Modified Nonlinear Schrodinger Equation Research Articles

Numerical simulation of modified nonlinear Schrodinger equation and turbulence generation

This article presents a numerical model to study wave turbulence in fluids. The model equation is derived by incorporating energy conservation (along with the usual fluid equations of a compressible flow), and the source of nonlinearity is the rise in temperature due to the acoustic wave's high amplitude. The nonlinear Schrödinger (NLS) and modified nonlinear Schrödinger (MNLS) equations have been derived and then solved numerically. A numerical simulation of the MNLS equation is used for investigating the turbulence generation and a semi-analytical method to understand the physics of localized structures of nonlinear waves. The numerical simulation is based on a pseudo-spectral approach to resolve spatial regimes, a finite difference method for temporal evolution. The results show a periodic pattern viz. Fermi–Pasta–Ulam (FPU) recurrence for NLS, while turbulence generation breaks down the FPU recurrence in MNLS. The turbulent power spectrum in the inertial sub-range approximately follows the Kolmogorov–Zakharov scaling (∼k−1.2).

Physics-informed Neural Network method for the Modified Nonlinear Schrödinger equation

In this paper, we apply the Physics-informed Neural Network (PINN) method to investigate some nonlinear wave solutions of the Modified Nonlinear Schrödinger (MNLS) equation, which characterizes the ultrashort pulse propagation well in optical fibers. We generate the training dataset by the Latin Hypercube Sampling (LHS) method under Dirichlet boundary conditions. We use the Xavier initialization method to generate the parameters of the neural network, and then use the Adam algorithm to optimize them. Finally, we derive the data-driven solutions of single soliton solution, double solitons solution, single breather solution, double breather solution, first-order rogue wave and second-order rogue wave solutions for the MNLS equation. Through loss and L2 error of the experimental results, the PINN method has been verified to have prominent stability and effectiveness.

Modified nonlinear Schrödinger equation for gravity waves with the influence of wind, currents, and dissipation

A new modified nonlinear Schrödinger (MNLS) equation is derived for gravity waves with the presence of wind, dissipation, and shear currents in finite water depth. Horizontal surface currents are assumed stationary and slowly varying spatially. Using the MNLS equation, the modulational instability (MI) of deep-water gravity wave trains affected by wind and dissipation is considered. It was demonstrated that the modulational perturbation of waves is unstable or becomes unstable after several wave periods, whereas the dissipation will suppress the MI. Then, a new theoretical formula for predicting the maximum amplitude is derived to take into account the effect of vorticity, dissipation, and wind. The effect of dissipation becomes significant in strong currents, while wind can increase the height amplification. Furthermore, an explicit analytical Peregrine breather (PB) solution that considers the effect of vorticity, dissipation, and wind is presented. Opposing currents and winds will increase the height of PB. However, following currents and dissipation have opposite effects. The effects of the shear current, dissipation, and wind on nondimensional maximum amplitudes during the evolution of the Akhmediev breather are similar to PB solution.

Shape-morphing reduced-order models for nonlinear Schrödinger equations

We consider reduced-order modeling of nonlinear dispersive waves described by a class of nonlinear Schrodinger (NLS) equations. We compare two nonlinear reduced-order modeling methods: (i) The reduced Lagrangian approach which relies on the variational formulation of NLS and (ii) The recently developed method of reduced-order nonlinear solutions (RONS). First, we prove the surprising result that, although the two methods are seemingly quite different, they can be obtained from the real and imaginary parts of a single complex-valued master equation. Furthermore, for the NLS equation in a stationary frame, we show that the reduced Lagrangian method fails to predict the correct group velocity of the waves whereas RONS predicts the correct group velocity. Finally, for the modified NLS equation, where the reduced Lagrangian approach is inapplicable, the RONS reduced-order model accurately approximates the true solutions.

