Soliton solutions of q-Toda lattice by Hirota direct method

Abstract

This paper presents the q-analogue of Toda lattice system of difference equations by discussing the q-discretization in three aspects: differential-q-difference, q-difference-q-difference and q-differential-q-difference Toda equation. The paper develops three-q-soliton solutions, which are expressed in the form of a polynomial in power functions, for the differential-q-difference and q-difference-q-difference Toda equations by Hirota direct method. Furthermore, it introduces q-Hirota D-operator and presents the q-differential-q-difference version of Toda equation. Finally, the paper presents its solitary wave like a solution in terms of q-exponential function and explains the nonexistence of further solutions in terms of q-exponentials by the virtue of Hirota perturbation.