A new integrable model of long wave-short wave interaction and linear stability spectra.

We consider the propagation of short waves which generate waves of much longer (infinite) wavelength. Model equations of such long wave-short wave (LS) resonant interaction, including integrable ones, are well known and have received much attention because of their appearance in various physical contexts, particularly fluid dynamics and plasma physics. Here we introduce a new LS integrable model which generalizes those first proposed by Yajima and Oikawa and by Newell.

Integrability and Linear Stability of Nonlinear Waves.

It is well known that the linear stability of solutions of partial differential equations which are integrable can be very efficiently investigated by means of spectral methods. We present here a direct construction of the eigenmodes of the linearized equation which makes use only of the associated Lax pair with no reference to spectral data and boundary conditions. This local construction is given in the general matrix scheme so as to be applicable to a large class of integrable equations, including the multicomponent nonlinear Schrödinger system and the multiwave resonant interaction system.

Vector rogue waves and baseband modulation instability in the defocusing regime.

We report and discuss analytical solutions of the vector nonlinear Schrödinger equation that describe rogue waves in the defocusing regime. This family of solutions includes bright-dark and dark-dark rogue waves. The link between modulational instability (MI) and rogue waves is displayed by showing that only a peculiar kind of MI, namely baseband MI, can sustain rogue-wave formation.

Rational solitons of wave resonant-interaction models.

Integrable models of resonant interaction of two or more waves in 1+1 dimensions are known to be of applicative interest in several areas. Here we consider a system of three coupled wave equations which includes as special cases the vector nonlinear Schrödinger equations and the equations describing the resonant interaction of three waves. The Darboux-Dressing construction of soliton solutions is applied under the condition that the solutions have rational, or mixed rational-exponential, dependence on coordinates.

Rogue waves emerging from the resonant interaction of three waves.

We introduce a novel family of analytic solutions of the three-wave resonant interaction equations for the purpose of modeling unique events, i.e., "amplitude peaks" which are isolated in space and time.

Solutions of the vector nonlinear Schrödinger equations: evidence for deterministic rogue waves.

We construct and discuss a semirational, multiparametric vector solution of coupled nonlinear Schrödinger equations (Manakov system). This family of solutions includes known vector Peregrine solutions, bright- and dark-rogue solutions, and novel vector unusual freak waves. The vector rogue waves could be of great interest in a variety of complex systems, from optics and fluid dynamics to Bose-Einstein condensates and finance.

Velocity-locked solitary waves in quadratic media.

We demonstrate experimentally the existence of three-wave resonant interaction solitary triplets in quadratic media. Stable velocity-locked bright-dark-bright spatial solitary triplets, determined by the balance between the energy exchange rates and the velocity mismatch between the interacting waves, are excited in a KTP crystal.

Frequency generation and solitonic decay in three wave interactions.

We consider experimentally three-wave resonant nonlinear interactions of fields propagating in nonlinear media. We investigate the spatial dynamics of two diffractionless beams at frequency omega1, omega2 which mix to generate a field at the sum frequency omega3. If the generated field at omega3 can sustain a soliton, it decays into solitons at omega1, omega2.

Parametric frequency conversion of short optical pulses controlled by a CW background.

We predict that parametric sum-frequency generation of an ultra-short pulse may result from the mixing of an ultra-short optical pulse with a quasi-continuous wave control. We analytically show that the intensity, time duration and group velocity of the generated idler pulse may be controlled in a stable manner by adjusting the intensity level of the background pump.

Inelastic scattering and interactions of three-wave parametric solitons.

We study the interactions of velocity-locked three-wave parametric solitons in a medium with quadratic nonlinearity and dispersion. We reveal that the inelastic scattering between three-wave solitons and linear waves may be described in terms of analytical solutions with dynamically varying group velocity, or boomerons. Moreover, we demonstrate the elastic nature of three-wave soliton-soliton collisions and interactions.

Stable control of pulse speed in parametric three-wave solitons.

We analyze the control of the propagation speed of three wave packets interacting in a medium with quadratic nonlinearity and dispersion. We find analytical expressions for mutually trapped pulses with a common velocity in the form of a three-parameter family of solutions of the three-wave resonant interaction. The stability of these novel parametric solitons is simply related to the value of their common group velocity.