Rational solutions for the potential nonlinear lumped self-dual network equation

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Abstract

In this letter we investigate the potential nonlinear lumped self-dual network equation, which is always used to model one-dimensional anharmonic chain of atoms. By the bilinear method we construct the rational solutions for the aforesaid equation. These solutions are presented in terms of Casoratian. The dynamical behaviors for the first three non-trivial rational solutions are analyzed with graphical illustration.

Introduction

The study of rational solutions for the discrete integrable systems has become in recent years a focus of attention within the theory of integrable systems. For instance, the spatial-temporary localized waves are used to describe rogue waves [1]. Up to now, a vast variety of works have been done to investigate the rational solutions of the discrete Korteweg–de Vries (KdV) type equations. Using the long wave limits procedure, Cârstea et al. studied non-singular rational solutions for the one-dimensional generalized Volterra system [2] and Toda lattice [3]. Applying Hirota’s bilinear formalism and Bäcklund transformation, Hu and Clarkson [4] gave the nonlinear superposition formulae of rational solutions for three discrete KdV type equations, including differential–difference KdV equation, Toda lattice and a discrete KdV equation. Without taking long wave limits in soliton solutions, rational solutions in the Casoratian form were derived through the Taylor expansions of the generating functions of soliton solutions of some discrete integrable equations, such as the Toda lattice and the differential–difference KdV equation [5], [6].

Compared with the case for continuous/discrete KdV type equations, it is more difficult to get rational solutions of the continuous/discrete modified KdV type equations. For the KdV type equations, in the construction of rational solutions by using the Wronskian/Casoratian technique, the element vector should satisfy a linear differential/difference equation set, in which the coefficient matrix always has only multiple zero eigenvalues (please see review paper [7] and references therein). Unfortunately, for the mKdV type equation no rational solutions arise from this perspective, which means that such coefficient matrix is not allowed to appear in the differential/difference equation set. To eliminate this defect, a Galilean transformation was applied to transform the mKdV equation into the KdV–mKdV equation, and the question about how to derive the rational solutions of the former equation turned to those of the latter one [8], [9].

The potential nonlinear lumped self-dual network equation (pNLSNE) reads [10] tun=tanun12un+12,which has also been proposed by discretizating bilinear formalism for the potential mKdV equation [11]. Thus Eq. (1.1) is also regarded to be a discrete version of the potential mKdV equation. Differentiation of the left and right sides of Eq. (1.1) over time yields the equation for the charge of the nth lattice site: ttun=(1+(tun)2)(tan(un1un)tan(unun+1)),which was initialed to describe the transmission line with the nonlinear capacitances and inductances. This equation was also used to model one-dimensional anharmonic chain of atoms, where un is the displacement of the nth atom in the chain. Eq. (1.2) has been got considerable attentions as evidenced by a series of papers [12], [13], [14], [15], [16]. Very recently, motivated by the way for constructing rational solutions proposed in Refs. [8], [9], rational solutions for (1.2) have been constructed [17].

In the present letter we are interested in the construction of rational solutions for the pNLSNE (1.1) with the help of bilinear method. Since the solutions to the pNLSNE (1.1) also solve Eq. (1.2), in this letter we will present some new solutions for Eq. (1.2), including soliton solutions, Jordan-block solutions and rational solutions. These solutions are different from those in [17]. The paper is organized as follows. In Section 2, we solve the pNLSNE equation (1.1) by means of bilinear method and derive its rational solutions in Casoratian form. In Section 3, dynamics for the first three non-trivial rational solutions are analyzed by using asymptotic analysis. Section 4 is devoted to the conclusions.

Section snippets

Casoratian and lower triangular Toeplitz matrices

Casoratian can be viewed as a discrete version of Wronskian, which is a determinant of a Casorati matrix C(ϕ1(n),ϕ2(n),,ϕN(n))=|ϕ(n),ϕ(n+1),,ϕ(n+N1)|,where ϕ(n)=(ϕ1(n),ϕ2(n),,ϕN(n))T is called the element vector of the Casoratian. Using the standard short-hand notations [18], we denote C(ϕ)=|ϕ(n),ϕ(n+1),,ϕ(n+N1)|=|0,1,,N1|=|N1̂|,where Nĵ indicates the set of consecutive columns 0,1,,Nj. With this notation it is easy to understand |N2̂,N|=|0,1,,N2,N|, etc.

The lower triangular

Dynamics

In this section, we investigate the dynamics for the first three non-trivial rational solutions, which are given by (2.19b), (2.19c), (2.19d). By the transformations (2.3), (2.5) and noting that i2lnfnfn=arctanImfnRefn=π2arctanRefnImfn,the solution for pNLSNE (1.1) can be expressed as un=π2+narctanc2+2cz4+c2arctanRefnImfn,z=1+c24t.

From (2.19), we know that the first three non-trivial rational solutions read un=π2+narctanc2+2cz4+c2arctan(c(1+2n+2z)),un=π2+narctanc2+2cz4+c2+arctan(2c(4+c2)(1+

Conclusions

By applying the bilinear method, in this letter we study the rational solutions for the pNLSNE (1.1), which is an integrable discrete version of the potential mKdV equation. The technique to construct Casoratian solutions in this paper was initially motivated by the procedure used to derive the rational solutions for the mKdV equation and Eq. (1.2). Except for rational solutions of Eq. (1.1), soliton solutions and Jordan-block solutions are also presented. Dynamics of some rational solutions

Acknowledgments

The authors are very grateful to the referees for their invaluable comments. This project is supported by the Natural Science Foundation of Zhejiang Province, China (No. LY18A010033) and the National Natural Science Foundation of China (No. 11401529).

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