New exact solutions for the two-dimensional Broadwell system

S. A. Dukhnovskii

Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 22 (2022) no. 1, pp. 4-14

Résumé

In this paper, we consider the discrete kinetic Broadwell system. This system is a nonlinear hyperbolic system of partial differential equations. The two-dimensional Broadwell system is the kinetic Boltzmann equation, and for this model momentum and energy are conserved. In the kinetic theory of gases, the system describes the motion of particles moving on a two-dimensional plane, the right-hand side is responsible for pair collisions of particles. For the first time, new traveling wave solutions are found using the $\exp(-\varphi(\xi))$-expansion method. This method is as follows. The solution is sought in the form of a traveling wave. In this case, the system is reduced to a system of ordinary differential equations. Further, the solution is sought according to this method in the form of an exponential polynomial, depending on an unknown function that satisfies a certain differential equation. Solutions of the differential equation themselves are known. The summation is carried out up to a certain positive number, which is determined by the balance between the highest linear and non-linear terms. Further, the proposed solution is substituted into the system of differential equations and coefficients at the same exponential powers are collected. Solving systems of algebraic equations, we find unknown coefficients and write the original solution. This method is universal and allows us to obtain a large number of solutions, namely, kink solutions, singular kink solutions, periodic solutions, and rational solutions. Corresponding graphs of some solutions are presented by the Mathematica package. With the help of computerized symbolic computation, we obtain new solutions. Similarly, it is possible to find exact solutions for other kinetic models.