Investigation of new solitons and other solutions to the modified nonlinear Schrödinger equation in ocean engineering

This article deals with the new solitons and other solutions to the modified nonlinear Schrödinger equation (MNLSE) that represents the propagation of rogue waves in deep water. The solutions are achieved in the shape of trigonometric and hyperbolic functions as well as solitary wave solutions by the mean of extended sinh-Gordon equation expansion and (G′G2)-expansion function methods. Furthermore, the singular periodic wave solutions are obtained and the constraint conditions for the existence of soliton solutions are also listed. We also plot 3D graphs for a better explanation of the achieved solutions by choosing the appropriate values of the parameters involved in solutions. We can assert from the obtained results that the applied techniques are simple, vibrant, and quite well, and will be helpful tool for addressing more highly nonlinear issues in various of fields, especially in ocean and coastal engineering. Furthermore, our findings are first step toward understanding the structure and physical behavior of complicated structures. We anticipate that our results will be highly valuable in better understanding the waves that occur in the ocean. We feel that this work is timely and will be of interest to a wide spectrum of experts working on ocean engineering models.

Spatiotemporal nonlinear evolution of the laser pulse and turbulence generation in laser produced plasmas

This study presents a model to understand the behavior of the turbulence generated in the magnetic field of mega gauss order during high-intensity laser interaction with magnetized plasma. The modified nonlinear Schrödinger (MNLS) equation is developed by contemplating the effect of the group velocity dispersion, diffraction, and nonlinearity induced by the relativistic variation of electron mass and the nonlinear ponderomotive force. Numerical simulation is carried out to solve the dimensionless MNLS equation. The simulation results show the generation of the solitary wave type coherent structures in the nonlinear spatiotemporal evolution of the laser pulse at the early stage, but subsequent turbulence generation has also been observed. The ensemble-averaged turbulent power spectrum has been studied and the power-law scaling is approximately ∼ k−1.85(a solid red line of scaling k−1.85 is given for reference). To get insight into the spatiotemporal nonlinear development of the laser pulse, while propagating in the plasma medium, a semi-analytical model has also been presented. The present study could be substantial in replicating astrophysical scenarios by laboratory simulations along with understanding the underlying quintessential physics of magnetic turbulence.

Impact of higher order dispersions and nonlinearities on instability criterion of oppositely directed coupler in the presence of negative index material channel

This paper intends to investigate the modulational instability (MI) criterion of two core positive-negative index coupler channel setup from the obtained analytical dispersion relation for MI gain profile. The governing equation, the modified nonlinear Schrodinger equation (MNLSE) is studied for the role of higher order dispersion (HOD) and higher order nonlinearity (HON) on instability spectra of two core positive indexed coupler (PIM) and negative indexed coupler (NIM) arrangement. The HOD comprises of the second, third, fourth and fifth order linear dispersion terms and HON is inclusive of cubic, quintic, septic, cubic-quintic-septic-self-steepening (SS), and cubic-quintic-septic-intrapulse Raman scattering (IRS). Our study reveals that it is the 6th order linear dispersion term which brings in the considerable changes of MI gain spectrum in both normal and anomalous dispersion regimes. In particular, the combined effect of septic SS and IRS in two core directional coupler with NIM channel proves to be very crucial in the generation of instability gain spectra. This study has higher prospects in generation of solitary wave in two core directional system for optical fiber communication.

Nonplanar dust acoustic waves in a four-component dusty plasma with double spectral distributed electrons: modulational instability and rogue waves

A theoretical investigation of nonplanar (cylindrical and spherical) amplitude modulation of dust acoustic waves (DAWs) in a four component dusty plasma including positive and negative dusts, Maxwellian ions and double spectral electron distribution (DSED) is carried out. The focusing modified nonlinear Schrödinger equation (MNLSE) is derived using the derivative expansion perturbation technique (DEPT). The limit of DSED to Maxwell distribution is discussed. The effects of the two spectral indices, r and q , on the modulational instability (MI) growth rate are examined and it is deduced that increasing the value of r and q raises the MI growth rate. Additionally, the structure of the first and second-order dust acoustic (DA) rogue waves in the modulational unstable region is analyzed. The influences of both of r and q on the rogue wave amplitude (energy) are investigated. We found that increasing the value of r enhances the rogue wave amplitude while the opposite behavior is found by increasing q values. The findings of the present study are useful to interpret some physical phenomena in various plasma situations that found in space regions, such as Titan (the largest moon) of Saturn, auroral region and asteroid zones, where the effectiveness of non-Maxwellian particles such as DSED is existed.

