On diffraction and dispersion effect on three wave interaction

The effect of diffraction and dispersion on three wave coupling is investigated. The instability leading to space modulation of packets is obtained. This instability and its nonlinear stage is similar to modulational instability of quasimonochromatic waves. The localized structures (solitons and waveguide) can be a result of the instability. We demonstrate that one-dimensional and two-dimensional such structures are unstable. The existence of stable three-dimensional solitons is shown.     

Conservation laws, soliton solutions and modulational instability for the higher-order dispersive nonlinear Schrödinger equation

In this paper, analytically investigated is a higher-order dispersive nonlinear Schrödinger equation. Based on the linear eigenvalue problem associated with this equation, the integrability is identified by admitting an infinite number of conservation laws. By using the Darboux transformation method, the explicit multi-soliton solutions are generated in a recursive manner. The propagation characteristic of solitons and their interactions under the periodic plane wave background are discussed. Finally, the modulational instability of solutions is analyzed in the presence of small perturbation.     

Propagation dynamics of relativistic electromagnetic solitary wave as well as modulational instability in plasmas

By one-dimensional particle-in-cell(PIC) simulations, the propagation and stability of relativistic electromagnetic(EM) solitary waves as well as modulational instability of plane EM waves are studied in uniform cold electron-ion plasmas.The investigation not only confirms the solitary wave motion characteristics and modulational instability theory, but more importantly, gives the following findings. For a simulation with the plasma density 10²³ m^-3 and the dimensionless vector potential amplitude 0.18, it is found that the EM solitary wave can stably propagate when the carrier wave frequency is smaller than 3.83 times of the plasma frequency. While for the carrier wave frequency larger than that, it can excite a very weak Langmuir oscillation, which is an order of magnitude smaller than the transverse electron momentum and may in turn modulate the EM solitary wave and cause the modulational instability, so that the solitary wave begins to deform after a long enough distance propagation. The stable propagation distance before an obvious observation of instability increases(decreases) with the increase of the carrier wave frequency(vector potential amplitude). The study on the plane EM wave shows that a modulational instability may occur and its wavenumber is approximately equal to the modulational wavenumber by Langmuir oscillation and is independent of the carrier wave frequency and the vector potential amplitude.This reveals the role of the Langmuir oscillation excitation in the inducement of modulational instability and also proves the modulational instability of EM solitary wave.     

Generation of Bragg solitons through modulation instability in a Bragg grating structure

In this article, we consider the continuous wave (cw) propagation through the nonlinear periodic structure that consists of alternating layers of both positive and negative Kerr coefficients along the propagation direction. We investigate the modulational instability (MI) conditions required for the generation of ultrashort pulses for the nonlinearity management system. We study the occurrence of MI at the top and bottom edges of the photonic band gap (PBG) where the forward and backward propagating waves are strongly coupled because of the presence of the grating structure. We also study the MI when cw is detuned from the edges of the PBG into the anomalous and normal dispersion regimes. In addition, we discuss the existence of gap solitons for the nonlinearity management system in the upper and lower branches of the dispersion curve through the MI gain spectra. We observe the generation of higher order solitons in the nonlinear periodic structure when the input power is increased beyond a certain critical level. Finally, we discuss the generation of higher order Bragg grating solitons through the intensity evolution of the forward and backward propagating fields.     

Modulation of dust acoustic waves with non-adiabatic dust charge fluctuations

The modulational instability of dust acoustic waves in a dusty plasma with non-adiabatic dust charge fluctuation is studied. Using the perturbation method, a modified nonlinear Schrödinger equation containing a damping term that comes from the effect of dust charge variation is derived. It is found that the modulational instability of the wave packet and the propagation characters of the envelope solitary waves are modified significantly by the non-adiabatic dust charge fluctuation.     

Modulational instability and moving gap soliton in Bose–Einstein condensation with Feshbach resonance management

We investigate the modulational instability of symmetric and asymmetric continuous wave solutions in Bose–Einstein condensates in optical lattices with Feshbach resonance managed atomic scattering length. The model is based on a pair of averaged coupled mode Gross–Pitaevskii equations. We analyze the characteristics of the modulational instability in the form of typical dependence of the instability growth rate on the perturbation wavenumber and system’s parameters. We have numerically solved the coupled mode equations by using the split step Fourier method. Convincing agreement has been obtained between analytical and numerical results. Furthermore, the moving and stationary gap solitons in the first spectral gap of the optical lattices for the same amplitude but different phases in the presence and absence of the mean atomic scattering length under the Feshbach resonance management are also constructed.     

Intense circularly polarized waves in plasmas

The nonlinear propagation of an intense circularly polarized electromagnetic wave in a cold plasma is investigated. A relativistic modulational instability and an exact localized solutions are found.     

