In this article, we consider the one–dimensional kinetic system of Carleman equations. The Carleman system is the kinetic Boltzmann equation. This system describes a monatomic rarefied gas consisting of two groups of particles. One particle from the first group, interacting with a particle of the first group, transforms into two particles of the second group. Similarly, two particles of the second group, interacting with themselves, transform into two particles of the first group, respectively. We found traveling wave solutions by using the tanh–function method for nonlinear partial differential system. The results of the work can be useful for mathematical modeling in various fields of science and technology: kinetic theory of gases, gas dynamics, autocatalysis. The obtained exact solutions are new.