A conjecture on the possible exact (1+1)-dimensional solutions to the discrete Boltzmann models I

Abstract

The exact multidimensional exponential type solutions to the discrete Boltzmann models, which have been found, are sums of unidimensional similarity waves. Our goal is to obtain, at a linear level, a physical signature of the class of possible exact rational solutions. In the discrete kinetic theory, contrary to the continuous theory, the macroscopic conservation laws are included into the set of nonlinear equations for the densities. These conservation laws, being linear differential relations, underline the above recalled result. For a class of (l+l)-dimensional rational solutions with two exponential variables, satisfying at least two conservation laws, we prove that the possible solutions are sums of similarity waves. We first consider the class of models with three independent densities and two conservation laws. Second we investigate what happens when the two conservation laws of mass and momentum exist. Third we sketch briefly results for five densities and three conservation laws. Finally we conjecture that all exact, bounded (l+l)-dimensional solutions are sums of similarity waves.