q-Analogue of Shock Soliton Solution

Details q-Analogue of Shock Soliton Solution q-Analogue of Shock Soliton Solution

2010-05-14

arxiv.org/abs/1005.2543v1

Nonlinear Sciences

Exactly Solvable and Integrable Systems

13 pages, 5 figures

Scientific paper

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.

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