A Relativistic Toda Lattice Hierarchy, Discrete Generalized (m,2N−m)-Fold Darboux Transformation and Diverse Exact Solutions

Abstract

This paper investigates a relativistic Toda lattice system with an arbitrary parameter that is a very remarkable generalization of the usual Toda lattice system, which may describe the motions of particles in lattices. Firstly, we study some integrable properties for this system such as Hamiltonian structures, Liouville integrability and conservation laws. Secondly, we construct a discrete generalized (m,2N−m)-fold Darboux transformation based on its known Lax pair. Thirdly, we obtain some exact solutions including soliton, rational and semi-rational solutions with arbitrary controllable parameters and hybrid solutions by using the resulting Darboux transformation. Finally, in order to understand the properties of such solutions, we investigate the limit states of the diverse exact solutions by using graphic and asymptotic analysis. In particular, we discuss the asymptotic states of rational solutions and exponential-and-rational hybrid solutions graphically for the first time, which might be useful for understanding the motions of particles in lattices. Numerical simulations are used to discuss the dynamics of some soliton solutions. The results and properties provided in this paper may enrich the understanding of nonlinear lattice dynamics.

Continuous Limit, Rational Solutions, and Asymptotic State Analysis for the Generalized Toda Lattice Equation Associated with 3 × 3 Lax Pair

“…, where t < 0, we can obtain the limit states of solutions u n , v n , w n as t → −∞, which are listed as follows: It should be noted here that the first-order rational solutions (12) of Equation ( 1) are significantly different from those of the equations with 2 × 2 Lax pairs in References [19,21]. The solutions of Equation ( 1) are obviously more complex than those of the equations with 2 × 2 Lax pairs in References [19,21] or even the 4 × 4 Lax pair in Reference [20], and the structures of the solutions (12) are not as symmetrical as those in References [19][20][21]. Remark 2.…”

“…In Reference [20], this generalized method is extended to obtain exact solutions of the discrete coupled Ablowitz-Ladik equation from a 2 × 2 matrix spectral problem to 4 × 4 matrix spectral problem. In Reference [21], this method is once again extended to the discrete generalized (m, 2N − m)-fold DT to obtain various exact solutions of a relativistic Toda lattice equation related to the 2 × 2 matrix spectral problem. However, this discrete generalized method has never been extended to solve the discrete integrable NDDEs related to the 3 × 3 Lax pair.…”