equation, multiple residual symmetries and n-th Bäcklund transformation are obtained. A direct result shows that one can derive special soliton solutions from some seed solutions. A brief summary is given in Section 4.


## 2. Nonlocal Symmetry and Finite Transformation of the nKdV Equation

First, under the infinitesimal transformationsu⟶u+εσ1 and v⟶v+εσ2 with the infinitesimal parameter ε, the symmetries σ1 and σ2 of the nKdV equation (1) can be expressed as a solution of their linear equations:(4a)σ1,x−σ2,t=0,(4b)σ1,xxx+4vσ1,x+2vxσ1+4uxσ2+2uσ2,x=0,which means that equation (1) is form invariant. At the same time, we can obtain the linear form of equation (2) under the infinitesimal transformation as follows:(5)ψ⟶ψ+εσ3.That is, the symmetric equations of the Lax pair (2) are(6a)σ3,xx−λσ3+vσ3+ψσ2=0,(6b)σ3,t+12λuσ3,x+ψxσ1−14λuxσ3+ψσ1,x=0.Second, supposing the symmetriesσ1 and σ2 with the auxiliary variable ψ as(7)σ1=ψψt,σ2=ψψx,the symmetry σ3 can be derived from equations (1), (6a)-(6b), and (7) as follows:(8)σ3=−fψ4,where the auxiliary variable ψ satisfies ψ2=fx, and the corresponding symmetric equation is 2ψσ3−σ4,x=0, where f is a auxiliary function and its infinitesimal transformation is f⟶f+εσ4. The direct result is(9)σ4=−f24.So, equation (1) can be expressed by the auxiliary variable ψ through equation (2) as(10a)u=−12ψψxψψxxt−ψxxψt+4λψψt,(10b)v=λψ−ψxxψ.The consistent condition of the auxiliary functionf is derived from equation (10) as(11)ft=14λψxψ2ψxxt−ψψxxψt+4λψ2ψt+4ψx2ψt−4ψψxψxt.Indeed, equation (11) has a typical Schwarzian form of the nKdV equation defined as follows:(12)St+4λCx=0,C=ftfx,S=fxxxfx−3fxx22fx2,which is invariant under the Möbius transformation(13)f⟶a+bfc+df,ad−bc≠0.The corresponding symmetric equation of equation (11) is(14)ψxxψt+4ψxψxt−2ψψxxt−8λψψtσ3+4λft+ψψxt−2ψxψtσ3,x+ψψxx−4ψx2−4λψ2σ3,t+ψψtσ3,xx+4ψψxσ3,xt−ψ2σ3,xxt+4λψxσ4,t=0.Since the nonlocal symmetry could not be used to construct the explicit solutions of a partial differential equation (PDE) directly, we need to transform these components to local ones. In this part, we will seek an enlarged system which possesses a Lie point symmetry for the nonlocal symmetry. For this purpose, we further introduce four auxiliary variablesg,h,p, and q, which need to obey the rule(15)ψt=g,ψx=h,ft=p,ht=q.The related symmetries are(16)σ3,t−σ5=0,σ3,x−σ6=0,σ4,t−σ7=0,σ6,t−σ8=0,under the infinitesimal transformations g⟶g+εσ5,h⟶h+εσ6,p⟶p+εσ7,andq⟶q+εσ8.It can be verified that equations (7)–(9), (14), and (16) have the following solution:(17)σ≡σ1,σ2,σ3,σ4,σ5,σ6,σ7,σ8=gψ,hψ,−fψ4,−f24,−14pψ+fg,−14ψ3+fh,−fp2,−143gψ2+ph+fq.This is a local Lie point symmetry of the prolonged equations (1), (2), and (15) with ψ2=fx.Correspondingly, the initial value problem can be written as follows:(18a)dUεdε=GεΨε,U0=u,dVεdε=HεΨε,V0=v,(18b)dΨεdε=−14FεΨε,Ψ0=ψ,dFεdε=−14F2ε,F0=f,(18c)dGεdε=−14PεΨε+FεGε,G0=g,dHεdε=−14Ψ3ε+FεHε,H0=h,(18d)dPεdε=−12FεPε,P0=p,dQεdε=−143GεΨ2ε+PεHε+FεQε,Q0=q.By solving this initial value problem, one can obtain the following finite transformation theorem.Theorem 1.
Ifu,v,ψ,f,g,h,p,q is a solution of the extended system (1), (2), and (15), so is(19a)Uε=u+4gψε4+fε−2pψ2ε24+fε2,Vε=v+4hψε4+fε−2ψ4ε24+fε2,(19b)Ψε=4ψ4+fε,Fε=4f4+fε,(19c)Gε=4g4+fε−4pψε4+fε2,Hε=h4+fε−ψ3ε4+fε2,(19d)Pε=16p4+fε2,Qε=4q4+fε−4ph+3gψ2ε4+fε2+8pψ3ε24+fε3,where ε is an arbitrary group parameter.


