q-Analogue of Shock Soliton Solution

TL;DR

This paper develops a q-analogue framework for shock soliton solutions, introducing q-Hermite polynomials and solving the q-heat and q-Burgers equations with explicit soliton solutions.

Contribution

It introduces a novel q-analogue approach to shock solitons, including new polynomial solutions and operator methods for the q-heat and q-Burgers equations.

Findings

Explicit q-shock soliton solutions derived

q-heat and q-Burgers equations solved analytically

Reduction to classical Burgers equation as q -> 1

Abstract

By using Jackson's q-exponential function we introduce the generating function, the recursive formulas and the second order q-differential equation for the q-Hermite polynomials. This allows us to solve the q-heat equation in terms of q-Kampe de Feriet polynomials with arbitrary N moving zeroes, and to find operator solution for the Initial Value Problem for the q-heat equation. By the q-analog of the Cole-Hopf transformation we construct the q-Burgers type nonlinear heat equation with quadratic dispersion and the cubic nonlinearity. In q -> 1 limit it reduces to the standard Burgers equation. Exact solutions for the q-Burgers equation in the form of moving poles, singular and regular q-shock soliton solutions are found.