N-Fold Darboux Transformation and Soliton Solutions for The relativistic Toda Lattice Equation

Abstract

Under investigation in this paper is the relativistic Toda lattice (RTL) equation which is a deformation of the Toda lattice (TL) equation and may simulate many physical phenomena. The N -fold Darboux transformation (DT) of the RTL equation is constructed in terms of determinants. Compared with the usual 1-fold DT, this kind of N-fold DT enables us to generate the multi-soliton solutions without complicated recursive process. Based on the N-fold DT, we obtain the N-fold explicit solutions from initial solutions. The figures of one-, two-, three-and four-soliton solutions with proper parameters are presented to illustrate the propagation of solitary waves, and elastic interactions between or among the two-, three- and four-soliton solutions are discussed: solitonic shapes and amplitudes have not changed after the interaction. What is more, the relationship between the structures of solutions and the parameters is generated with N = 1, from which we find that the 1-fold solutions may be one-soliton solutions or periodic solutions. Results in this paper might be helpful for understanding and interpreting some physical phenomena.