Research paper
Positive and negative integrable lattice hierarchies: Conservation laws and Nfold Darboux transformations

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Highlights

  • Construct positive and negative integrable lattice hierarchies and corresponding Hamiltonian structures.

  • Derive the infinite number of conservation laws.

  • Establish N-fold Darboux transformations.

  • Present explicit exact solutions and plot figures to illustrate the propagation of solitary waves.

Abstract

Positive and negative nonlinear integrable lattice hierarchies are established starting from a discrete matrix spectral problem with three potential functions, particularly the corresponding Hamiltonian structures are presented respectively with the help of the trace identity, all these facts show that these two hierarchies are integrable in Liouville sense. By using Lax pair we derive infinite number of conservation laws and Nfold Darboux transformation (DT) for the first nontrivial system in the two hierarchies. Comparing with the usual 1fold DT, this kind of Nfold DT enables us to generate the multi-soliton solutions without complicated recursive process. As applications, we derive Nfold explicit exact solutions from seed solutions and plot their figures with properly parameters to analyze and illustrate the propagation of solitary waves.

Introduction

It is well known that the integrable lattice equations have many applications in various scientific contexts ranging from mathematical physics, numerical analysis to quantum physics, as well as statistical physics [1], [2], [3], [4]. In the soliton theory, one of the central concerns is to generate as much new integrable lattice equations as possible and study their associated properties, which will provide clues for classifying integrable models. Up to now, many methods and theories of constructing integrable lattice equations have been proposed and developed, such as the discrete Zakharov-Shabat eigenvalue problem [2], [5], the r-matrix approach [6], [7] and the Tu scheme [8], [9]. Among these existing methods of construction, the Tu scheme has turned out to be one of efficient methods to derive integrable lattice equations. Following this technique, the Hamiltonian structures and Liouville integrability of the lattice equations can also be obtained by standard method. Since the Tu method was proposed, some new hierarchies of integrable lattice equations have been constructed and investigated [10], [11], [12], [13], [14].

For an integrable lattice equation, it is always a fundamental and significant topic to construct explicit exact solutions. The Darboux transformation (DT) has turned out to be one of efficient methods to derive the exact solutions of some nonlinear Lax integrable equations, the main idea of DT is to construct a gauge transformation to keep the Lax pair of equations invariant, and the exact solutions can be obtained in a recursive way [15], [16]. Specifically speaking, beginning with a seed solution of the equations and using the 1fold DT, one can get a new solution. Taking this new solution as another new seed solution, and using the 1fold DT once again, one can derive other new solution. Hence a series of explicit solutions can be generated by iteration step by step. However, with iterations increases, it is very difficult to carry out the iterative process to generate higher-order solution, thus the Nfold DT is developed [17], [18], [19], [20]. Although the Nfold DT can be interpreted as a nonlinear superposition of 1-fold DT, it doesn’t need to change the seed solution and enables us to generate the multi-soliton solutions without complicated recursive process. With the help of Nfold DT, the problem of solving a nonlinear equation can be finally reduced to that of solving a linear system. Based on Nfold DT, Wen, Yang and Yan [21]proposed generalized perturbation (n,M)fold DT and studied multi-rogue-wave structures for the modified self-steepening nonlinear Schro¨dinger equation. Recently, DT is also presented for more general integrable systems, called integrable couplings, associated with non-seminsimple Lie algebras [22].

Various lattice equations possessing such integrable properties as the Lax pair [23], conservation laws [24], [25], [26] and DT [27], [28], [29] are related to the discrete matrix spectral problem. A number of important spectral problems have been proposed and systematically investigated, for example, the Ablowitz-Ladik spectral problem [2], [30], the modified Toda spectral problem [12], the relativistic Toda spectral problem [13] and so on. In 2006, Sun, Chen and Xu [31] considered the following discrete spectral problemEφn=Unφn,Un=(0λpnλqn1+λ2pnqn).Starting from (1.1), they constructed a hierarchy of nonlinear differential-difference equations and established their Hamiltonian structures. Further, they presented a new Bargmann-type integrable systems which are completely integrable Hamiltonian system in Liouville sense. In 2011, R. Sahadevan and S. Balakrishnan [32] studied the following matrix spectral problemEφn=Unφn,Un=(0λpnλqn1+λ2snqn).Based on (1.2), they established a new three coupled nonlinear differential-difference equation and reported that the equation admits Lax representation, possesses infinitely many generalized (nonpoint) symmetries, conservation laws and a recursion operator. In 2012, Xue, Tian and Ai [33] investigated the following discrete matrix spectral problemEφn=Unφn,Un=(0λpnλqn1+λ2rn).From (1.3), an integrable nonlinear differential-difference hierarchy and its reductions were studied. In particular, when rn=pnqn, they constructed the infinitely many conservation laws, DT and some explicit exact solutions for the first nontrivial system in the hierarchy.

