N-Fold Darboux transformation and solitonic interactions for a Volterra lattice system

Xiaoyong Wen,Xiaoyan Hu

doi:10.1186/1687-1847-2014-213

Abstract

Under consideration in this paper is a Volterra lattice system. Through symbolic computation, the Lax pair and conservation laws are derived, an integrable lattice hierarchy and an N-fold Darboux transformation (DT) are constructed for this system. Furthermore, N-soliton solutions in terms of determinant are generated with the resulting N-fold DT. Structures of the one-, two- and three-soliton solutions are shown graphically. Overtaking inelastic solitonic interactions between/among the two and three solitons are discussed by figures plotted.

Highlights

Explicit solutions of the nonlinear partial differential equations (NPDEs), in particular the soliton solutions, describe certain phenomena
A soliton is a localized nonlinear wave which has particle-like properties [ ]
Dynamical behaviors of the solitons in the continuous and discrete cases are described by the NPDEs and Nonlinear differentialdifference equations (NDDEs), respectively [ ]

Summary

Introduction

Explicit solutions of the nonlinear partial differential equations (NPDEs), in particular the soliton solutions, describe certain phenomena (see [ ] and references therein). A soliton is a localized nonlinear wave which has particle-like properties [ ]. Nonlinear differentialdifference equations (NDDEs), taken as spatially discrete analogues of the NPDEs, have received certain attention [ – ]. Studies on the solitons might be divided into two categories, i.e., the continuous and discrete (lattice) cases [ ]. Dynamical behaviors of the solitons in the continuous and discrete cases are described by the NPDEs and NDDEs, respectively [ ]. The Toda lattice [ ] is the discrete approximation of the Korteveg-de Vries (KdV) equation in fluids; the discrete nonlinear Schrödinger equation [ ] can describe the interaction and propagation of optical pulses in a nonlinear waveguide array; the Volterra lattice system [ , – ] is in connection with the spectrum of Langmuir wave in plasma dynamics

Objectives

Conclusion