Group analysis of the one-dimensional Boltzmann equation: I. symmetry groups

Abstract

We consider the one-dimensional Boltzmann equation f_t + cf_x + Ff_c = 0, where the functions f and F are assumed to depend on three variables t, x, and c. We obtain relations defining the symmetry algebra in the general case and also under the additional conditions of conservation of the relations dx = c dt and dc = F dt, which arise from physical considerations. We show that the widest symmetry algebra is obtained in the case of conservation of both relations. This algebra is infinite-dimensional, and its structure is independent of the form of the function F.