Construction of positive (2+1)-dimensional exact solutions for the Broadwell model

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Abstract
In the first part of the paper, for the three dimensional Broadwell model, we construct (2+1)-dimensional (two spatial coordinates plus time) semi-periodic exact solutions. As usual such solutions are sums of similarity shock waves solutions. These semi-periodic solutions represent periodic waves submitted to a strong perturbation and rebuilt after the shock. To this aim we build up positive periodic (1+1)-dimensional solutions with a time dependence and add a component which is a real similarity shock-wave. For the whole solution we choose the time dependence of the periodic part such that the positivity problem is reduced to the positivity of the shock component. These solutions depend on two arbitrary parameters and we prove that there exists a domain for which the constructed solutions are positive. In the second part of the paper we consider a class of discrete multidimensional models generalizing both the 4-velocity model and the Broadwell model. We write down the equations which must be satisfied by the parameters of the solutions sums of similarity shock waves. From a counting argument it seems that there exists few hopes to construct (3+1)-dimensional positive solutions.

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