q-Viscous Burgers' Equation: Dynamical Symmetry, Shock Solitons and q-Semiclassical Expansion

TL;DR

This paper introduces a novel q-viscous Burgers' equation with exact polynomial solutions, explores its dynamical symmetry, constructs shock solitons, and analyzes how the parameter q influences soliton interactions and speeds.

Contribution

It presents a new q-diffusive heat equation, derives exact polynomial solutions, and investigates the effects of q on shock solitons and their dynamics, including a q-semiclassical expansion.

Findings

Exact polynomial solutions in generalized Kampe de Feriet form.

q modifies soliton speeds and interactions, especially for q<1.

Discovery of a new soliton fission phenomenon at high amplitude.

Abstract

We propose new type of $q$-diffusive heat equation with nonsymmetric $q$-extension of the diffusion term. Written in relative gradient variables this system appears as the $q$- viscous Burgers' equation. Exact solutions of this equation in polynomial form as generalized Kampe de Feriet polynomials, corresponding dynamical symmetry and description in terms of Bell polynomials are derived. We found the generating function for these polynomials by application of dynamical symmetry and the Zassenhaus formula. We have constructed and analyzed shock solitons and their interactions with different $q$. We obtain modification of the soliton relative speeds depending on value of $q$.For $q< 1$ the soliton speed becomes bounded from above and as a result in addition to usual Burgers soliton process of fusion, we found a new phenomena, when soliton with higher amplitude but smaller velocity is fissing to two solitons. q-Semiclassical expansion of these equations are found in terms of Bernoulli polynomials in power of $\ln q$.