Sengul Nalci (Izmir Institute of Technology, Izmir, Turkey)

q-Kampe de Feriet Polynomials applied to nonlinear q-heat and q-Schrodinger equations

Abstract:
   By using Jackson's q-exponential function we introduce the generating function, the recursive formulas and the second order q-difference 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. Exact solutions for the q-Burgers equation in the form of moving poles, singular and regular q-shock soliton solutions are obtained. Novel, self-similar property for stationary regular q-shock soliton solution is found. The results are extended to complex polynomials for the linear q-Schrodinger equation and the complex Burgers-Madelung hydrodynamic system. (joint work with O.K. Pashaev).