In this paper, motivated by the works mentioned above, we mainly pay attention to construct the three coupled nonlinear lattice equations starting from (1.3) and investigate their corresponding integrable properties, especially the infinite number of conservation laws, the Nfold DT and soliton solutions. The rest of the paper is organized as follows. In Section 2, from the discrete matrix spectral problem (1.3), we derive the positive and negative integrable lattice hierarchies and construct their Hamiltonian structures. Infinitely many conservation laws and the Nfold DT of the first nontrivial system in the two hierarchies are constructed in Sections 3 and 4, respectively. In Section 5, we generate the exact solutions of the system and present their figures with properly parameters to show the propagation of solitary waves. Some remarks and summary will be presented in the last section.

Section snippets

Positive and negative integrable lattice hierarchies and Hamiltonian structures

First, we specify some fundamental conceptions, please refer for details. The shift operator E, the inverse of E, the difference operator Δ and (E1)1 are defined as follows:Ef(n,t)=f(n+1,t)=fn+1,E1f(n,t)=f(n1,t)=fn1,f(n,t)=(E1)f(n,t)=fn+1fn,(E1)1=12(k=1Ekk=0Ek),where f(n, t) is an arbitrary function.

Infinitely Many Conservation Laws of (2.6) and (2.13)

Using the Lax pair, a systematic method which is called as the Riccati method is developed to establish infinitely many conservation laws for an integrable lattice equation, see [25], [26] for details. First, we review the definition of conservation law. For a lattice equationF(u˙n,u¨n,,un1,un,un+1,)=0,if there exist scalar functions ρn and Jn, such thatρ˙n|F=0=(E1)Jn,then (3.2) is called the conservation law of (3.1), where ρn is the conserved density and Jn is the associated flux.

For the

Nfold Darboux transformation

For an integrable lattice equations with Lax pair Eφn=Unφn, φn,t=Vnφn, if there is a gauge transformationφ˜n=Tn(λ)φnand a transformation between potential functions,r˜n=f(rn,sn),s˜n=g(rn,sn),such that φ˜n satisfiesEφ˜n=U˜nφ˜n,φ˜n,t=V˜nφ˜n,and the matrixes U˜n and V˜n with the potential functions r˜n,s˜n have the same structures as Un and Vn respectively, then the gauge transformation (4.1) together with the transformation (4.2) are called as a DT of the integrable lattice equations. Obviously,

Explicit exact solutions of (2.6)

Taking the seed solution pn=1,qn=1,rn=1 of (2.6) and solving the corresponding Lax pair Eφn=Unφn,φn,t=Vn(1)φn, we have two linearly independent solutions as(φ1,n(λ)φ2,n(λ))=(λ2n1e(λ221)tλ2ne(λ221)t),(ψ1,n(λ)ψ2,n(λ))=(λeλ22teλ22t).It follows from (4.9) and (4.14) thatσi,n=λi2ne(λi221)t+γieλi22tλi2n1e(λi221)t+γiλieλi22t,σi,n+1=1σi,n+1+λi2λi,i=1,2,,2N.Then, by Theorem 4.1, the Nfold solutions of (2.6) are obtained as followsp˜n=xn+1(N1)+wn+1(N)zn(N),q˜n=yn(N1)zn+1(N)+zn(N)zn+1(N)zn(N

Conclusions

In this paper, we derive the positive and negative nonlinear integrable lattice hierarchies starting from a discrete matrix spectral problem (1.3), and the Hamiltonian structures of the two hierarchies are constructed respectively by using the discrete trace identity, these facts mean that the two hierarchies are integrable in Liouville sense. Based on the Lax pair, we derive the infinite number of conservation laws and Nfold DT for the first nontrivial system (2.6) and (2.13) in the two

Credit author statement

Fangcheng Fan and Shaoyun Shi constructed the positive and negative non- linear integrable lattice hierarchies and Hamiltonian structures.

Fangcheng Fan and Zhiguo Xu derived infinite number of conservation laws and N-fold Darboux transformation.

Fangcheng Fan established N-fold explicit exact solutions from seed solu- tions and plotted their figures.

Fangcheng Fan wrote and edited the manuscript, Shaoyun Shi and Zhiguo Xu helped to revise the manuscript.

All authors gave final approval for

Declaration of Competing Interest

The authors declare that they have no known competing nancial interests or personal relationships that could have appeared to in uence the work reported in this paper.

Acknowledgments

This work was completed with the support by NSFC grant (No. 11771177, 11501242), China Automobile Industry Innovation and Development Joint Fund (No. U1664257), Program for Changbaishan Scholars of Jilin Province and Program for JLU Science, Technology Innovative Research Team (No. 2017TD-20), Science and Technology Development Project of Jilin Province (20200201264JC). The scientific research project of The Education Department of Fujian Province (JAT190369). President’s fund of Minnan Normal

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