On the exact solution of the evolution equations for two interacting narrow wave packets propagating in a non-Abelian plasma

In this paper, we present and discuss a self-consistent system of kinetic equations of the Boltzmann type, which takes into account the time evolution of soft non-Abelian plasma excitations and the mean value of the color charge in the interaction of a high-energy color-charged particle with a plasma. Based on these equations, we examine a model problem of interaction of two infinitely narrow wave packets and obtain a system of first-order nonlinear ordinary differential equations, which governs the dynamics of interacting the colorless $N^{l}_{\mathbf \kappa}$ and color $W^{l}_{\mathbf \kappa}$ components of the density of the number collective bosonic excitations. Due to the autonomy of the right-hand sides, we reduce this system to a single nonlinear Abel differential equation of the second kind. Finally, we show that at a certain ratio between the constants involved in this nonlinear equation, one can obtain an exact solution in the parametric form.