250116 VO Nonlinear Schrödinger and Wave Equations (2025W)

Nonlinear Schrödinger equations (NLS : "dispersive") and Nonlinear Wave equations (NLW : "hyperbolic") are fundamental classes of Partial Differential Equations (PDE), with many important applications. To deal with NLS and NLW in one course (in the spirit of e.g. Terry Tao’s book) reveals an interesting mutual crossover of ideas between these 2 different types of PDEs.

In this lecture we deal with all aspects of "Applied Mathematics”, i.e. Modeling, Analysis and Numerics,
based on lecture notes that are handed out to students before the lectures.

1) Modeling: motivation / derivation of NLS :
a) quantum physics, where “one particle” NLS occur as approximate models for the linear N-body Schrödinger equation.
Quantum HydroDynamics.
b) nonlinear optics, where the paraxial approximation of the Helmholtz (wave) equation yields 2+1 dimensional cubic NLS

2) Analysis:
Existence and Uniqueness (“Local/Global WellPosedness) of NLS and NLW
with local and non-local nonlinearities, scattering, finite(-time) Blow-up; asymptotic results e.g. for the (semi-)classical limit of NLS.

Methods:
Functional analysis, Semigroup theory, Sobolev embeddings, Strichartz estimates, energy estimates, linear PDE theory,...,

3) Numerics:
Finite Element Methods for NLS,
Time Splitting,
Spectral methods,
Boundary conditions