Integrability,multi-soliton and rational solutions,and dynamical analysis for a relativistic Toda lattice system with one perturbation parameter

Under investigation in this paper is a relativistic Toda lattice system with one perturbation parameter $alpha$ abbreviated as RTL_($alpha$) system by Suris, which may describe the motions of particles in lattices interacting through an exponential interaction force. First of all, an integrable lattice hierarchy associated with an RTL_($alpha$) system is constructed, from which some relevant integrable properties such as Hamiltonian structures, Liouville integrability and conservation laws are investigated. Secondly, the discrete generalized (m, 2N − m)-fold Darboux transformation is constructed to derive multi-soliton solutions, higher-order rational and semi-rational solutions, and their mixed solutions of an RTL_($alpha$) system. The soliton elastic interactions and details of rational solutions are analyzed via the graphics and asymptotic analysis. Finally, soliton dynamical evolutions are investigated via numerical simulations, showing that a small noise has very little effect on the soliton propagation. These results may provide new insight into nonlinear lattice dynamics described by RTL_($alpha$) system.