Embedded solitons: solitary waves in resonance with the linear spectrum

It is commonly held that a necessary condition for the existence of soliton s in nonlinear-wave systems is that the soliton's frequency (spatial or tem poral) must not fall into the continuous spectrum of radiation modes. Howev er, this is not always true. We present a new class of codimension-one soli tons (i.e., those existing at isolated frequency values) that are embedded into the continuous spectrum. This is possible if the spectrum of the linea rized system has (at least) two branches, one corresponding to exponentiall y localized solutions, and the other to radiation modes. An embedded solito n (ES) is obtained when the latter component exactly vanishes in the solita ry-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 severa l different mechanism underlying the existence of ESs. We also consider the distinctive properties of semi-stability of ESs, and moving ESs. Results a re presented for four different physical models, including an extended fift h-order KdV equation describing surface waves in inviscid fluids, and three models from nonlinear optics. One of them pertains to a resonant Bragg gra ting in an optical fiber with a cubic nonlinearity, while two others descri be second-harmonic generation (SHG) in the temporal or spatial domain (i.e. , respectively, propagating pulses in nonlinear-optical fibers, or stationa ry patterns in nonlinear planar waveguides). Special attention is paid to t he SHG model in the temporal domain for a case of competing quadratic and c ubic nonlinearities. In particular, a new result is that when both harmonic s have anomalous dispersion, an ES can exist which is, virtually, completel y stable. (C) 2001 Elsevier Science B.V. All rights reserved.