Modulational instability of a strip beam in a bulk type I quadratic medium

The evolution of a strip (one-dimensional) fundamental beam with propagation distance owing to spatial modulational instabilities was analyzed in a quadratic medium near type I phase matching. We obtained the gain coefficient for the modulational instability and showed that the wave evolves into a clean periodic sequence of solitary waves and does not reproduce the incident beam.     

Modulational instability and gap solitons in periodic ferromagnetic films

The modulational instability and gap solitons are theoretically studied in the ferromagnetic films under a periodic magnetic field. By multiple scale expansion, the envelope soliton solutions are obtained naturally. Due to the periodic modulation of dispersion, the solitons may be pushed into the gap region. For the easy-axis magnetic film, the red-shift of frequency leads to a modulational instability in the bottom of band and generates a bright gap soliton. For the easy-plane case, the blue-shift leads to an instability in the top of band and a dark gap soliton emerges. The weak damping produces an attenuation factor and a small oscillation.     

Optical solitons due to quadratic nonlinearities: from basic physics to futuristic applications

We present an overview of nonlinear phenomena related to optical quadratic solitons—intrinsically multi-component localized states of light, which can exist in media without inversion symmetry at the molecular level. Starting with presentation of a few derivation schemes of basic equations describing three-wave parametric wave mixing in diffractive and/or dispersive quadratic media, we discuss their continuous wave solutions and modulational instability phenomena, and then move to the classification and stability analysis of the parametric solitary waves. Not limiting ourselves to the simplest spatial and temporal quadratic solitons we also overview results related to the spatio-temporal solitons (light bullets), higher order quadratic solitons, solitons due to competing nonlinearities, dark solitons, gap solitons, cavity solitons and vortices. Special attention is paid to a comprehensive discussion of the recent experimental demonstrations of the parametric solitons including their interactions and switching. We also discuss connections of quadratic solitons with other types of solitons in optics and their interdisciplinary significance.     

Soliton transverse instabilities in anisotropic nonlocal self-focusing media

We study, both numerically and experimentally, the transverse modulational instability of spatial stripe solitons in anisotropic nonlocal photorefractive media. We demonstrate that the instability scenarios depend strongly on the stripe orientation, but the anisotropy-induced features are largely suppressed for spatial solitons created by self-trapping of partially incoherent light.     

Optical vortex solitons in parametric wave mixing

We analyze two-component spatial optical vortex solitons supported by parametric wave mixing processes in a nonlinear bulk medium. We study two distinct cases of such localized waves, namely, parametric vortex solitons due to phase-matched second-harmonic generation in an optical medium with competing quadratic and cubic nonlinear response, and vortex solitons in the presence of third-harmonic generation in a cubic medium. We find, analytically and numerically, the structure of two-component vortex solitons, and also investigate modulational instability of their plane-wave background. In particular, we predict and analyze in detail novel types of vortex solitons, a "halo-vortex," consisting of a two-component vortex core surrounded by a bright ring of its harmonic field, and a "ring-vortex" soliton which is a vortex in a harmonic field that guides a ring-like localized mode of the fundamental-frequency field.     

Electrostatic dust-acoustic envelope solitons in an electron-depleted plasma

A standard nonlinear Schrödinger equation has been established by using the reductive perturbation method to investigate the propagation of electrostatic dust-acoustic waves, and their modulational instability as well as the formation of localized electrostatic envelope solitons in an electron-depleted unmagnetized dusty plasma system comprising opposite polarity dust grains and super-thermal positive ions. The relevant physical plasma parameters (viz., charge, mass, number density of positive and negative dust grains, and super-thermality of the positive ions, etc.) have rigorous impact to recognize the stability conditions of dust-acoustic waves. The present study is useful for understanding the mechanism of the formation of dust-acoustic envelope solitons associated with dust-acoustic waves in the laboratory and space environments.     

Transverse instability of vector solitons and generation of dipole arrays

We develop a theory of modulational instability of multiparameter solitary waves and analyze the transverse instability of composite (or vector) optical solitons in a saturable nonlinear medium. We demonstrate theoretically and experimentally that a soliton stripe breaks up into an array of ( 2+1)-dimensional dipole-mode vector solitons, thus confirming the robust nature of those solitons as fundamental composite structures of incoherently coupled fields.     

