Two nonlinear systems from mathematical physics

Abstract

The dissertation is divided into two chapters. In the first one, we consider the 2-Vortex problem for two point vortices in a complex domain. The Hamiltonian of the system contains the regular part of a hydrodynamic Green’s function, the Robin function h and two coefficinets which are the strengths of the point vortices. We prove the existence of infinitely many periodic solutions with minimal period T which are a superposition of a slow motion of the center of vorticity along a level line of h and of a fast rotation of the two vortices around their center of vorticity. These vortices move in a prescribed subset of the domain that has to satisfy a geometric condition. The minimal period can be any T in a certain interval. Subsets to which our results apply can be found in any generic bounded domain. The proofs are based on a recent higher dimensional version of the Poincaré-Birkhoff theorem due to Fonda and Ureña. In the second part, we study bifurcations of a multi-component Schrödinger system. We construct a solution branch synchronized to a positive solution of a simpler system. From this branch, we find a sequence of local bifurcation values in the one dimensional case and also in the general case provided that the positive solution is nondegenerate