A New Derivation of Exact Solutions for Incompressible Magnetohydrodynamic Plasma Turbulence

The objective of this paper is to study the propagation of nonlinear, quasi-parallel, magnetohydrodynamic waves of small-amplitude in a cold Hall plasma of constant density. Magnetohydrodynamic equations, along with the cold plasma were expanded using the reductive perturbation method, which leads to a nonlinear partial differential equation that complies with a modified form of the derivative nonlinear evolution Schrödinger equation. Exact solutions of the turbulent magnetohydrodynamic model in plasma were formulated for the physical quantities that describe the problem completely by using the complex ansatz method. In addition, the complete set of equations was used taking into account the magnetic field, which can be considered to be constant in the x-direction to create stable vortex waves. Vortex solutions of the modified nonlinear Schrödinger equation (MNLS) were compared with the solutions of incompressible magnetohydrodynamic equations. The obtained equations differ from the standard NLS equation by one additional term that describes the interaction of the nonlinear waves with the constant density. The behavior of both the velocity profile and the waveform of pressure were studied. The results showed that there are clear disturbances in the identity of the velocity and thus pressure. The identity of both velocity and pressure results from that a magnetic field is formed.

New visions of the soliton solutions to the modified nonlinear Schrodinger equation

In this article, we will study the modified nonlinear Schrödinger equation which represents the propagation of rogue waves in ocean engineering that occur in deep water, usually far out at sea, and are a threat even to capital ships and ocean liners. In related subject for all like waves which occurred at different branches of science which have the same behaviors such as fluid dynamics waveguides that have unexpected large displacements, long-wave limit of a breather (a pulsing mode), telecommunications engineering devices, etc. We will implement new multiple visions for the soliton solution to the modified nonlinear Schrödinger equation (MNLSE) that describes successfully these phenomenon. Studying this important model and its own behavior of the resulting solitons description will contribute effectively in marine safety and improvement the pulse devices which cooperate to protect human life effectively and technology for high-capacity optical communications through which the modern asocial media was built…etc. Three different techniques are invited for this purpose. The first technique is the Paul-Painlevé approach method (PPAM), while the second technique is the Ricatti-Bernolli Sub ODE method (RBSOM). In related subject, the variational iteration method VIM is implemented in the same vein and parallel to construct the numerical solutions corresponding to the achieved soliton solutions via each one of these two methods individually. A good comparison not only between these three visions of soliton solutions but also with that achieved previously by other authors has been demonstrated.

Riemann–Hilbert approach to the modified nonlinear Schrödinger equation with non-vanishing asymptotic boundary conditions

The modified nonlinear Schrödinger (NLS) equation was proposed to describe the nonlinear propagation of the Alfven waves and the femtosecond optical pulses in a nonlinear single-mode optical fiber. In this paper, we present the inverse scattering transform for the modified NLS equation iut+uxx+2|u|2u+i1α(|u|2u)x=0,α>0,with nonvanishing boundary values at infinity u(x,t)∼u±e−4iα2t+2iαx,x→±∞.An appropriate two-sheeted Riemann surface is introduced to map the original spectral parameter k into a single-valued parameter z. The direct scattering problem is shown to be well posed for potentials u such that u−u±∈L1,2(R±), for which existence and analyticity properties of eigenfunctions and scattering data are established. Their asymptotic behaviors and the symmetries are analyzed in details based on the Lax pair for the modified NLS equation. Then the inverse scattering problem is formulated as a Riemann–Hilbert (RH) problem associated with the problem of nonzero boundary conditions. The existence and uniqueness of solution for the mixed RH problem for t>0 are strictly proved by decomposing it into a pure RH problem and a scattering RH problem. The N-soliton solutions for the modified NLS equation are obtained via a reconstruction formulae between solution of the modified NLS equation and the solution of above mixed RH problem. As an illustrate example of N-soliton formula, for N=1 and N=2, two kinds of one-soliton solutions and three kinds of two-soliton solutions are explicitly presented, respectively according to different distribution of the spectrum. The dynamical feature of those solutions are characterized in the particular case with a quartet of discrete eigenvalues. It is shown that distribution of the spectrum and non-vanishing boundary also affect feature of soliton solutions. Finally, we analyze the differences between our results and those on zero boundary case.

