Regions Of Modulational Instability Research Articles

Modified three-dimensional nonlinear Schrödinger equation in finite water depth for gravity waves with influence of slowly varying currents

A new modified three-dimensional (3D) nonlinear Schrödinger equation (3DMNLSE) is derived for gravity waves in the presence of wind, dissipation, and two-dimensional slowly varying currents, which include transverse and longitudinal currents in finite water depth. The effect of currents on modulational instability (MI) is investigated. The divergence (convergence) effect of longitudinal favorable (adverse) currents decreases (increases) the MI growth rate and region of instability while suppressing (enhancing) the formation of freak waves. Meanwhile, the transverse currents hardly affect the occurrence of MI. Furthermore, some results from the simulations with the space evolution of 3DMNLSE are presented. The results show the ubiquitous occurrence of freak waves in 3D wave fields under certain sets of initial conditions. We demonstrate that larger waves can be triggered when a weakly modulated wave train enters a region of adverse currents. The maximum amplitude of a freak wave depends on the ratio of the current velocity to the wave phase velocity.

Modulational instability of dust-ion acoustic waves in generalized (r, q) distributed electron dissipative plasmas

The modulational instability of the dust-ion acoustic waves in a dissipative dusty plasma system containing warm ions, generalized ( r , q ) distributed electrons, and immobile dust grains has been studied. The Krylov–Bogoliubov–Mitropolsky (KBM) perturbation method is employed to analyze the amplitude modulation of a monochromatic plane wave through the derivation of a nonlinear Schrödinger equation. This equation is solved analytically to determine the modulational instability region, and the corresponding dissipative rogue wave solution is obtained. The effects of various system parameters, such as the kinematic viscosity of the ions (υ), the wavenumber (k), the ion to electron temperature ratio ( σ i ), the unperturbed ion to electron density ratio (γ) as well as the electron distribution parameters: r and q, are investigated. The dependence of the amplitude of dissipated rogue waves on υ is discussed. It is found that various system parameters have significant effects on the features of dissipated rogue waves. The present investigation can be relevance to the electrostatic rogue structures in various cosmic dust-laden plasmas such as Saturn's F-ring.

Formation, propagation, and excitation of matter solitons and rogue waves in chiral BECs with a current nonlinearity trapped in external potentials.

In this paper, we investigate formation and propagation of matter solitons and rogue waves (RWs) in chiral Bose-Einstein condensates modulated by different external potentials, modeled by the chiral Gross-Pitaevskii (GP) equation with the current nonlinearity and external potentials. On the one hand, the introduction of two potentials (Pöschl-Teller and harmonic-Gaussian potentials) enables the discovery of exact soliton solutions in both focusing and defocusing cases. We analyze the interplay effects of current nonlinearity and potential on soliton stability via associated Bogoliubov-de Gennes equations. Moreover, multiple families of numerical solitons (ground-state and dipole modes) trapped in potentials are found, exhibiting distinctive structures. The interactions between solitons trapped in potentials are studied, which exhibit the inelastic trajectories and repulsive interactions. On the other hand, we introduce the time-dependent potentials such that the controlled RWs are found in both focusing and defocusing GP equations with current nonlinearity. Furthermore, through the interaction between potentials and current nonlinearity, it is possible to enlarge the region of modulational instability, leading to the generation of RWs and chiral solitons. High-order RWs are generated from several Gaussian perturbations on a continuous wave. The presence of current nonlinearity disrupts the structures of these high-order RWs, causing them to undergo a transform into chiral lower-amplitude solitons. Finally, various types of soliton excitations are investigated by varying the strengths of potential and current nonlinearity, showing the abundant dynamic transforms of chrital matter solitons.

Modulation instability generation with blue-detuned pump laser in coupled microcavities

Optical frequency combs based on microcavities with Kerr nonlinearity are promising frequency comb sources for many applications. A typical Kerr soliton comb is generated in a nonlinear microcavity with anomalous dispersion pumped by a red-detuned continuous-wave laser. Modulation instability (MI) is the basis for Kerr soliton comb generation. In a microcavity with nearly zero dispersion, the first pair of MI modes can grow only with a red-detuned pump laser. In this paper, we find that MI generation is possible with blue-detuned pump lasers for coupled microcavities with nearly zero dispersion. We study a microcavity with Kerr nonlinearity coupled with an auxiliary microcavity, which has negligible nonlinearity. By theoretical analysis, we show that the coupled microcavities can create a region supporting MI generation in the blue-detuned side of the resonances of the nonlinear main cavity, whereas there is no blue-detuned MI generation in a single nonlinear microcavity. The properties of the blue-detuned MI region are determined by the coupling coefficient between the two microcavities, the loss of the auxiliary cavity, and the detuning between the modes of the two microcavities. The size and location of the blue-detuned MI region can be varied by tuning these parameters. Numerical simulations of MI generation based on the blue-detuned MI region in the coupled microcavities are presented. By considering more modes, MI comb generations with coupled microcavities having anomalous and normal dispersion are also numerically simulated.

