Three-wave Solitons Research Articles

On multiple soliton solutions of the extended (3+1)-dimensional Jimbo–Miwa equations

In this paper, we deal with a set of equations, namely the extended (3 + 1)-dimensional Jimbo–Miwa (JM) equations. We focus on the calculations of soliton solutions to the aforementioned equations using the multiple exponential function method (MEF). As a result, we derive up the one to three-wave soliton solution to the both equations, while presenting the profiles obtained through the constant values related to those solutions. It is noteworthily to point out that, all the calculations in this paper are made with software package Maple. Also, the profiles of the waves depend on the arbitrary values of autocephalous parameters.

Two-dimensional Airy waves and three-wave solitons in quadratic media

We address the dynamics of two-dimensional (2D) truncated Airy waves (TAWs) and three-component solitons in the system of two fundamental-frequency (FF) and second-harmonic (SH) fields, coupled by quadratic () terms. The system models second–harmonic-generating optical media and atomic-molecular mixtures in Bose–Einstein condensates. In addition to stable solitons, the system maintains truncated-Airy-waves states in either one of the FF components, represented by exact solutions, which are stable, unlike the Airy waves in the degenerate (two-component) system. It is also possible to imprint vorticity onto the 2D Airy modes. By means of systematic simulations, we examine interactions between TAWs originally carried by different FF components, which are bending in opposite directions, through the SH field. The interaction leads to fusion of the input into a pair of narrow solitons. This is opposed to what happens in the 1D system, where the interacting Aity waves split into a large number of solitons. The interaction of truncated Aity waves carrying identical imprinted vorticities creates an additional pair of solitons, while opposite vorticities create a set of small-amplitude ‘debris’ in the output. Slowly moving solitons colliding with a heavy TAW bounce back, faster ones are absorbed by it, and collisions are quasi-elastic for fast solitons. Soliton-soliton collisions lead to merger into a single mode, or elastic passage, for lower and higher velocities, respectively.

The interaction of Airy waves and solitons in a three-wave system

We employ the generic three-wave system, with the interaction between two components of the fundamental frequency (FF) wave and the second-harmonic (SH) wave, to consider the collisions of truncated Airy waves (TAWs) and three-wave solitons in a setting which is not available in other nonlinear systems. The advantage of this is that single-wave TAWs, carried by either one of the FF components, are not distorted by the nonlinearity and are stable, three-wave solitons being stable too in the same system. The collision between mutually symmetric TAWs, carried by the different FF components, transforms them into a set of solitons, the number of which decreases with the increase of the total power. The TAW absorbs an incident small-power soliton, and a high-power soliton absorbs the TAW. Between these limits, the collision with an incident soliton converts the TAW into two solitons, with a remnant of the TAW attached to one of them, or leads to the formation of a complex TAW-soliton bound state. At large velocities, the collisions become quasi-elastic.

Three-Wave Trapponic Solitons for Tunable High-Repetition Rate Pulse Train Generation

We demonstrate that the parametric mixing of an ultrashort Zakharov-Manakov (ZM) soliton pulse with a quasi-continuous-wave (CW) signal in a weakly quadratic nonlinear and dispersive medium may lead to the generation of a three-wave soliton with variable speed, or trappon. We show that trappons may provide a novel mechanism of cavityless ultrashort pulse train generation. The timewidth, the amplitude and the repetition rate of the generated pulse trains may be controlled in a stable manner by adjusting the intensity level of the CW signal and/or the ZM short soliton pulse.

Collisions of three-wave solitons in media with competing quadratic and cubic nonlinearities

We report results of analysis of collisions between in-phase spatial solitons in a three-wave system, which includes competing ${\ensuremath{\chi}}^{(2)}:{\ensuremath{\chi}}^{(3)}$ nonlinearities and birefringence. The character of the collisions is quantified by a critical value of the initial tilt of the colliding solitons, $Q$, which separates quasielastic passage and merger (solitons with phase shift $\ensuremath{\pi}$ bounce from each other). If $Q$ is very close to ${Q}_{\text{cr}}$, the solitons may fuse into a double-peak state. ${Q}_{\text{cr}}$ is smaller for relatively large negative mismatch of the ${\ensuremath{\chi}}^{(2)}$ interactions. With the increase of the ${\ensuremath{\chi}}^{(3)}/{\ensuremath{\chi}}^{(2)}$ ratio, the system quickly transits from the behavior governed by the ${\ensuremath{\chi}}^{(2)}$ nonlinearity to a regime dominated by the ${\ensuremath{\chi}}^{(3)}$ terms.

