Family of asymptotic solutions to the two-dimensional kinetic equation with a nonlocal cubic nonlinearity
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Publication:6392802
DOI10.3390/SYM14030577arXiv2203.02333MaRDI QIDQ6392802
Sergei A. Siniukov, A. V. Shapovalov, A. E. Kulagin
Publication date: 4 March 2022
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.
Integro-partial differential equations (45K05) Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory (81Q20) Statistical mechanics of plasmas (82D10) Kinetic theory of gases in equilibrium statistical mechanics (82B40)
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