Development Of Modulational Instability Research Articles

Vector breathers in the Manakov system

AbstractWe study theoretically the nonlinear interactions of vector breathers propagating on an unstable wavefield background. As a model, we use the two‐component extension of the one‐dimensional focusing nonlinear Schrödinger equation—the Manakov system. With the dressing method, we generate the multibreather solutions to the Manakov model. As shown previously in [D. Kraus, G. Biondini, and G. Kovačič, Nonlinearity 28(9), 3101, (2015)], the class of vector breathers is presented by three fundamental types I, II, and III. Their interactions produce a broad family of the two‐component (polarized) nonlinear wave patterns. First, we demonstrate that the type I and the types II and III correspond to two different branches of the dispersion law of the Manakov system in the presence of the unstable background. Then, we investigate the key interaction scenarios, including collisions of standing and moving breathers and resonance breather transformations. Analysis of the two‐breather solution allows us to derive general formulas describing phase and space shifts acquired by breathers in mutual collisions. The found expressions enable us to describe the asymptotic states of the breather interactions and interpret the resonance fusion and decay of breathers as a limiting case of infinite space shift in the case of merging breather eigenvalues. Finally, we demonstrate that only type I breathers participate in the development of modulation instability from small‐amplitude perturbations withing the superregular scenario, while the breathers of types II and III, belonging to the stable branch of the dispersion law, are not involved in this process.

Modulational Interaction in a Dusty Plasma of Meteoroid Wakes

This is a study of the possible modulational instability of electromagnetic waves in meteoroid wakes associated with the dust acoustic mode at altitudes of 80–120 km, which is a linear stage of modulational interaction. The parameters of meteoroid wakes at different altitudes in the Earth’s ionosphere are considered. It is shown that the charging of dust particles of meteoric matter creates conditions for the occurrence of dust acoustic waves. Dust acoustic disturbances are excited due to the modulational instability of electromagnetic waves from the meteoric trail. The influence of neutrals on the development of modulational interaction is taken into account. The concentration of neutrals in meteoric wakes is higher than the concentration of neutrals in the Earth’s ionosphere. It has been found that the condition for the excitation of a dust acoustic wave is satisfied for the typical parameters of dusty plasma of meteoroid wakes at altitudes of 100–120 km. Due to collisions between dust and neutrals, the development of modulation instability is suppressed at altitudes of 80–90 km, while inelastic collisions of neutrals with electrons and ions do not affect the development of modulational instability. The modulational instability of electromagnetic waves can explain the occurrence of low-frequency noise during the passage of meteoric bodies in a frequency range characteristic of dust acoustic waves. It is shown that the modulation instability has time to develop for characteristic temperatures and particle concentrations in meteoroid wakes. Equations for the charging of dust particles in meteoroid wakes are given. It has been found that the dust is positively charged, both in the daytime and at night, due to intense emission currents from the surface of dust particles.

Some aspects of modulational interaction in Earth's dusty ionosphere

The development of modulational instability involving dust acoustic perturbations in dusty ionospheric plasma and in dusty plasmas of meteor tails in Earth's ionosphere was considered. The effect of collisions of electrons, ions, and dust grains with neutrals at different altitudes was estimated. It is shown that, in this case, the influence of collisions of electrons and ions with neutrals is usually less significant than the influence of collisions between dust grains and neutrals. It is demonstrated that the effect of the modulational instability on the propagation of electromagnetic waves in the dusty ionospheric plasma is the most significant at heights of 100–120 km. The values of the wave vectors of the electromagnetic pump wave at which inelastic collisions with neutrals are important for the development of modulational interaction are calculated. The modulational interaction in the dusty ionosphere is important for the explanation of different phenomena such as ground-based observations of low-frequency ionospheric radio noises with frequencies below 60 Hz. The absence of observations of low-frequency ionospheric radio noise during such phenomena as noctilucent clouds and polar mesosphere summer echoes caused by the presence of dusty plasmas at heights of 80–95 km is explained by suppression of the development of the modulational instability at these heights. The role of inelastic collisions with neutrals in meteor tails is also discussed. It is shown that for typical parameters of dusty plasmas of meteor tails such collisions do not influence the development of the modulation instability in meteor tails.

