Jeremie Szeftel

Design of Absorbing Boundary Conditions for Schrödinger Equations in

We construct a family of absorbing boundary conditions for the linear Schrodinger equation on curved boundaries in any dimension which are local both in space and time. We give a convergence result and show that the corresponding initial boundary value problems are well posed. We also prove the nonlinear Schrodinger equation with the absorbing boundary conditions of the linearized problem to be well posed. We finally present numerical results both for the linear and the nonlinear case.

\mathbbR

Consider a nonlinear Klein-Gordon equation on a compact manifold M with smooth Cauchy data of size e → 0. Denote by T e the maximal time of existence of a smooth solution. One always has T e ≥ c e −k+1 if the nonlinearity vanishes at order k at 0. We prove that when M=Td, k=2, and the equation is quasilinear, T e ≥ ce −2 . When M=Sd−1, the equation is semilinear, and k = 2p, we show that T e ≥ ce −2p for almost every mass. The proof of these results relies on a systematic use of normal forms methods.

d

We consider the 2-dimensional focusing mass critical NLS with an inhomogeneous nonlinearity:

Long-time existence for small data nonlinear Klein-Gordon equations on tori and spheres

i\partial_tu+\Delta u+k(x)|u|^{2}u=0

Existence and uniqueness of minimal blow-up solutions to an inhomogeneous mass critical NLS

. From standard argument, there exists a threshold

LONG-TIME EXISTENCE FOR SEMI-LINEAR KLEIN-GORDON EQUATIONS WITH SMALL CAUCHY DATA ON ZOLL MANIFOLDS

M_k>0

Absorbing Boundary Conditions for One-dimensional Nonlinear Schrödinger Equations

such that

Optimized and Quasi-optimal Schwarz Waveform Relaxation for the One Dimensional Schrödinger Equation

H^1

A nonlinear approach to absorbing boundary conditions for the semilinear wave equation

solutions with

Discontinuous Galerkin and Nonconforming in Time Optimized Schwarz Waveform Relaxation

\|u\|_{L^2} M_k