When   Friday, October 30, 2009   Time   2:00 PM - 3:00 PM  
Where   Technological Instit M 416 2145 Sheridan Rd.   map it
Audience   - Faculty/Staff - Student - Public
Contact   Molly E Scanlon   +1 847 491 5586  
Group   McCormick-Colloquia Engineering Sciences and Applied Mathematics

Applied Math Colloquium

Title: Numerical Solution of the nonlinear Helmholtz equation

Speaker: Guy Baruch, Tel Aviv University, Israel

Abstract:

The nonlinear Helmholtz equation models the propagation of intense laser beams in Kerr media such as water, silica and air. It is a semilinear elliptic equation which requires non-selfadjoint radiation boundary-conditions, and remains unsolved in many configurations.

Its commonly-used parabolic approximation, the nonlinear Schrodinger equation (NLS), is known to possess singular solutions.  We therefore consider the question, which has been open since the 1960s:  Do nonlinear Helmholtz solutions exists, under conditions for which the NLS solution becomes singular?
In other words, is the singularity removed in the elliptic model?

In this work we develop a numerical method which produces such solutions in some cases, thereby showing that the singularity is indeed removed in the elliptic equation.

We also consider the subcritical case, wherein the NLS has stable
solitons. For beams whose width is comparable to the optical wavelength, the NLS model becomes invalid, and so the existence of such "nonparaxial solitons'' requires solution of the Helmholtz model.

Numerically, we consider the case of grated material, that has material discontinuities in the direction of propagation. We develop a high-order discretization which is ``semi-compact'', i.e., compact only in the direction of propagation, that is optimal for this case.

Joint work with Gadi Fibich and Semyon Tsynkov.

This talk is part of the RTG Seminar Series.