Electrophonic noises from meteors and dust acoustic modulational perturbations

The article discusses dusty plasmas processes in the Earth’s ionosphere associated with the passages of meteoroids and the spread of the meteor tail. Mechanisms explaining the occurrence of electrophonic noises recorded simultaneously with the passage of meteor bodies are suggested for extremely low frequencies range. It is explained by modulational instability of electromagnetic waves from meteor associated with the dust acoustic mode with characteristic frequencies 0.03–60 Hz. As electromagnetic waves propagate in the dusty plasmas of the meteor tail or in the dusty ionospheric plasmas, dust acoustic waves are excited as a result of the development of modulational instability of electromagnetic waves at characteristic frequencies and as a result, the electromagnetic waves become modulated. At the surface, these perturbations can be converted into sound waves by means of receivers. The growth rates of modulational instability at which the modulational excitation of extremely low-frequency dust acoustic disturbances occur are calculated. The conditions of the development of modulational instability are given. The correlation between observations of ionospheric radio noises and passages of meteors is noticed. Observed data of electrophonic noises include discussing range of frequencies 0.03–60 Hz.

Effect of Discrete Breathers on the Specific Heat of a Nonlinear Chain

A nonlinear chain with six-order polynomial on-site potential is used to analyze the evolution of the total to kinetic energy ratio during development of modulational instability of extended nonlinear vibrational modes. For the on-site potential of hard-type (soft-type) anharmonicity, the instability of $q=\pi$ mode ($q=0$ mode) results in the appearance of long-living discrete breathers (DBs) that gradually radiate their energy and eventually the system approaches thermal equilibrium with spatially uniform and temporally constant temperature. In the hard-type (soft-type) anharmonicity case, the total to kinetic energy ratio is minimal (maximal) in the regime of maximal energy localization by DBs. It is concluded that DBs affect specific heat of the nonlinear chain and for the case of hard-type (soft-type) anharmonicity they reduce (increase) the specific heat.

On the Plasma–Dust Processes Accompanying Meteor Showers

Plasma–dust processes in the Earth’s ionosphere related to transit of meteor bodies and spreading of meteor trail are analyzed. Processes that occur in the meteor trail itself and those taking place in the Earth’s atmosphere upon transit of a meteor body, along with their influence on observers and operation of various technical systems, are investigated. In particular, mechanisms explaining acoustic phenomena accompanying meteor transit during meteor showers are proposed. The mechanism of formation of dusty plasma in meteoroid trails is described. It is demonstrated that charging of dust particles of the meteoroid material creates conditions for the appearance of dust acoustic waves. Dust acoustic perturbations are driven as a result of development of modulation instability of electromagnetic waves emitted by the meteor trail and have frequencies typical of dust sound (0.003–60 Hz). The reports of witnesses revealed that observers on the Earth can hear low-frequency sounds appearing during meteor showers. It is suggested that the sound reaches the Earth’s surface as a result of transfer of oscillations from dust particles to neutrals the concentration of which in the atmosphere is high. The mechanisms of fragmentation of meteor bodies and charging of formed dust particles are analyzed. Characteristic charges of dust fragments for nano- and micrometer-scale particles are estimated. Particles of different shape are considered. The growth rates that give rise to modulation excitation of low-frequency dust acoustic perturbations are calculated. Conditions leading to development of a modulation instability are found.

Effect of discrete breathers on macroscopic properties of the Fermi-Pasta-Ulam chain

The effect of discrete breathers (DBs) on macroscopic properties of the Fermi-Pasta-Ulam chain with symmetric and asymmetric potentials is investigated. The total to kinetic energy ratio (related to specific heat), stress (related to thermal expansion), and Young's modulus are monitored during the development of modulational instability of the zone boundary mode. The instability results in the formation of chaotic DBs followed by the transition to thermal equilibrium when DBs disappear due to energy radiation in the form of small-amplitude phonons. It is found that DBs reduce the specific heat for all the considered chain parameters. They increase the thermal expansion when the potential is asymmetric and, as expected, thermal expansion is not observed in the case of symmetric potential. The Young's modulus in the presence of DBs is smaller than in thermal equilibrium for the symmetric potential and for the potential with a small asymmetry, but it is larger than in thermal equilibrium for the potential with greater asymmetry. Our results can be useful for setting experiments on the identification of DBs in crystals by measuring their macroscopic properties.

