"Embedded solitons": solitary waves in resonance with the linear spectrum

Abstract

It is commonly held that a necessary condition for the existence of solitons in nonlinear-wave systems is that the soliton's frequency (spatial or temporal) must not fall into the continuous spectrum of radiation modes. However, this is not always true. We present a new class of {\it % codimension-one} solitons (i.e., those existing at isolated frequency values) that are {\em embedded} into the continuous spectrum. This is possible if the spectrum of the linearized system has (at least) two branches, one corresponding to exponentially localized solutions, and the other to radiation modes. An embedded soliton (ES) is obtained when the latter component exactly vanishes in the solitary-wave's tail. The paper contains both a survey of recent results obtained by the authors and some new results, the aim being to draw together several different mechanism underlying the existence of ESs. We also consider the distinctive property of {\em semi-stability} of ES, and {\em moving} ESs. Results are presented for four different physical models, including an extended 5th-order KdV equation describing surface waves in inviscid fluids, and three models from nonlinear optics. One of them pertains to a resonant Bragg grating in an optical fiber with a cubic nonlinearity, while two others describe second-harmonic generation (SHG) in the temporal or spatial domain (i.e., respectively, propagating pulses in nonlinear optical fibers, or stationary patterns in nonlinear planar waveguides). Special attention is paid to the SHG model in the temporal domain for a case of competing quadratic and cubic nonlinearities. An essential new result is that ES is, virtually, fully stable in the latter model in the case when both harmonics have anomalous dispersion.