9 research outputs found
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
q-Analogue of Shock Soliton Solution
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.Comment: 13 pages, 5 figure