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.
- Multiple scales analysis
- Nonlinear Schrödinger equation
- Singular perturbation
ASJC Scopus subject areas
- Applied Mathematics