## 3. Bäcklund Transformation Related to Multiple Residual Symmetries

Considering the intrusion of the spectral parameterλ in the nonlocal symmetry of equation (7), we can derive infinitely many residual symmetries of the fields u and v, that is,(20)σ1=∑i=1nciψiψi,t,σ2=∑i=1nciψiψi,x,where ψii=1,2,…,n are spectral functions of the Lax pair in equation (2) with different spectral parameters λi≠λji≠j.Just as the case ofn=1, to find the finite transformation of equation (20), we have to introduce a suitable prolonged system such that the symmetry can be localized to a Lie point symmetry. The corresponding finite transformation can be summarized as the following theorem.Theorem 2.
Ifu,v,ψi,fi,gi,hi,pi,qi,i=1,2,…,n is a solution of the enlarged system(21a)ux−vt=0,uxxx+4uxv+2uvx=0,(21b)ψi,xx+vψi−λiψi=0,(21c)ψi,t+12λiuψi,x−14λiuxψi=0,(21d)ψi2−fi,x=0,4λifi,tψi,x−ψi2ψi,xxt+ψiψi,xxψi,t−4λiψi2ψi,t−4ψi,x2ψi,t+4ψiψi,xψi,xt=0,(21e)ψi,t−gi=0,ψi,x−hi=0,fi,t−pi=0,hi,t−qi=0,then the symmetry (20) is localized to a Lie point symmetry as follows:(22a)σ1=∑i=1ncigiψi,σ2=∑i=1ncihiψi,(22b)σ3i=−14cifiψi−14∑j≠icjψjhiψj−hjψiλi−λj,σ4i=−14cifi2−14∑j≠icjhiψj−hjψi2λi−λj2,(22c)σ5i=−14cipiψi−figi−14∑j≠icjgjhiψj−hjψi+ψjqiψj−qjψi+ψjgjhi−gihjλi−λj,(22d)σ6i=−14ciψi3+fihi−14∑j≠icjhjhiψj−hjψiλi−λj+ψiψj2,(22e)σ7i=−12cifipi−12∑j≠icjhiψj−hjψiqiψj−qjψi+gjhi−gihjλi−λj2,(22f)σ8i=−14ci3giψi2+pihi+fiqi−14∑j≠icjqjhiψj−hjψi+hjgjhi−gihj+hjqiψj−qjψiλi−λj+ψj2gi+2ψjψigj.Proof.
The enlarged system (21a)–(21e) has the following linearized form:(23a)σ1,x−σ2,t=0,σ1,xxx+4vσ1,x+2vxσ1+4uxσ2+2uσ2,x=0,(23b)σ3i,xx−λiσ3i+vσ3i+ψiσ2=0,(23c)σ3i,t+12λiuσ3i,x+ψi,xσ1−14λiuxσ3i+ψiσ1,x=0,(23d)ψi,xxψi,t+4ψi,xψi,xt−2ψiψi,xxt−8λiψiψi,tσ3i+4λifi,t+ψiψi,xt−2ψi,xψi,tσ3i,x+ψiψi,xx−4ψi,x2−4λiψi2σ3i,t+ψiψi,tσ3i,xx+4ψiψi,xσ3i,xt−ψi2σ3i,xxt+4λiψi,xσ4i,t=0,(23e)2ψiσ3i−σ4i,x=0,σ3i,t−σ5i=0,σ3i,x−σ6i=0,σ4i,t−σ7i=0,σ6i,t−σ8i=0,i=1,2,…,n.
We first consider the special case, i.e., for any fixedi=k,ck≠0 while cj=0j≠k in equations (22a)–(22f). In this case, we obtain the localized symmetry for u,v,ψk,fk,gk,hk,pk, and qk from equation (17) as follows:(24)σ≡σ1,σ2,σ3k,σ4k,σ5k,σ6k,σ7k,σ8k=ckgkψk,ckhkψk,−ckfkψk4,−ckfk24,−ck4pkψk+fkgk,−ck4ψk3+fkhk,−ckfkpk2,−ck43gkψk2+pkhk+fkqk.
Forj≠k, we eliminate v through equation (21b) by taking i=k and i=j, respectively. Then, we have(25)ψj,xx=ψjψk,xxψk−λk−λjψj.
Substitutingσ2=ckhkψk, into equation (23b) with i=j and vanishing ψj,xx through equation (25), we have(26)λk−λj+ψj,xxψjσ3j−σ3j,xx−cjψkψjψj,x=0.
It can be easily verified that equation (26) has the following solution:(27)σ3j=−cjψjhkψj−hjψk4λk−λj.
