Title: q-Viscous Burgers' Equation: Dynamical Symmetry, Shock Solitons and q-Semiclassical Expansion,
Abstract: We propose new type of $q$-diffusive heat equation with nonsymmetric
$q$-extension of the diffusion term. Written in relative gradient variables
this system appears as the $q$- viscous Burgers' equation. Exact solutions of
this equation in polynomial form as generalized Kampe de Feriet polynomials,
corresponding dynamical symmetry and description in terms of Bell polynomials
are derived. We found the generating function for these polynomials by
application of dynamical symmetry and the Zassenhaus formula. We have
constructed and analyzed shock solitons and their interactions with different
$q$. We obtain modification of the soliton relative speeds depending on value
of $q$.For $q< 1$ the soliton speed becomes bounded from above and as a result
in addition to usual Burgers soliton process of fusion, we found a new
phenomena, when soliton with higher amplitude but smaller velocity is fissing
to two solitons. q-Semiclassical expansion of these equations are found in
terms of Bernoulli polynomials in power of $\ln q$.