Dissipative electrostatic wave modulation in warm multi-ion dusty plasmas with superthermal electrons

Abstract A theoretical investigation has been carried out to explore the modulational instability (MI) of electrostatic waves in a warm multi-ion dusty plasma system containing positive ions, negative ions and positively or negatively charged dust in presence of superthermal electrons. With the help of the standard perturbation technique, it is found that the dynamics of the modulated wave is governed by a damped nonlinear Schrödinger equation (NLSE). Regions of MI of the electrostatic wave are precisely determined and the analytical solutions predict the formation of dissipative bright and dark solitons as well as dissipative first- and second-order rogue wave solutions. It is found that the striking features (viz., instability criteria, amplitude and width of rogue waves, etc.) are significantly modified by the effects of relevant plasma parameters such as degree of the electron superthermality, dust density, etc. The time dependent numerical simulations of the damped NLSE reveal that modulated electrostatic waves exhibit breather like structures. Moreover, phase plane analysis has been performed to study the dynamical behaviors of NLSE by using the theory of dynamical system. It is remarked that outcome of present study may provide physical insight into understanding the generation of several types of nonlinear structures in dusty plasma environments, where superthermal electrons, positive and negative ions are accountable (e.g. Saturn’s magnetosphere, auroral zone, etc.).

Generation of matter waves in Bose-Bose mixtures with helicoidal spin-orbit coupling

The paper studies the modulational instability (MI), both theoretically and numerically, of the helicoidal spin-orbit coupled Bose-Bose mixture. An expression of the MI growth rate is found through the linear stability analysis of continuous wave, followed by a comprehensive parametric study of the MI regions, emphasizing the effects of the spin-orbit coupling, the helicoidal gauge potential, and interatomic interactions. Direct numerical simulations concur with the analytical predictions. Under suitable balance between nonlinear and dispersive effects, trains of solitonlike objects are obtained, and their behaviors are very sensitive to parameter variations. Attention is particularly paid to the impact of the left- and right-handed helicoidal spin-orbit couplings on the appearance of matter waves that have the form soliton molecules in the Bose-Bose mixture. Additionally, for qualitative support of the obtained structures, the formation of a bright solitons train is also reported numerically using two neighboring solitons subjected to a fixed phase difference. Their behavior under the action of the helicoidal spin-orbit coupling is also debated, especially when left- and right-handed helicoidal couplings are interchanged.

Rogue wave solutions and modulation instability for the mixed nonlinear Schrödinger equation

Via Darboux transformation (DT) algorithm, rogue wave and semi-rational solutions of the mixed nonlinear Schrödinger equation (MNLSE) will be derived. Meanwhile, the dynamic features of those solutions will be graphically analyzed. In addition, the modulation instability (MI) of the MNLSE will be discussed and it will be found that the existence range of rogue waves is strictly consistent with the zero frequency modulation instability region.

Saturable higher nonlinearity effects on the modulational instabilities in three-core triangular configuration couplers

This paper demonstrates the possibility to observe the modulational instability (MI) regions due to modified nonlinear saturability over and above other high-order nonlinearities for the triangular configuration of a three-core oppositely directed coupler. In this configuration, two channels are positive-refractive-index material, and one is a negative-index material. The governing equations are the modified coupled nonlinear Schrödinger equations, to which we add the higher-order nonlinearity terms and coupling terms. This equation is further modified with the saturable nonlinearity term. In the presence of several nonlinearities, we derive the dispersion relation for the optical coupler under consideration. We use a linear stability analysis to study the modulation stability characteristic gain in both the normal and anomalous group-velocity dispersion regimes. We have widely varied the parameter range of the physical system parameters to accommodate different possibilities. In the normal region, the saturable nonlinearity aids in increasing the bandwidth of the MI region, while in the anomalous regime, the bandwidth reduces. In the normal dispersion case, the simultaneous presence of all nonlinearities in the optical coupler does not provide a suitable scope for pulse propagation. On the other hand, in the anomalous regime, these nonlinearities favor for pulse propagation.

