250076 VO Nonlinear Schrödinger and Wave Equations (2018S)

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 them jointly (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 3 aspects of "Applied Mathematics”, i.e. = “Modeling + Analysis + Numerics", based on lecture notes that are handed out to students.

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.

3) Numerics:
Spectral methods, finite difference and relaxation schemes, Absorbing Boundary Conditions, ...

Methods:
functional analysis, semigroup theory, Sobolev embeddings, Strichartz estimates, energy estimates, linear PDE theory, …
Numerical schemes: Finite Difference schemes, spectral methods, time splitting, Absorbing Boundary Layers ("optical potential")