Semi-Classical Analysis For Nonlinear Schrodinger Equations

These lecture notes review recent results on the high-frequency analysis of nonlinear Schrödinger equations in the presence of an external potential. The book consists of two relatively independent parts: WKB analysis, and caustic crossing. In the first part, the basic linear WKB theory is constructed and then extended to the nonlinear framework. The most difficult supercritical case is discussed in detail, together with some of its consequences concerning instability phenomena. Applications of WKB analysis to functional analysis, in particular to the Cauchy problem for nonlinear Schrödinger equations, are also given. In the second part, caustic crossing is described, especially when the caustic is reduced to a point, and the link with nonlinear scattering operators is investigated.

These notes are self-contained and combine selected articles written by the author over the past ten years in a coherent manner, with some simplified proofs. Examples and figures are provided to support the intuition, and comparisons with other equations such as the nonlinear wave equation are provided.

Contents: WKB Analysis: Preliminary Analysis; Weak Nonlinear Geometric Optics; Convergence of Quadratic Observables via Modulated Energy Functionals; Pointwise Description of the Wave Function; Some Instability Phenomena; Caustic Crossing: The Case of Focal Points: Caustic Crossing: Formal Analysis; Focal Point without External Potential; Focal Point in the Presence of an External Potential; Some Ideas for Supercritical Cases.