q-Shock Soliton Evolution

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.Comment: 15 pages, 6 figure

Dynamic Transitions of Quasi-Geostrophic Channel Flow

The main aim of this paper is to describe the dynamic transitions in flows described by the two-dimensional, barotropic vorticity equation in a periodic zonal channel. In \cite{CGSW03}, the existence of a Hopf bifurcation in this model as the Reynolds number crosses a critical value was proven. In this paper, we extend the results in \cite{CGSW03} by addressing the stability problem of the bifurcated periodic solutions. Our main result is the explicit expression of a non-dimensional number $\gamma$ which controls the transition behavior. We prove that depending on $\gamma$, the modeled flow exhibits either a continuous (Type I) or catastrophic (Type II) transition. Numerical evaluation of $\gamma$ for a physically realistic region of parameter space suggest that a catastrophic transition is preferred in this flow