Nonlinear modulation of quantum electron acoustic waves in a Thomas–Fermi plasma with effects of exchange-correlation

The amplitude modulation of electron acoustic waves (EAWs) is investigated in an unmagnetized four-component quantum plasma containing dynamic cold electron fluid, Thomas–Fermi distributed hot electrons and positrons and immobile ions in presence of exchange-correlation potentials. For this purpose, basic set of quantum hydrodynamic equations is reduced to a nonlinear Schrodinger equation (NLSE) adopting a reductive perturbation technique. The regions of the stable and unstable wave packets are obtained precisely in terms of the intrinsic plasma parameters. It is shown that the NLSE leads to the modulational instability (MI) of EAWs, and the formation of EA rogue waves, which are due to the effects of nonlinearity and dispersion in the propagation of EAWs. The effects of relevant plasma parameters on the MI criterion are numerically investigated. It is also found that the modulated EAW packets can propagate in the form of bright envelope solitons as well as dark envelope solitons. Within the modulational instability region, the dependence of first- and second-order EA rogue wave profiles on the system parameters is discussed.

Nonlinear states and dynamics in a synthetic frequency dimension

Recent advances in the study of synthetic dimensions revealed a possibility to employ the frequency space as an additional degree of freedom which allows for investigating and exploiting higher-dimensional phenomena in a priori low-dimensional systems. However, the influence of nonlinear effects on the synthetic frequency dimensions was studied only under significant restrictions. In the present paper, we develop a generalized mean-field model for the optical field envelope inside a single driven-dissipative resonator with quadratic and cubic nonlinearities, whose frequencies are coupled via an electro-optical resonant temporal modulation. The leading-order equation takes the form of a driven Gross-Pitaevskii equation with a cosine potential. We numerically investigate the nonlinear dynamics in such a microring resonator with a synthetic frequency dimension in the regime where parametric frequency conversion occurs. We observe that the modulation brings additional control to the system, enabling one to readily create and manipulate bright and dark dissipative solitons inside the cavity. In the case of anomalous dispersion, we find that the presence of electro-optical mode coupling confines and stabilizes the chaotic modulation instability region. This leads to the appearance of an unconventional type of stable coherent structure which emerges in the synthetic space with restored translational symmetry, in a region of parameters where conventionally only chaotic modulation instability states exist. This structure appears in the center of the synthetic band and, therefore, is referred to as the band soliton. Finally, we extend our results to the case of multiple modulation frequencies with controllable relative phases creating synthetic lattices with nontrivial geometry. We show that an asymmetric synthetic band leads to the coexistence of chaotic and coherent states of the electromagnetic field inside the cavity, i.e., dynamics that can be interpreted as chimeralike states. Recently developed ${\ensuremath{\chi}}^{(2)}$ microresonators can open the way to experimentally exploring our findings.

Modulational instability of Bose-Einstein condensates with helicoidal spin-orbit coupling

We theoretically study the modulation instability (MI) of the two-component helicoidal spin-orbit coupled Bose-Einstein condensates (BECs). The effects of spin-orbit coupling, the helicoidal gauge potential, and atomic interactions on MI are investigated. The results indicate that the presence of the helicoidal gauge potential breaks the symmetric properties of MI, strongly modifies the distribution of the MI region and the MI gain in parameters space, and the MI can be excited even when the miscibility condition for the atomic interactions is satisfied. Furthermore, the effect of the helicoidal gauge potential on MI is strongly coupled with the intra and intercomponent atomic interactions. Particularly, with the increase of the helical gauge potential, the MI gain increases for the repulsive atomic interaction case, however, the MI gain decreases for the attractive atomic interaction case. The direct numerical simulations are performed to support the analytical predictions, and a good agreement is found. Our results provide a potential way to manipulate the MI in BECs with helicoidal gauge potential.

Discrete modulational instability and bright localized spin wave modes in easy-axis weak ferromagnetic spin chains involving the next-nearest-neighbor coupling**Project supported by the National Natural Science Foundation of China (Grant Nos. 11604121 and 11875126), the Natural Science Fund Project of Hunan Province, China (Grant No. 2017JJ3255), the National College Students’ Innovation Entrepreneurship Training Program, China (Grant No. 201810531014), and the Scientific

We report a theoretical work on the properties of modulational instability and bright type nonlinear localized modes in one-dimensional easy-axis weak ferromagnetic spin lattices involving next-nearest-neighbor couplings. With a linear stability analysis, we calculate the growth rates of the modulational instability, and plot the instability regions. When the strength of the next-nearest-neighbor coupling is large enough, two new asymmetric modulational instability regions appear near the boundary of the first Brillouin zone. Furthermore, analytical forms of the bright nonlinear localized modes are constructed by means of a quasi-discreteness approach. The influence of the next-nearest-neighbor coupling on the Brillouin zone center mode and boundary mode are discussed. In particular, we discover a reversal phenomenon of the propagation direction of the Brillouin zone boundary mode.

