TY - JOUR

T1 - Nonlinearity saturation as a singular perturbation of the nonlinear Schrödinger equation

AU - Glasner, Karl

AU - Allen-Flowers, Jordan

N1 - Publisher Copyright:
© 2016 Society for Industrial and Applied Mathematics.

PY - 2016

Y1 - 2016

N2 - Saturation of a Kerr-type nonlinearity in the nonlinear Schrödinger equation (NLSE) can be regarded as a singular perturbation which regularizes the well-known blow-up phenomenon in the cubic NLSE. An asymptotic expansion is proposed which takes into account multiple scale behavior both in the longitudinal and transverse directions. In one dimension, this leads to a free boundary problem reduction where the solitary wave acts solely to reflect impinging waves and is accelerated by an elastic transfer of momentum. In two dimensions, we find that interaction of a solitary wave and an adjacent wave field is governed by behavior of certain eigenfunctions of the linearized fast-scale operator. This leads to an outer solution with a free logarithmic singularity, whose position evolves by virtue of a large transfer of momentum between the ambient and solitary waves. However, for a certain value of wave power, we find there is essentially no interaction and the solitary wave is asymptotically transparent to the ambient field. We test our results by numerical simulation of both the full equation and free boundary reductions.

AB - Saturation of a Kerr-type nonlinearity in the nonlinear Schrödinger equation (NLSE) can be regarded as a singular perturbation which regularizes the well-known blow-up phenomenon in the cubic NLSE. An asymptotic expansion is proposed which takes into account multiple scale behavior both in the longitudinal and transverse directions. In one dimension, this leads to a free boundary problem reduction where the solitary wave acts solely to reflect impinging waves and is accelerated by an elastic transfer of momentum. In two dimensions, we find that interaction of a solitary wave and an adjacent wave field is governed by behavior of certain eigenfunctions of the linearized fast-scale operator. This leads to an outer solution with a free logarithmic singularity, whose position evolves by virtue of a large transfer of momentum between the ambient and solitary waves. However, for a certain value of wave power, we find there is essentially no interaction and the solitary wave is asymptotically transparent to the ambient field. We test our results by numerical simulation of both the full equation and free boundary reductions.

KW - Multiple scales analysis

KW - Nonlinear Schrödinger equation

KW - Singular perturbation

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U2 - 10.1137/15M1024974

DO - 10.1137/15M1024974

M3 - Article

AN - SCOPUS:84964801009

VL - 76

SP - 525

EP - 550

JO - SIAM Journal on Applied Mathematics

JF - SIAM Journal on Applied Mathematics

SN - 0036-1399

IS - 2

ER -