Soliton excitations and interaction in alpha helical protein with interspine coupling in modified nonlinear Schrödinger equation**Project supported by the National Natural Science Foundation of China (Grant Nos. 11874324 and 11705164), the Natural Science Foundation of Zhejiang Province of China (Grant Nos. LY17A040011, LY17F050011, and LR20A050001), the Foundation of “New Century 151 Talent Engineering” of Zhejiang Province of China, and the

The three-coupling modified nonlinear Schrödinger (MNLS) equation with variable-coefficients is used to describe the dynamics of soliton in alpha helical protein. This MNLS equation with variable-coefficients is firstly transformed to the MNLS equation with constant-coefficients by similarity transformation. And then the one-soliton and two-soliton solutions of the variable-coefficient-MNLS equation are obtained by solving the constant-coefficient-MNLS equation with Hirota method. The effects of different parameter conditions on the soliton solutions are discussed in detail. The interaction between two solitons is also discussed. Our results are helpful to understand the soliton dynamics in alpha helical protein.

Bifurcation analysis of dissipative rogue wave in electron-positron-ion plasma with relativistic ions and superthermal electrons

Abstract In this manuscript, a modified nonlinear Schrodinger equation (MNLSE) is derived for an unmagnetized collisionless three components plasma containing superthermal electrons, Boltzmann distribution of positrons and relativistic ions. By bifurcation of dynamical system, we determined the stable and unstable regions and predicted the kinds of solutions of MNLSE. This solutions reveal dark soliton in heteroclinic areas and rogue wave in unstable regions. A novel form of rogue wave is obtained also the effects of viscosity, superthermal electron κ, ratio of electron to positron temperature, ratio of ion to electron temperature and the density of positron are illustrated with some graphics in two-dimensional and three-dimensional. The derived results have numerous applications in plasma physics and it could be much importance in predicting and enriching rogue wave happening in dissipation plasma physics.

Modulation of cylindrical (spherical) waves in a plasma with vortex electron distribution

In the present work, employing cylindrically (spherically) symmetric field equations of a plasma consisting of a cold electron fluid, hot electrons obeying a trapped/vortex-like distribution and stationary ions, we studied the amplitude modulation of electron-acoustic waves. Due to the physical nature of the problem under investigation, the nonlinearity of the field equations is of order (3/2), which causes considerable difficulty in the analysis of modulation problems. To solve this difficulty, we expanded this nonlinear term into the Fourier cosine series of the phase function and obtained the modified cylindrical (spherical) nonlinear Schrödinger (NLS) equation. A consistent analysis for the modulational instability is presented and a criterion between the time parameter τ and the wave number K is established. In addition, motivated with the solitonic solution of modified NLS equation for planar case and utilizing the “weighted residual method,” we proposed a harmonic wave of variable frequency with progressive wave amplitude to the evolution equation. It is found that the modified cylindrical (spherical) NLS equation assumes an envelope type of progressive wave solution in the sense weighted residual. The numerical results reveal that the amplitude of spherical wave is much larger than that of the cylindrical wave and that both amplitudes decrease with increasing time parameter τ. It is further observed that the wave profiles get distorted with progressing time.

Enhancement in self-compression due to co-propagating laser pulse in plasma

The study proposes enhancement in self-compression of the laser pulse with relativistic nonlinearity in a plasma. It is presented that, a low-intensity laser pulse can tune the pulse duration and peak intensity of the high-intensity laser pulse. The coupled modified nonlinear Schrodinger equations (MNLS) have been established for both laser pulses in plasma. The enhancement in compression of the high-intensity laser pulse is solved by numerical methods. The nonlinearity of plasma, self-phase modulation (SPM), and group velocity dispersion (GVD) compress the laser pulse to the extremely short duration high peak intensity laser pulse. As a result, it is observed that the compression of the high-intensity laser pulse is highly sensitive to the combined intensity of both laser pulses, plasma density and even a small walkoff between two pulses can affect the high-intensity laser pulse compression.

Modulational instability: Conservation laws and bright soliton solution of ion-acoustic waves in electron-positron-ion-dust plasmas

We consider the nonlinear propagation of non-planar (cylindrical and spherical) ion-acoustic (IA) envelope solitary waves in an unmagnetized plasma consisting of electron-positron-ion-dust plasma with two-electron temperature distributions in the context of the non-extensive statistics. The basic set of fluid equations is reduced to the modified nonlinear Schrodinger (MNLS) equation in cylindrical and spherical geometry by using the reductive perturbation method (RPM). It is found that the nature of the modulational instabilities would be significantly modified due to the effects of the non-extensive and other plasma parameters as well as cylindrical and spherical geometry. Conservation laws of the MNLS equation are obtained by Lie symmetry and multiplier method. A new exact solution (envelope bright soliton) is obtained by the extended homogeneous balance method. Finally, we study the results of this article.