Dynamics of the plane and solitary waves in a Noguchi network: Effects of the nonlinear quadratic dispersion

We consider a modified Noguchi network and study the impact of the nonlinear quadratic dispersion on the dynamics of modulated waves. In the semi-discrete limit, we show that the dynamics of these waves are governed by a nonlinear cubic Schrödinger equation. From the graphical analysis of the coefficients of this equation, it appears that the nonlinear quadratic dispersion counterbalances the effects of the linear dispersion in the frequency domain. Moreover, we establish that this nonlinear quadratic dispersion provokes the disappearance of some regions of modulational instability in the dispersion curve compared to the results earlier obtained by Pelap et al. (Phys. Rev. E 91 022925 (2015)). We also find that the nonlinear quadratic dispersion limit considerably affects the nature, stability, and characteristics of the waves which propagate through the system. Furthermore, the results of the numerical simulations performed on the exact equations describing the network are found to be in good agreement with the analytical predictions.

Modulational instability, quantum breathers and two-breathers in a frustrated ferromagnetic spin lattice under an external magnetic field**Project supported by the National Natural Science Foundation of China (Grant No. 11604121), the Scientific Research Fund of Hunan Provincial Education Department, China (Grant Nos. 16B210 and 16A170), and the Natural Science Fund Project of Jishou University, China (Grant No. jdx17036).

The modulational instability, quantum breathers and two-breathers in a frustrated easy-axis ferromagnetic zig-zag chain under an external magnetic field are investigated within the Hartree approximation. By means of a linear stability analysis, we analytically study the discrete modulational instability and analyze the effect of the frustration strength on the discrete modulational instability region. Using the results from the discrete modulational instability analysis, the presence conditions of those stationary bright type localized solutions are presented. On the other hand, we obtain the analytical expressions for the stationary bright localized solutions and analyze the effect of the frustration on their emergence conditions. By taking advantage of these bright type single-magnon bound wave functions obtained, quantum breather states in the present frustrated ferromagnetic zig-zag lattice are constructed. What is more, the analytical forms for quantum two-breather states are also obtained. In particular, the energy level formulas of quantum breathers and two-breathers are derived.

Quantum breathers and intrinsic localized excitation associated with the modulational instability in 1D Bose–Hubbard chain

In this work, the energy spectrum and intrinsic localized modes associated with the modulational instability of boson chains is described by a Bose–Hubbard model. By using of number states method combined with the numerical diagonalization, we show that the energy spectrum of the system exhibits bound states signature. With the help of multiple scales method in addition to a quasidiscreteness approximation, we obtain bright and dark type intrinsic localized modes at the center and at the edges of the Brillouin zone respectively and their appearance conditions. We found that the inclusion of interaction of two bosons at the same site can influence the appearance conditions of bright and dark soliton solutions. Considering the fact that nowadays, the forthright way to predict the formation of intrinsic localized modes in nonlinear systems is the modulational instability. We investigate analytically, through the linear stability of plane wave solutions, the existence of localized structures in the Bose–Hubbard chain. It is found that the shape of the region of modulational instability and the instability growth rates of the plane wave can be more affected dramatically when the interaction of two bosons at the same site is involved in the system. Furthermore, we calculate the instability growth rates of plane wave at the center and at the edge of the Brillouin zone for different values of the interaction of two bosons to analyze their formation conditions which are in full agreement with the method of multiple scale in addition to a quasi-discreteness approximation analysis.

Modulated wave formation in myocardial cells under electromagnetic radiation

We exclusively analyze the onset and condition of formation of modulated waves in a diffusive FitzHugh–Nagumo model for myocardial cell excitations. The cells are connected through gap junction coupling. An additive magnetic flux variable is used to describe the effect of electromagnetic induction, while electromagnetic radiation is imposed on the magnetic flux variable as a periodic forcing. We used the discrete multiple scale expansion and obtained, from the model equations, a single differential-difference amplitude nonlinear equation. We performed the linear stability analysis of this equation and found that instability features are importantly influenced by the induced electromagnetic gain. We present the unstable and stable regions of modulational instability (MI). The resulting analytic predictions are confirmed by numerical experiments of the generic equations. The results reveal that due to MI, an initial steady state that consisted of a plane wave with low amplitude evolves into a modulated localized wave patterns, soliton-like in shape, with features of synchronization. Furthermore, the formation of periodic pulse train with breathing motion presents a disappearing pattern in the presence of electromagnetic radiation. This could provide guidance and better understanding of sudden heart failure exposed to heavily electromagnetic radiation.