Deep learning-based classification of high intensity light patterns in photorefractive crystals

In this paper, we establish a new scheme for identification and classification of high intensity events generated by the propagation of light through a photorefractive SBN crystal. Among these events, which are the inevitable consequence of the development of modulation instability, are speckling and soliton-like patterns. The usual classifiers, developed on statistical measures, such as the significant intensity, often provide only a partial characterization of these events. Here, we try to overcome this deficiency by implementing the convolution neural network method to relate experimental data of light intensity distribution and corresponding numerical outputs with different high intensity regimes. The train and test sets are formed of experimentally obtained intensity profiles at the crystal output facet and corresponding numerical profiles. The accuracy of detection of speckles reaches maximum value of 100%, while the accuracy of solitons and caustic detection is above 97%. These performances are promising for the creation of neural network based routines for prediction of extreme events in wave media.

MODULATION INSTABILITY IN TWO COMPONENT BOSE-EINSTEIN CONDENSATE WITH RELATIVE COMPONENT MOTION

The development of modulation instability in a spatially homogeneous two-component Bose-Einstein condensate (BEC), in which the interacting components move through each other at a relative speed, is investigated. It is shown that nonlinear dynamics, leading to modulation instability, is determined by both the values of the constant interaction and the relative velocity between the components. The maximum oscillation increment is found and the limits of the existence of modulation instability in the space of wave numbers are determined.

Rogue waves generation through multiphase solutions degeneration for the derivative nonlinear Schrödinger equation

The generation of rogue waves from the development of modulational instability of the plane waves due to small perturbations or interactions of solitons is usually modeled by some breather solutions, which are considered to as solitons on a finite background. The instability of small-amplitude perturbations is usually chaotic and may contain many frequencies in their spectra. Thus, a more general description of rogue wave generation can be achieved via the consideration of multiphase solutions and their interactions. The general N-order phase solutions of the derivative nonlinear Schrodinger (DNLS) equation are constructed from the trivial seed (zero solution) by using the determinant representation of the N-fold Darboux transformation. By adjusting the relative phases of the multiphase solutions in the interacting area, namely taking the limit $$\lambda _k \rightarrow \lambda _\mathrm{c}$$ for some of $$\lambda _k$$ (not for all of them), where $$\lambda _\mathrm{c}$$ is a critical eigenvalue associated with the synchronization of each phase in the multiphase nonlinear waves of the DNLS equation, it is possible to obtain different types of quasi-rational solutions from the multiphase solutions degeneration mechanism: quasi-rational traveling wave solutions, rogue waves, and quasi-rogue waves with periodic conditions. Hence, this multiphase solution degeneration represents a new mechanism of rogue wave generation.

Role of Collisions with Neutrals in the Process of Modulational Excitation of Dust Acoustic Perturbations in Dusty Ionosphere

The role of inelastic collisions of electrons and ions with neutrals during the development of modulational instability involving dust acoustic perturbations in dusty ionospheric plasma, as well as the effect of collisions of electrons, ions, and dust grains with neutrals on the manifestations of modulational interaction in the dusty ionosphere, are estimated. It is shown that, in this case, the influence of collisions of electrons and ions with neutrals is usually less significant than the influence of collisions between dust grains and neutrals. It is demonstrated that the effect of modulational instability on the propagation of electromagnetic waves in the dusty ionospheric plasma is the most significant at altitudes of 100–120 km. The modulational interaction in the dusty ionosphere is important for the explanation of such effects as ground-based observations of low-frequency ionospheric radio noises with frequencies below 50 Hz, generation of infrasonic waves in the ionosphere and the possibility to detect them near the Earth’s surface, enhancement of the green nightglow emission at a wavelength of 557.7 nm from the lower ionosphere layer at altitudes of 110–120 km, and modulational excitation of inhomogeneities in the electron and ion densities in the ionosphere at altitudes of 100–120 km. The absence of observations of low-frequency ionospheric radio noise during such phenomena as noctilucent clouds and polar mesosphere summer echoes caused by the presence of dusty plasma at altitudes of 80–95 km is explained. It is shown that the latter phenomenon is related to the suppression of modulational processes at these altitudes.