The symmetry forσ4j,σ5j,σ6j,σ7j, and σ8j can be easily obtained from equation (23e) with i=j:(28a)σ4j=−cjhjψk−hkψj24λk−λj2,σ5j=−cjgjhkψj−hjψk+ψjqkψj−qjψk+ψjgjhk−gkhj4λk−λj,(28b)σ6j=−14cjhjhkψj−hjψkλk−λj+ψkψj2,σ7j=−12cjhkψj−hjψkqkψj−qjψk+gjhk−gkhjλi−λj2,(28c)σ8j=−14cjqjhkψj−hjψk+hjgjhk−gkhj+hjqkψj−qjψkλk−λj+ψj2gk+2ψjψkgj.
After taking the linear combination of the above results for allk=1,2,…,n, Theorem 2 is proved.
When a nonlocal symmetry is localized to a Lie point symmetry, searching for its finite transformation is inevitable according to Lie’s first principle. For the Lie point symmetry (22a)–(22f), its initial value problem has(29a)dUεdε=∑i=1nciGiεΨiε,U0=u,dVεdε=∑i=1nciHiεΨiε,V0=v,(29b)dΨiεdε=−14ciFiεΨiε−14∑j≠icjΨjεHiεΨjε−HjεΨiελi−λj,Ψi0=ψi,(29c)dFiεdε=−14ciFi2ε−14∑j≠icjHiεΨjε−HjεΨiε2λi−λj2,Fi0=fi,(29d)dGiεdε=−14ciPiεΨiε−FiεGiε−14∑j≠icjλi−λiGjεHiεΨjε−HjεΨiε+ΨjεQiεΨjε−QjεΨiε+ΨjεGjεHiε−GiεHjε,Gi0=gi,(29e)dHiεdε=−14ciΨi3ε+FiεHiε−14∑j≠icjHjεHiεΨjε−HjεΨiελi−λj+ΨiεΨj2ε,Hi0=hi,(29f)dPiεdε=−12ciFiεPiε−12∑j≠icjλi−λj2HiεΨjε−HjεΨiεQiεΨjε−QjεΨiε+GjεHiε−GiεHjε,Pi0=pi,(29g)dQiεdε=−14ci3GiεΨi2ε+PiεHiε+FiεQiε−14∑j≠icj1λi−λjQjεHiεΨjε−HjεΨiε+HjεGjεHiε−GiεHjε+HjεQiεΨjε−QjεΨiε,dQiεdε=−14ci3GiεΨi2ε+PiεHiε+FiεQiε−14∑j≠icjQjεHiεΨjε−HjεΨiελi−λj+HjεGjεHiε−GiεHjελi−λj+HjεQiεΨjε−QjεΨiελi−λj+Ψj2εGiε+2ΨjεΨiεGjε,Qi0=qi.
Then, one can get the followingn-th Bäcklund theorem for the enlarged system (21a)–(21e) by solving (29a)–(29g).Theorem 3.
Ifu,v,ψi,fi,gi,hi,pi,qi,i=1,2,…,n is a solution of the prolonged nKdV equations (21a)–(21e), so is Uε,Vε,Ψiε,Fiε,Giε,Hiε,Piε,Qiε,i=1,2,…,n, where(30a)Uε=u+2lnΔxt,Vε=v+2lnΔxx,(30b)Ψiε=2ΓiΔ,Fiε=2ΔiΔ,Giε=Ψi,tε,(30c)Hiε=Ψi,xε,Piε=Fi,xε,Qiε=Ψi,xtε,where Δ,Δi, and Γi are three determinants of the matrices D,Di, and Ei, respectively, which are defined as follows:(31)D=c1εf1+4c1εμ1,2⋯c1εμ1,j⋅⋅⋅c1εμ1,nc2εμ1,2c2εf2+4⋅⋅⋅c2εμ2,j⋅⋅⋅c2εμ2,n⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅cjεμ1,jcjεμ2,j⋅⋅⋅cjεfj+4⋅⋅⋅cjεμj,n⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅cnεμ1,ncnεμ2,n⋅⋅⋅cnεμj,n⋅⋅⋅cnεfn+4,μi,j=hjψi−hiψjλi−λj,Di=c1εf1+4c1εμ1,2⋯c1εμ1,i−1c1εμ1,ic1εμ1,i+1⋯c1εμ1,nc2εμ1,2c2εf2+4⋯c2εμ2,i−1c2εμ2,ic2εμ2,i+1⋯c2εμ2,n⋯⋯⋯⋯⋯⋯⋯⋯ci−1εμ1,i−1ci−1εμ2,i−1⋯ci−1εfi−1+4ci−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+4⋯ci+1εμi+1,n⋯⋯⋯⋯⋯⋯⋯⋯cnεμ1,ncnεμ2,n⋯cnεμi−1,ncnεμi,ncnεμi+1,n⋯cnεfn+4,Ei=c1εf1+4c1εμ1,2⋯c1εμ1,i−1c1εμ1,ic1εμ1,i+1⋯c1εμ1,nc2εμ1,2c2εf2+4⋯c2εμ2,i−1c2εμ2,ic2εμ2,i+1⋯c2εμ2,n⋯⋯⋯⋯⋯⋯⋯⋯ci−1εμ1,i−1ci−1εμ2,i−1⋯ci−1εfi−1+4ci−1εμi−1,ici−1εμi−1,i+1⋯ci−1εμi−1,nψ1ψ2⋯ψi−1ψiψi+1⋯ψnci+1εμ1,i+1ci+1εμ2,i+1⋯ci+1εμi−1,i+1ci+1εμi,i+1ci+1εfi,i+1+4⋯ci+1εμi+1,n⋯⋯⋯⋯⋯⋯⋯⋯cnεμ1,ncnεμ2,n⋯cnεμi−1,ncnεμi,ncnεμi+1,n⋯cnεfn+4.From Theorem 