Family of asymptotic solutions to the two-dimensional kinetic equation with a nonlocal cubic nonlinearity

TL;DR

This paper develops a semiclassical method to construct asymptotic solutions for a two-dimensional kinetic ionization equation with nonlocal cubic nonlinearity, enabling analysis of ion distribution evolution in complex media.

Contribution

It introduces a novel semiclassical approach using symmetry operators and the Maslov germ method to solve a nonlinear kinetic equation with nonlocal effects.

Findings

Constructed a family of asymptotic solutions for the kinetic equation.

Established a nonlinear superposition principle for the solutions.

Applied the method to analyze ion distribution in active media.

Abstract

We apply the original semiclassical approach to the kinetic ionization equation with the nonlocal cubic nonlinearity in order to construct the family of its asymptotic solutions. The approach proposed relies on an auxiliary dynamical system of moments of the desired solution to the kinetic equation and the associated linear partial differential equation. The family of asymptotic solutions to the kinetic equation is constructed using the symmetry operators acting on functions concentrated in a neighborhood of a point determined by the dynamical system. Based on these solutions, we introduce the nonlinear superposition principle for the nonlinear kinetic equation. Our formalism based on the Maslov germ method is applied to the Cauchy problem for the specific two-dimensional kinetic equation. The evolution of the ion distribution in the kinetically enhanced metal vapor active medium is obtained as the nonlinear superposition using the numerical-analytical calculations.