Dark and anti-dark vector solitons of the coupled modified nonlinear Schrödinger equations from the birefringent optical fibers

Coupled modified nonlinear Schr?dinger (CMNLS) equations describe the pulse propagation in the picosecond or femtosecond regime of the birefringent optical fibers. A new type of the Lax pair and another hierarchy of the infinitely many conservation laws are derived based on the Wadati-Konno-Ichikawa system. By means of the Hirota method, soliton solutions in the normal dispersion regime are obtained. Parametric regions for the existence of dark and anti-dark vector soliton solutions are given. Asymptotic analysis shows that the collision between two solitons (two anti-dark solitons, two dark solitons, or dark and anti-dark solitons) in each polarization direction is elastic. Moreover, there is no energy transfer between two polarization components of each vector soliton, whether dark or anti-dark vector soliton. In addition, dark and anti-dark solitons can coexist on the same background seen from the collision between the dark and anti-dark solitons in one polarization direction. Our graphical analysis shows that the parameters in the CMNLS equations not only determine the regions for the existence of dark and anti-dark soliton solutions but also control the phase and direction of the propagation of the solitons. Finally, through the linear stability analysis, the modulational instability condition is given.     

Coupled modulational instability of copropagating spin waves in magnetic thin films

This letter reports the first results on the coupled modulational instability of copropagating spin waves in a magnetic film. Strong instability was observed for the two waves with either attractive or repulsive nonlinearity. If the two waves have attractive nonlinearity, the instability leads to the formation of bright solitons. If the two waves have repulsive nonlinearity, the process results in the formation of black solitons. The instability was also observed for the two waves in separated attractive-repulsive nonlinearity regimes.     

Instability and evolution of nonlinearly interacting water waves

We consider the modulational instability of nonlinearly interacting two-dimensional waves in deep water, which are described by a pair of two-dimensional coupled nonlinear Schr?dinger equations. We derive a nonlinear dispersion relation. The latter is numerically analyzed to obtain the regions and the associated growth rates of the modulational instability. Furthermore, we follow the long term evolution of the latter by means of computer simulations of the governing nonlinear equations and demonstrate the formation of localized coherent wave envelopes. Our results should be useful for understanding the formation and nonlinear propagation characteristics of large-amplitude freak waves in deep water.     

Modulation instability of lower hybrid waves leading to cusp solitons in electron–positron(hole)–ion Thomas Fermi plasma

Following the idea of three‐wave resonant interactions of lower hybrid waves, it is shown that quantum‐modified lower hybrid (QLH) wave in electron–positron–ion plasma with spatial dispersion can decay into another QLH wave (where electron and positrons are activated, whereas ions remain in the background) and another ultra‐low frequency quantum‐modified ultra‐low frequency Lower Hybrid (QULH) (where ions are mobile). Quantum effects like Bohm potential and Fermi pressure on the lower hybrid wave significantly reshaped the dispersion properties of these waves. Later, a set of non‐linear Zakharov equations were derived to consider the formation of QLH wave solitons, with the non‐linear contribution from the QLH waves. Furthermore, modulational instability of the lower hybrid wave solitons is investigated, and consequently, its growth rates are examined for different limiting cases. As the growth rate associated with the three‐wave resonant interaction is generally smaller than the growth associated with the modulational instability, only the latter have been investigated. Soliton solutions from the set of coupled Zakharov and NLS equations in the quasi‐stationary regime have been studied. Ordinary solitons are an attribute of non‐linearity, whereas a cusp soliton solution featured by nonlocal nonlinearity has also been studied. Such an approach to lower hybrid waves and cusp solitons study in Fermi gas comprising electron positron and ions is new and important. The general results obtained in this quantum plasma theory will have widespread applicability, particularly for processes in high‐energy plasma–laser interactions set for laboratory astrophysics and solid‐state plasmas.     

Propagation and modulational instability of Rossby waves in stratified fluids

Perturbation analysis and scale expansion are used to derive the (2+1)-dimensional coupled nonlinear Schrödinger (CNLS) equations that can describe interactions of two Rossby waves propagating in stratified fluids. The (2+1)-dimensional equations can reflect and describe the wave propagation more intuitively and accurately. The properties of the two waves in the process of propagation can be analyzed by the solution obtained from the equations using the Hirota bilinear method, and the influence factors of modulational instability are analyzed. The results suggest that, when two Rossby waves with slightly different wave numbers propagate in the stratified fluids, the intensity of bright soliton decreases with the increases of dark soliton coefficients. In addition, the size of modulational instable area is related to the amplitude and wave number in y direction.     

Modulational instability and complex dynamics of confined matter-wave solitons

We study the formation of bright solitons in a Bose-Einstein condensate of 7Li atoms induced by a sudden change in the sign of the scattering length from positive to negative, as reported in a recent experiment [Nature (London) 417, 150 (2002)]]. The numerical simulations are performed by using the Gross-Pitaevskii equation with a dissipative three-body term. We show that a number of bright solitons is produced and this can be interpreted in terms of the modulational instability of the time-dependent macroscopic wave function of the Bose condensate. In particular, we derive a simple formula for the number of solitons that is in good agreement with the numerical results. We find that during the motion of the soliton train in an axial harmonic potential the number of solitonic peaks changes in time and the density of individual peaks shows an intermittent behavior.