APPLIED MATH SEMINAR

Name:   Guy Baruch, Tel-Aviv University

Title:     Numerical Solution of the Nonlinear Helmholtz Equation

When/where:    Monday, November 17, 4:15PM, AKW 200

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 Schrödinger equation (NLS), is known to possess singular solutions. We therefore consider the question, which has been open since the 1960s: do nonlinear Helmholtz solutions exist, 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 'rsemi compact", i.e., compact only in the direction of propagation, that is optimal for this case.

Joint work with Gadi Fibich and Semyon Tsynkov.