Quantum breathers associated with modulational instability in 1D ultracold boson in optical lattices involving next-nearest neighbor interactions

The dynamics and modulation instability of an ultracold gas of bosonic atoms in an optical lattice can be portrayed by a Bose–Hubbard model and the system parameters are mastered by laser light. Based on the time dependent Hartree approximation combined with the semi-discrete multiple-scale method, the equation of motion for single-boson wave function is found analytically, the existence conditions of appearance of bright stationary localized solitons solutions of this quantum Bose–Hubbard model are discussed. We find that the introduction of the next nearest neighbor interactions (NNNI) may change the stability property of the plane waves and may predict the formation of modulational instability in the wave number k = kmax and k = keBZ in the system. With the help of stationary localized single-boson wave functions obtained, the quantized energy level and the quantum breather state are determined. The performance of the analytical results are checked by numerical calculations. Furthermore, we have shown that the presence of the NNNI affect significatively the shape of the region of modulational instability and it is responsible of the appearance of new region of modulational instability that occurs for the k = kmax carrier wave. The formation conditions of the modulational instability region predicted by the analytical analysis of stationary localized solutions in the wave number k = kmax and k = keBZ are in good agreement with the forecast respectively.

A general nonlocal nonlinear Schrödinger equation with shifted parity, charge-conjugate and delayed time reversal

A general nonlocal nonlinear Schrodinger equation with shifted parity, charge-conjugate and delayed time reversal is derived from the nonlinear inviscid dissipative and equivalent barotropic vorticity equation in a $$\beta $$ -plane. The modulational instability (MI) of the obtained system is studied, which reveals a number of possibilities for the MI regions due to the generalized dispersion relation that relates the frequency and wavenumber of the modulating perturbations. Exact periodic solutions in terms of Jacobi elliptic functions are obtained, which, in the limit of the modulus approaches unity, reduce to soliton, kink solutions and their linear superpositions. Representative profiles of different nonlinear wave excitations are displayed graphically. These solutions can be used to model different blocking events in climate disasters. As an illustration, a special approximate solution is given to describe a kind of two correlated dipole blocking events.

Solitons in multi-body interactions for a fully modulated cubic–quintic Gross–Pitaevskii equation

• Dynamics of solitons in a fully modulated cubic–quintic Gross–Pitaevskii equation is analyzed. • Role of quintic parameter on the size of wave numbers of carrier and envelope is analyzed. • Lax-pair for the model with varying coefficients and external potential is given. • Regions of modulational instability and linear and nonlinear stabilities are investigated. A phase imprint approach is applied to a cubic Gross–Pitaevskii equation (GPE) in order to obtain a modified non-autonomous derivative cubic–quintic nonlinear GPE with fully variable coefficients. This model describes the dynamics of condensates in Bose–Einstein condensates, with both two- and three-body interatomic interactions with an external potential when the coefficient of the dispersion term is constant. This model is also applicable to fiber optics media in the absence of external potential. We show that this modified GPE model has a Lax-pair when all of its coefficients depend on time. However, the external potential depends on both the time and space variables. We obtain two classes of exact analytical solutions. These classes of solutions contain four different types of solitary-like wave solutions in the form of kink, anti-kink, bright, dark solitary, and periodic wave solutions. We reveal the effects of a quintic term on wave numbers, for both the carrier and envelope waves. Stability analysis is carried out, and conditions on parameters that determine regions with linear stability are discussed. Graphical analysis of some solutions are presented.

Modulational instability and discrete breathers in a nonlinear helicoidal lattice model

We investigate the problem on the discrete modulation instability of plane waves and discrete breather modes in a nonlinear helicoidal lattice model, which is described by a discrete nonlinear Schrödinger equation with the first-, second-, and third-neighbor coupling. By means of the linear stability analysis, we present an analytical expression of the instability growth rate and identify the regions of modulational instability of plane waves. It is shown that the introduction of the third-neighbor coupling will affect the shape of the areas of modulational instability significantly. Based on the results obtained by the modulational instability analysis, we predict the existence conditions for the stationary breather modes. Otherwise, by making use of the semidiscrete multiple-scale method, we obtain analytical solutions of discrete breather modes and analyze their properties for different types of nonlinearities. Our results show that the discrete breathers obtained are stable for a long time only when the system exhibits the repulsive nonlinearity. In addition, it is found that the existence of the stable bright discrete breather closely relates to the presence of the third-neighbor coupling.