Nonlinear stability, excitation and soliton solutions in electrified dispersive systems

A weakly nonlinear theory of wave propagation in two superposed dielectric fluids in the presence of a horizontal electric field is investigated in (2+1)-dimensions. The equation governing the evolution of the amplitude of the progressive waves is obtained in the form of a two-dimensional nonlinear Schrödinger equation. A three-wave resonant interaction for nonlinear excitations created from electrohydrodynamic capillary-gravity waves is observed to be possible in a dispersive medium with a self-focusing cubic nonlinearity. Under suitable conditions, the nonlinear envelope equations for the resonant interaction are derived by using multiple scales and inverse scattering methods, and an explicit three-wave soliton solution is discussed. Both the dynamic properties and the modulational instability of finite amplitude electrohydrodynamic wave are studied for the cubic nonlinear Schrödinger equation by means of linearized stability analysis and the nonlinear interaction coefficient. We show that the trajectories in phase space exhibit different behavior with the increase of nonlinear perturbations, and we determine the electric field and wavenumber ranges at which the original point is elliptic or hyperbolic, respectively. It is found also that the presence of the electric field in the equation modifies the nature of wave stability and soliton structures, and that the amplitude and width of the soliton are decreased and increased, respectively, when the electric field value increases.

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.

THREE-WAVE SOLITON EXCITATIONS IN A DISK-SHAPED BOSE–EINSTEIN CONDENSATE

A three-wave resonant interaction of collective modes and related soliton excitations in a disk-shaped Bose–Einstein condensate are investigated. The phase-matching conditions for the resonant interaction are satisfied by suitably choosing the wavevectors and the frequencies of the collective modes. A set of nonlinearly coupled envelope equations describing the spatio-temporal evolution of the three-wave resonant interaction are derived by using a method of multiple-scales, and some explicit (2+1)-dimensional three-wave soliton solutions are also presented and discussed.

Three-wave solitons and continuous waves in media with competing quadratic and cubic nonlinearities.

We formulate a general model of three-wave optical interactions (in the spatial domain), which combines quadratic (chi(2)) and cubic (chi(3)) nonlinearities, the latter including four-wave mixing. The model can be realized in chi(2) materials where an effective chi(3) nonlinearity is engineered by means of the quasi-phase-matching technique. Both self-focusing and self-defocusing chi(3) nonlinearities are considered. The birefringence of the two fundamental-frequency (FF) waves is taken into regard. Several types of solitons in this system are found, by means of the variational approximation and numerical methods. These are exact single-component solitons and generic three-wave (3W) ones, which are classified by relative signs of their components. Stability of the solitons is investigated by means of the Vakhitov-Kolokolov (VK) criterion, and then tested by direct simulations. One type of the single-component FF solitons (the "fast" one, in terms of the known two-component birefringent chi(3) model) is, chiefly, unstable, as in that model, but nevertheless a stability interval is found for it, which provides for the first example of stable fast solitons. The other FF soliton (the "slow" one, in terms of the same chi(3) model, where it is always stable) has its stability and instability regions. A single-component soliton in the second harmonic (SH) is found too; it also has its stability region, contrary to the common belief that such a soliton must always be unstable due to the parametric interaction. The 3W solitons are stable indeed if this is predicted by the VK condition, in the case when all the three components are positive. Following variation of the chi(2) mismatch parameter, the 3W soliton bifurcates from the SH one, and at another point it bifurcates back into the slow-FF single-component soliton; conjectured normal forms of the respective bifurcations are given. 3W solitons with different signs of their components may be unstable contrary to the VK criterion, which is explained by consideration of the chi(2) term in the system's Hamiltonian. In direct simulations, unstable solitons evolve into stable breathers. A different instability takes place in the case of the self-defocusing chi(3) nonlinearity, when all the solitons blow up into a turbulent state. Parallel to the solitons, continuous-wave solutions are studied too. In terms of the existence and stability, they resemble solitons of similar types.

Temporal walk-off for self-structuration of three-wave solitons in CW-pumped backward optical parametric oscillators

Stability analysis of the non-degenerate backward optical parametric oscillator in quasi-phase-matching decay interaction, between a CW-pump and counter-propagating signal and idler waves, proves that inhomogeneous stationary solutions exhibit a Hopf bifurcation, in contrast with the degenerate configuration. More importantly, we show that this Hopf instability gives rise to the generation of a backward three-wave soliton whose stable self-similar symbiotic structure is ensured by the group-velocity delay, or temporal walk-off, between the signal and the idler waves.

