q-Shock Soliton Evolution

TL;DR

This paper introduces a new q-analogue of Hermite and Kampe-de Feriet polynomials, derives associated q-heat and Burgers equations, and studies their shock soliton solutions and self-similarity properties.

Contribution

It develops a novel q-analogue framework for polynomials and differential equations, including new solutions and properties of q-shock solitons.

Findings

Regular q-shock soliton solutions are constructed.

A self-similarity property of q-shock solitons is identified.

Extension to q-Schrodinger and fluid equations is achieved.

Abstract

By generating function based on the Jackson's q-exponential function and standard exponential function, we introduce a new q-analogue of Hermite and Kampe-de Feriet polynomials. In contrast to standard Hermite polynomials, with triple recurrence relation, our polynomials satisfy multiple term recurrence relation, derived by the q-logarithmic function. It allow us to introduce the q-Heat equation with standard time evolution and the q-deformed space derivative. We found solution of this equation in terms of q-Kampe-de Feriet polynomials with arbitrary number of moving zeros, and solved the initial value problem in operator form. By q-analog of the Cole-Hopf transformation we find a new q-deformed Burgers type nonlinear equation with cubic nonlinearity. Regular everywhere single and multiple q-Shock soliton solutions and their time evolution are studied. A novel, self-similarity property of these q-shock solitons is found. The results are extended to the time dependent q-Schr\"{o}dinger equation and the q-Madelung fluid type representation is derived.