A Relativistic Toda Lattice Hierarchy, Discrete Generalized (<i>m</i>,2<i>N</i>−<i>m</i>)-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.

Keywords

• relativistic Toda lattice system
• discrete generalized (<i>m</i>,2<i>N</i>−<i>m</i>)-fold Darboux transformation
• soliton and rational solutions
• hybrid solutions
• asymptotic analysis