8 - Bifurcation of Stationary Solutions of the Vlasov-Maxwell System

Abstract

Since the bifurcation of the stationary solutions of Vlasov–Maxwell (VM) system always had a special importance both for theoretical study and practical applications, this chapter translates it into the bifurcation problem of the semi-linear elliptic system and studies it as an operator equation in Banach space. It derives the branching equation by using a classical approach by Lyapunov–Schmidt and studies asymptotic of nontrivial branches of solutions. Here, the principal idea is to study a potential BEq, since the system of elliptic equations is potential. Further investigation established the existence theorem for the bifurcation points and revealed the asymptotic properties of nontrivial branches of the solutions of VM system. The problem of the bifurcation analysis of a VM system, formulated for the first time by Vlasov, proved to be very complicated against the general background of the progress of the bifurcation theory in other directions, and it remains open at the present moment. There exist only separate results. One simple theorem about the point of bifurcation is covered by Sidorov and Sinitsyn and another is proven in paper for the stationary VM system.