EMBEDDED SOLITONS IN SECOND-HARMONIC-GENERATING SYSTEMS

We present a new type of soliton, found in models characterized by opposing dispersions and competing nonlinearities at fundamental and second harmonics. They are isolated solitary waves, existing at discrete values of the propagation constant inside the system’s continuous spectrum. We show analytically, and verify by simulations, that the fundamental solitons are linearly stable. They can be nonlinearly stable or unstable, depending on the sign of the energy perturbation, which could make these pulses useful for switching applications. Higher-order solitons are found, too, but they are linearly unstable. We report a new type of soliton, in the form of an isolated (codimension-one) solitary-wave solution whose intrinsic frequency resides inside the continuous spectrum of the radiation modes. It is a special member of a family of delocalized solitons, which are solitary waves with nonvanishing oscillating tails. In terms of the dynamicalsystems theory, these are trajectories homoclinic to cycles, whereas ordinary solitons are homoclinic to fixed points. Delocalized solitons are known in various models of the hydrodynamic [1] and optical [2] origin. We demonstrate that the amplitudes of the oscillating tails can exactly vanish at a discrete set of frequencies, resulting in a delocalized soliton becoming truly localized, and with finite energy. We call these solutions embedded (in the continuous spectra) solitons (ES). Because the vanishing of the tail’s amplitude is an additional condition, these solitons (in contrast to familiar gap solitons [3]), never exist in continuous families, but only as isolated solutions. A physical model giving rise to ES describes an optical medium with quadratic (x 2 ) and cubic (x 3 ) nonlinearities. Various systems of this type have been recently considered [4]. We start with a general one, iuz 1 12utt 1 u y1g 1juj 2 1 2jyj 2 u 0, (1) iyz 2 12dytt 1 q y1 12u 2 1