Comparison of pulse propagation and gain saturation characteristics among different input pulse shapes in semiconductor optical amplifiers

This paper presents the pulse propagation and gain saturation characteristics for different input optical pulse shapes with different energy levels in semiconductor optical amplifiers (SOAs). A finite-difference beam propagation method (FD-BPM) is used to solve the modified nonlinear Schrödinger equation (MNLSE) for the simulation of nonlinear optical pulse propagation and gain saturation characteristics in the SOAs. In this MNLSE, the gain spectrum dynamics, gain saturation are taken into account those are depend on the carrier depletion, carrier heating, spectral hole-burning, group velocity dispersion, self-phase modulation and two photon absorption. From this simulation, we obtained the output waveforms and spectra for different input pulse shapes considering different input energy levels. It has shown that the output pulse shape has changed due to the variation of input parameters, such as input pulse shape, input pulse width, and input pulse energy levels. It also shown clearly that the peak position of the output waveforms are shifted toward the leading edge which is due to the gain saturation of the SOA. We also compared the gain saturation characteristics in the SOA for different input pulse shapes.

Monopole excitations of a harmonically trapped one-dimensional Bose gas from the ideal gas to the Tonks-Girardeau regime.

Using a time-dependent modified nonlinear Schrödinger equation (MNLSE)-where the conventional chemical potential proportional to the density is replaced by the one inferred from Lieb-Liniger's exact solution-we study frequencies of the collective monopole excitations of a one-dimensional Bose gas. We find that our method accurately reproduces the results of a recent experimental study [E. Haller etal., Science 325, 1224 (2009)] in the full spectrum of interaction regimes from the ideal gas, through the mean-field regime, through the mean-field Thomas-Fermi regime, all the way to the Tonks-Giradeau gas. While the former two are accessible by the standard time-dependent NLSE and inaccessible by the time-dependent local density approximation, the situation reverses in the latter case. However, the MNLSE is shown to treat all these regimes within a single numerical method.

Generalized perturbation (n,M)-fold Darboux transformations and multi-rogue-wave structures for the modified self-steepening nonlinear Schrödinger equation.

In this paper, a simple and constructive method is presented to find the generalized perturbation (n,M)-fold Darboux transformations (DTs) of the modified nonlinear Schrödinger (MNLS) equation in terms of fractional forms of determinants. In particular, we apply the generalized perturbation (1,N-1)-fold DTs to find its explicit multi-rogue-wave solutions. The wave structures of these rogue-wave solutions of the MNLS equation are discussed in detail for different parameters, which display abundant interesting wave structures, including the triangle and pentagon, etc., and may be useful to study the physical mechanism of multirogue waves in optics. The dynamical behaviors of these multi-rogue-wave solutions are illustrated using numerical simulations. The same Darboux matrix can also be used to investigate the Gerjikov-Ivanov equation such that its multi-rogue-wave solutions and their wave structures are also found. The method can also be extended to find multi-rogue-wave solutions of other nonlinear integrable equations.

Controllable Nonautonomous Rogue Waves in the Modified Nonlinear Schrödinger Equation with Distributed Coefficients in Inhomogeneous Fibers

Abstract Under investigation in this paper is an inhomogeneous modified nonlinear Schrödinger (MNLS) equation describing the ultrashort pulse dynamics with the distributed dispersion, self-phase modulation, self-steepening, and linear gain/loss. Nonautonomous rogue waves for the inhomogeneous MNLS equation are constructed via the modified Darboux transformation with the inhomogeneous parameters. The dynamical behaviours and main characteristics of the nonautonomous rogue waves in inhomogeneous fibers are discussed by analysing certain physical quantities. The inhomogeneous effects on the evolution of rogue waves are considered by virtue of the presented rational solutions with distributed coefficients. It is found that the group velocity dispersion (GVD) and linear gain/loss coefficients have effects on the trajectories and amplitudes of the rogue waves, respectively. Additionally, an intriguing composite rogue wave of the inhomogeneous MNLS equation is revealed, owing to the proper choice of the GVD coefficient. Our results could be useful for controlling the rogue waves in the dispersion-managed fiber system.