Integrability on generalized $q$-Toda equation and hierarchy

TL;DR

This paper introduces a new integrable generalization of the $q$-Toda equation, constructs its soliton solutions, and establishes its integrability through Lax pairs, bi-Hamiltonian structure, and tau symmetry.

Contribution

It presents a novel generalized $q$-Toda equation, its hierarchy, and demonstrates integrability via Lax pairs, soliton solutions, and bi-Hamiltonian structure.

Findings

Construction of a new integrable generalized $q$-Toda equation

Derivation of soliton solutions confirming integrability

Establishment of Lax pairs and bi-Hamiltonian structure

Abstract

In this paper, we construct a new integrable equation which is a generalization of $q$-Toda equation. Meanwhile its soliton solutions are constructed to show its integrable property. Further the Lax pairs of the generalized $q$-Toda equation and a whole integrable generalized $q$-Toda hierarchy are also constructed. To show the integrability, the Bi-hamiltonian structure and tau symmetry of the generalized $q$-Toda hierarchy are given and this leads to the tau function.