Modulation Instability in Two Component Bose-Einstein Condensate with Dissipation

In this paper, we consider the dynamic evolution of a binary mixture of a Bose-Einstein condensate taking into account the presence of dissipation inside the components. Using the introduction of the dissipative function, the modified Gross-Pitaevskii equations are obtained. These equations, in contrast to the usual Gross-Pitaevskii equations for two-component condensate, allow us to take into account the dissipation in the system. The influence of dissipative processes on the development of modulation instability in a spatially homogeneous two-component Bose-Einstein condensate is investigated. In contrast to the one-component Bose-Einstein condensate, in which modulation instability arises only when there are forces of attraction between atoms, in a two-component Bose-Einstein condensate nonlinear dynamics, leading to modulation instability is more complex. It essentially depends on the signs and values of the constant interaction of the components, which leads to a greater variety of possible scenarios for the development of modulation instability. The paper considers two cases. The first case is when repulsive forces act inside the components, and the second is when repulsive forces act in the first component, and in the second one - attractive forces. At the same time, the situation when there is a repulsion in the first component, and attraction between the particles in the second component differs significantly from the case of only positive interaction inside the components. The relations between the interaction constants that determine the development of the modulation instability turn out to be different. Given the relations between the interaction constants, taking into account dissipation processes, the occurrence of modulation instability in two-component Bose-Einstein condensates was studied, the maximum growth rate of oscillations was found, and the limits of the existence of modulation instability in the space of wave numbers were found. It is shown that the small effect of dissipation on the modulation instability in the Bose – Einstein condensate is explained not only by the smallness of the friction forces. For wave vectors corresponding to a mode with a maximum increment, the contribution of dissipation in the linear approximation with respect to the dissipative parameter is strictly zero. Thus, the condition for the development of the most rapidly growing mode of oscillations, which determines the beginning of the modulation instability, remains the same as in the nondissipative case.

Generation of subpicosecond pulses due to the development of modulation instability of whispering-gallery-mode wave packets in an optical waveguide with a travelling refractive-index wave

We consider the transformation of a quasi-continuous (with the duration greater than 1 ns) background tunnelling wave of the whispering gallery mode (WGM) type into ultrashort (pico- and subpicosecond) laser pulses, arising due to the development of modulation instability in a cylindrical optical waveguide with travelling refractive-index wave. It is shown that the peak power of the generated laser pulses can exceed the power of the background WGM by a few orders of magnitude.

Axisymmetric Nonlinear Modulated Waves in a Cylindrical Shell

A quasi-hyperbolic equation is derived that simulates the axisymmetric propagation of bending waves in a cylindrical shell, which interacts with a nonlinearly elastic medium. With the correct asymptotic procedure, the study of a wave process reduces to analysis of a nonlinear Schrodinger equation. It is established that the development of modulation instability requires a “soft” nonlinearity of the medium surrounding the shell. Operating modes that allow the propagation of stable light envelope solitons are revealed.

Formation of rogue waves from a locally perturbed condensate.

The one-dimensional focusing nonlinear Schrödinger equation (NLSE) on an unstable condensate background is the fundamental physical model that can be applied to study the development of modulation instability (MI) and formation of rogue waves. The complete integrability of the NLSE via inverse scattering transform enables the decomposition of the initial conditions into elementary nonlinear modes: breathers and continuous spectrum waves. The small localized condensate perturbations (SLCP) that grow as a result of MI have been of fundamental interest in nonlinear physics for many years. Here, we demonstrate that Kuznetsov-Ma and superregular NLSE breathers play the key role in the dynamics of a wide class of SLCP. During the nonlinear stage of MI development, collisions of these breathers lead to the formation of rogue waves. We present new scenarios of rogue wave formation for randomly distributed breathers as well as for artificially prepared initial conditions. For the latter case, we present an analytical description based on the exact expressions found for the space-phase shifts that breathers acquire after collisions with each other. Finally, the presence of Kuznetsov-Ma and superregular breathers in arbitrary-type condensate perturbations is demonstrated by solving the Zakharov-Shabat eigenvalue problem with high numerical accuracy.

