Nonlinear waves and related nonintegrable and integrable systems

Abstract

Spectral method related to Lame equation with finite-gap potential is used to study the optical cascading equations. These equations are known not to be integrable by inverse scattering method. Due to "partial integrability" two-gap solutions are obtained in terms of products of elliptic functions and are classified in five different families related to eigenvalues of appropriate spectral problem. In special cases, when periodic solutions reduce to localized solitary waves, previously known phase-locked solutions are recovered, and additional one solution is obtained. For vector nonlinear Schrodinger equation n=3 we present exact solutions in a form of multicomponent cnoidal waves.