Title：Nonlinear PDEs on metric graphs
Abstrct：I will give an overview of results for stationary states for the nonlinear Schrödinger equation on different metric graphs, which include stars, tadpoles, dumbbell, and periodic graphs. For the subcritical powernonlinearity and for a star graph, a half-soliton state is a degenerate critical point of the action functional under the mass constraint such that thesecond variation is nonnegative. By using normal forms, we prove that the degenerate critical point is a saddle point, for which the small perturbations to the half-soliton state grow slowly in time resulting in the nonlinear instability of the half-soliton state. This phenomenon is just one example of a nontrivial interplay between the dynamical properties of the nonlinear PDEs and geometric properties of the metric graphs.