3, we can obtain an infinite number of new solutions from a suitable seed solution of the nKdV equation (1). Especially, one can obtain recursive soliton solutions of this system from the known one. For example, taking the seed solution u=αandv=β for equation (1), it is not difficult to verify that equations (21a)–(21e) possess the following solution:(32)ψi=expξi2,fi=12kiexpξi,λi=ki2+β,μi,j=expξi+ξj/2ki+kj,ξi=2kix−αkitki2+β,i,j=1,2,…,n.The corresponding first three multiple wave solutions for equation (1) are(33)Δ1≡Δ=c1εf1+4,(34a)u=α+2lnΔ1xt=α−32c1k13αεexpξ1k12+β8k1+c1εexpξ12,(34b)v=β+2lnΔ1xx=β+64c1k13εexpξ18k1+c1εexpξ12,for the line soliton solution:(35)Δ2≡Δ=c1εf1+4c2εf2+4−c1c2ε2μ1,22=16+4c1εf1+4c2εf2+c1c2ε2k1−k22f1f2k1+k22,(36)u=α+2lnΔ2xt,v=β+2lnΔ2xx,(37)u=α+2lnΔ3xt,v3=β+2lnΔ3xx,(38)Δ3=c1εf1+4c1εμ1,2c1εμ1,3c2εμ1,2c2εf2+4c2εμ2,3c3εμ1,3c3εμ2,3c3εf3+4=64+16c1εf1+16c2εf2+16c3εf3+4c1c2ε2k1−k22f1f2k1+k22+4c1c3ε2k1−k32f1f3k1+k32+4c2c3ε2k2−k32f2f3k2+k32+c1c2c3ε3k1−k22k1−k32k2−k32f1f2f3k1+k22k1+k32k2+k32.For illustrating more details, the parameters are set as follows:a=ε=c1=c2=c3=k1=1,k2=1/2,andk3=2. Figure 1 displays the bell-like bright and dark solitons for the above condition of equations (34a)–(34b). Figure 1(a) shows a line dark soliton for u1 with the amplitude 1, while Figure 1(b) shows a bright one for v1 with the amplitude 2. Similarly, Figure 2 shows the collision of two-resonant solitons in equation (36), and Figure 3 shows interactions of three-resonant solitons expressed in equation (37).Figure 1
Plots of two line solitons expressed by equations (34a)–(34b) for solution of the nKdV equation (1) with the parameters a=ε=c1=k1=1.
(a)
(b)Figure 2
Interactions of two-resonant solitons expressed by equation (36) for solution of the nKdV equation (1) with the parameters a=ε=c1=c2=k1=1andk2=1/2.
(a)
(b)Figure 3
Interactions of three-resonant solitons expressed by equation (37) for solution of the nKdV equation (1) with the parameters a=ε=c1=c2=c3=k1=1,k2=1/2,andk3=2.
(a)
(b)


## 4. Summary

The nonlocal symmetry of the nKdV equation is obtained with the aid of its Lax pair. After introducing four auxiliary variablesg, h, p, and q, 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. Furthermore, we can localize the linear superposition of multiple residual symmetries and construct the infinite transformation for the nKdV equation. From Theorem 3, the n-th Bäcklund transformation can be expressed in a compact way of determinants. According to this conclusion, one can derive special soliton solutions from some seed solutions.



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