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arXiv · 1010.2675

q-Shock Soliton Evolution

Oktay K. Pashaev·Sengul Nalci
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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-Schrödinger equation and the q-Madelung fluid type representation is derived.

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Oktay K. Pashaev, Sengul Nalci. 2010-10-13. q-Shock Soliton Evolution. https://arxiv.org/abs/1010.2675

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