Talk by Nataliia Goloshchapova
On February 1st, 2023, Prof. Dr. Nataliia Goloshchapova (Sao Paolo) gave a talk about "Stability of standing waves for NLS equation on certain metric graphs" as part of the research seminar Analysis of the FernUniversität in Hagen. This lecture is partially supported by the COST action Mathematical models for interacting dynamics on networks.
Abstract
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An orbital stability of a solitary wave to a Hamiltonian model
(1) du/dt = JE'(u(t))
means that a solution to the Cauchy problem stays close to an orbit generated by the wave profile when an initial data is close to the profile. Solitary waves are of special interest by preserving their shape in time. Stability study mainly involves investigation of a spectrum of an operator L associated with the linearization of (1). Firstly, we will discuss how to apply the Grillakis-Jones theory and the Grillakis-Shatah-Strauss theory to establish stability/instability of the standing wave solutions to nonlinear Schrödinger equations on star graphs (with different vertex conditions). Secondly, we will consider standing waves with constant profiles on certain compact graphs with Kirchhoff vertex conditions. We will demonstrate how to use the boundary triplets theory and the surgery principles in spectral analysis of an operator L, thus making conclusions about their stability properties.