Phase-dependent interactions between three-wave spatial solitons in bulk quadratic media

We report the first experimental observation, to our knowledge, of phase-controlled three-wave interactions between two spatial solitons in a bulk quadratic nonlinear crystal. We demonstrate that the nature of the interaction can be attractive or repulsive, depending on the phase relationships between the input beams at the fundamental frequency and their power imbalance. Full control of the interaction is achieved by the adjustment of the relative phases and/or the orientation of the input waves, leading to scattering, fusion, spiraling, and power-exchange phenomena. Experimental observations are in good qualitative agreement with the corresponding theoretical and numerical results.

Quadratic solitons resulting from double-resonance wave mixing

The existence and stability of three-wave solitons, both (1+1) and (2+1) dimensional, that result from a double-resonance (type I plus type II) parametric interaction in a purely quadratic nonlinear medium are investigated. We demonstrate the existence of a family of stable solitons for a broad parameter range in the double-resonance model. Further, these solitons exhibit multistability, a property that is potentially useful for optical switching applications. We introduce a way to measure the quality of multistability and use this measure to compare the double-resonance model with single-resonance models in χ(2) media. We also discuss the modulational instability of the double-resonance system and present physical estimates of the power required for soliton generation.

Stable self-guided propagation of two optical harmonics coupled by a microwave or a terahertz wave

We propose a new scheme for self-trapping of (2+1)-dimensional optical beams: parametric interaction between self-guided waves of optical and microwave (or terahertz) frequencies, which relies on the combination of optical rectification and the electro-optic effect in a bulk noncentrosymmetric medium. We show that, in spite of strong domination of quasi-cubic terms, quadratic nonlinearity provides an effective stabilization mechanism to prevent collapse in a large range of parameters where three-wave solitons exist.

Soliton multistability as a result of double-resonance wave mixing in ?^(2) media

We investigate analytically and numerically the existence and stability properties of three-wave solitons resulting from double-resonance (type I plus type II) parametric interaction in a purely quadratic nonlinear medium. The existence of a family of stable solitons for the double-resonance model is demonstrated in a broad parameter range. Moreover, these solitons are shown to exhibit multistability, a feature that is potentially useful for optical switching applications. Finally, we find and present a novel family of quasi solitons.

Spatial optical solitons resulting from multistep cascading

We introduce a novel class of parametric optical solitons supported simultaneously by two second-order nonlinear cascading processes, second-harmonic generation and sum-frequency mixing. We obtain analytical and numerical solutions for three-wave spatial solitons and show that the presence of an additional cascading mechanism can change dramatically the properties and stability of two-wave quadratic solitary waves.

Three-wave gap solitons in waveguides with quadratic nonlinearity

Three-wave gap solitons in waveguides with quadratic nonlinearity

Self-induced phase-matching and three-wave bright spatial solitons in quadratic media

The family of three-wave bright spatial solitons due to non-degenerate two-wave mixing in quadratic media is characterised and calculated, which includes the previously calculated sech 2 solution as a special case. The notion of self-induced phase-matching is explained; the similarity of this three-wave soliton family with that due to second-harmonic generation, the reduction to third-order-like cascaded nonlinearity, and stability to propagation perturbations are also discussed.

HIERARCHY OF INSTABILITIES AND SOLITONS IN AN ACOUSTIC DISTRIBUTED SYSTEM WITH FEEDBACK

We have considered the hierarchy of instabilities, the transition to chaos, and the periodic generation of three-wave solitons in an acoustic distributed system with additional feedback and pumping. At positive feedback only steady-state conditions are shown to be stable. Self-oscillations, period-doubling bifurcations, and transition from quasiperiodicity to chaos of parametrically coupled waves have been found at negative feedback. “Hard” excitation of the generator leads to the formation of solitons. Established effects have been studied on dependence with physical parameters of a TeO 2 crystal.

Three-wave polariton solitons of new type in polar semiconductors

Abstract The formation of coupled three-wave solitons of new type under resonant interaction of two polariton waves by means of LO-phonons in polar semiconductors is first discussed. The consistent consideration of polariton effects enables to give classification of the soliton solutions by their dispersion. The possibility of the formation of quasi-phonon solitons with unusual properties is first reported.

Origin and interaction of three-wave solitons

Origin and interaction of three-wave solitons