Solutions of the Carleman system via the Painlevé expansion

The one-dimensional discrete kinetic system of Carleman equations is considered. This system describes a monatomic rarefied gas consisting of two groups of particles. These groups of particles move along a straight line, in opposite directions at a unit speed. Particles interact within one group, i. e. themselves, changing direction. Recently, special attention has been paid to the construction of exact solutions of non-integrable partial differential equations using the truncated Painlevé series. Applying the Painlevé expansion to non-integrable partial differential equations, we obtain the conditions in resonance that must be satisfied. Solution of the system is sought using the truncated Painlevé expansion. This system does not satisfy the Painlevé test. It leads to the singularity manifold constraints, one of which is the Bateman equation. Knowing the implicit solution of the Bateman equation, one can find new particular solutions of the Carleman system. Also, the solution is constructed using the rescaling ansatz, which allows us to reduce the problem to finding solutions to the corresponding Riccati equation.