Numerial Solution of the
Nonlinear Helmholtz Equation
Guy Baruh
Tel Aviv University, Israel
De. 5th, 2008
The nonlinear Helmholtz equation models the propagation of intense laser
beams in Kerr media suh as water, silia and air. It is a semilinear ellipti
equation whih requires non-selfadjoint radiation boundary-onditions, and
remains unsolved in many ongurations.
Its ommonly-used paraboli approximation, the nonlinear Shrodinger
equation (NLS), is known to possess singular solutions. We therefore onsider
the question, whih has been open sine the 1960s: do nonlinear Helmholtz
solutions exists, under onditions for whih the NLS solution beomes singular ? In other words, is the singularity removed in the ellipti model ?
In this work we develop a numerial method whih produes suh solutions in some ases, thereby showing that the singularity is indeed removed
in the ellipti equation.
We also onsider the subritial ase, wherein the NLS has stable solitons.
For beams whose width is omparable to the optial wavelength, the NLS
model beomes invalid, and so the existene of suh nonparaxial solitons
requires solution of the Helmholtz model.
Numerially, we onsider the ase of grated material, that has material
disontinuities in the diretion of propagation.
We develop a high-order
disretization whih is "semi-ompat", i.e., ompat only in the diretion
of propagation, that is optimal for this ase.
Joint work with Gadi Fibih and Semyon Tsynkov.