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.
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