On mathematical models for describing parametric and modulation instabilities

The development of modulation instability of a finite amplitude wave is considered. A modified S-theory, previously developed in the works of V. Ye. Zakharov, was used to describe unstable modes. It is noted that near the instability threshold a self-similar spatial wave structure can form at different scales. The dynamics of the instability is analyzed in a model of a high intensity practically undamped oceanic wave, the mechanism of the formation of anomalous amplitude waves is shown. The modes of excitation of parametric instabilities are discussed. It is noted, that there is a similarity in the development of modulation and parametric instabilities, however there are also differences related to the nature of the dynamics of the wave packets of unstable perturbations. The nature of parametric instabilities is associated with the presence of a high-frequency, spatially homogeneous component of one or more parameters of the medium. Since the wave number of such HF oscillations is zero, or extremely small, in comparison with the characteristic lengths of the system, the spectrum of unstable disturbances is also located symmetrically, resembling the unstable spectrum of modulation instability. However, the growing perturbations practically do not move in space. Far from the threshold of parametric instability, a strong small-scale modulation of environmental parameters can form, which often takes on the character of a process with aggravation. The formation of self-similar spatial structures in the developed convection of a thin liquid or gas layer due to the development of modulation instability. The toroidal convection vortices generate poloidal vortices of large scale - the effect of a hydrodynamic dynamo and the experimental results of the investigation self-similar structures on the graphite are presented. The nature of self-consistent parametric instabilities is similar to the processes of modulation instability.

Fermi–Pasta–Ulam recurrence and modulation instability

We give a qualitative explanation of the analog of the Fermi-Pasta-Ulam (FPU) recurrence in a one-dimensional focusing nonlinear Schrodinger equation (NLSE). That recurrence can be considered as a result of the nonlinear development of modulation instability. All known exact localized solitons-type solutions describing propagation on the background of the modulationally unstable condensate show the recurrence to the condensate state after its interaction with solitons. The condensate state locally recovers its original form with the same amplitude but a different phase after soliton leave its initial region. This is the analog of the FPU recurrence for the NLSE. Based on the integrability of the NLSE, we demonstrate that the FPU recurrence takes place not only for condensate but for more general solution in the form of the cnoidal wave. This solution is periodic in space and can be represented as a solitonic lattice. That lattice reduces to isolated soliton solution in the limit of large distance between solitons. The lattice transforms into the condensate in the opposite limit of dense soliton packing. The cnoidal wave is also modulationally unstable due to soliton overlapping. This instability at the linear stage does not provide the cnoidal wave recurrence. The recurrence happens at the nonlinear stage of the modulation instability. From the practical point of view the latter property is very important, especially for the fiber communication systems which use soliton as an information carrier.

Evolution of surface plasmon–polariton wave in a thin metal film: The modulation‐instability effect

The modulation instability development of intensive surface plasmon–polariton waves in a thin metal film is studied. It is shown both analytically and numerically that the modulation‐instability effect can give rise to spatial redistribution and longitudinal localization of surface plasmon–polariton wave energy on the subwavelength scale. Analytical expressions for the driving parameters of the modulation instability process − nonlinearity and dispersion − are derived. The impact of the film thickness and dielectric permittivities of constituents on the dynamics of surface plasmon–polariton wave transformation is considered. Numerical simulations show that in the layer structure comprising a silver film of subwavelength thickness a train of subpicosecond optical pulses with high repetition rate can be generated. image

Localized spin excitations in a disordered antiferromagnetic chain with biquadratic interactions

Dynamical theory of soliton excitation in one dimensional antiferromagnet (AFM) is studied by a revised Hamiltonian in which biquadratic interaction is taken into account in addition to the uniaxial anisotropy and exchange energy. By using Holstein-Primakoff transformation, the coherent state ansatz and the time-dependent variational principle, we obtain a set of two coupled nonlinear partial differential equations that governs the dynamics of the system. Sine-cosine function method is used to study the complete nonlinear soliton excitation and the effect of inhomogeneity in the system. The presence of inhomogeneity is found to cause a disorder in the AFM system. Finally, the evolution of Modulational Instability (MI) is analyzed in the presence of small perturbations.

Experimental study on modulational instability and evolution of crescent waves

A series of experiments on the instability of steep water wave trains in water with finite water depths and infinite water depths in a wide wave basin were performed. It was found that under the coupled development of modulational instability and class-II instability, the initial two-dimensional steep wave trains evolved into three-dimensional crescent waves, followed by the occurrence of disordered water surfaces, and that the wave energy transferred to sidebands in the amplitude spectrum of the water surface elevation. The results also show that water depth has a significant effect on the growth of modulational instability and the evolution of crescent waves. The larger the water depth, the more quickly the modulational instability suppresses class-II instability.