Exact exponential type solutions to the discrete Boltzmann models

Abstract

We review the recently found (1+1) and (2+1)-dimensional exact solutions to the discrete Boltzmann models. They are sums of respectively two and three similarity shock waves with exponential variables. Further we prove original results: (i) Similarity waves for the 6-velocity planar model with ternary collisions included and without spurious conservation law (ii) (1+1)-dimensional rational solutions for binary collisions and denominators of the fourth order in two exponential variables (iii) For the (2+1)-dimensional shock waves, positive densities can be constructed once the asymptotic shock-limits are positive.