Nonlinear Schrodinger Equation Research Articles

PT-symmetric PINN for integrable nonlocal equations: Forward and inverse problems.

Since the PT-symmetric nonlocal equations contain the physical information of the PT-symmetric, it is very appropriate to embed the physical information of the PT-symmetric into the loss function of PINN, named PTS-PINN. For general PT-symmetric nonlocal equations, especially those equations involving the derivation of nonlocal terms due to the existence of nonlocal terms, directly using the original PINN method to solve such nonlocal equations will face certain challenges. This problem can be solved by the PTS-PINN method, which can be illustrated in two aspects. First, we treat the nonlocal term of the equation as a new local component so that the equation is coupled at this time. In this way, we successfully avoid differentiating nonlocal terms in neural networks. On the other hand, in order to improve the accuracy, we make a second improvement, which is to embed the physical information of the PT-symmetric into the loss function. Through a series of independent numerical experiments, we evaluate the efficacy of PTS-PINN in tackling the forward and inverse problems for the nonlocal NLS equation, the nonlocal derivative NLS equation, the nonlocal (2+1)-dimensional NLS equation, and the nonlocal three-wave interaction systems. The numerical experiments demonstrate that PTS-PINN has good performance. In particular, PTS-PINN has also demonstrated an extraordinary ability to learn large space-time scale rogue waves for nonlocal equations.

Role of polarized force on dust-acoustic structures in the presence of suprathermal electrons

The modulational instability (MI) of dust-acoustic wave (DAW) in an unmagnetized dusty plasma including dust, ions, and electrons is studied. Electrons are kappa-distributed, ions follow a Maxwellian distribution, and negatively charged dust particles display dynamic behavior. A nonlinear Schrödinger equation (NLSE) is generated using the Krylov–Bogoliubov–Mitropolsky approach to explore MI. Using the essential physical characteristics, the steady and unstable portions of the modulated wave packets are properly characterized. The impact of polarization force, suprathermal index (κ), effective temperature of electron to ion temperature ratios, and other physical factors in stable and unstable DAW zones is numerically studied. In unstable zones, unpredictable amplitude perturbations cause freak/rogue waves for which the essential studies on the localized solutions of the NLSE are done extensively.

Normalized ground state of a mixed dispersion nonlinear Schrodinger equation with combined power-type nonlinearities

We study the existence of normalized ground state solutions to a mixed dispersion fourth-order nonlinear Schrodinger equation with combined power-type nonlinearities. By analyzing the subadditivity of the ground state energy with respect to the prescribed mass, we employ a constrained minimization method to establish the existence of ground state that corresponds to a local minimum of the associated functional. Under certain conditions, by studying the monotonicity of ground state energy as the mass varies, we apply the constrained minimization arguments on the Nehari-Pohozaev manifold to prove the existence of normalized ground state solutions.
 For more information see https://ejde.math.txstate.edu/Volumes/2024/29/abstr.html

Limiting behaviors of constrained minimizers for the mass subcritical fractional NLS equations

Limiting behaviors of constrained minimizers for the mass subcritical fractional NLS equations

A design methodology for nonlinear oscillator chains enabling energy localization tuning and soliton stability enhancement with optimal damping

In this paper, the vibration energy localization in coupled nonlinear oscillators is investigated, based on the creation of standing solitons. The main objective is to establish a design methodology for mechanical lattices using the Nonlinear Schrödinger Equation (NLSE) as a guide strategy, even in the presence of damping. A three-dimensional diagram is used to illustrate stable parameter regions for damped stationary solitons. Moreover, an analysis of the influence of the number of oscillators in the system, and a numerical investigation regarding the stability of solitonic behavior is done. Through numerical analyses, it is observed that the developed algorithm not only has the capability to locate the highest amplitudes in the chain of oscillators, but also to control the intensity at which these amplitudes are located according to design requirements. The outcomes of the proposed methodology elucidate the impact that the coupling stiffness has on the stabilization of the NLSE, as well as the influence of the number of oscillators on the continuity hypothesis. The developed algorithm holds potential for practical applications in mechanical engineering since the NLSE is used as a design line rather than as a consequence of the phenomenon description.

Bifurcation, phase plane analysis and exact soliton solutions in the nonlinear Schrodinger equation with Atangana's conformable derivative

The nonlinear Schrodinger equation (NLSE) with Atangana's conformable fractional derivative (ACFD) is an equation that describes how the quantum state of a physical system changes in time. This present study examines the exact soliton results (ESRs) and analyze their bifurcations and phase plane (PP) of the NLSE with ACFD through the modified (G′/G)-expansion scheme (MG′/GES). Firstly, ACFD and its properties are included and then by MG′/GES, the exact soliton results and analyze their bifurcations and phase plane of NLSE, displayed with the ACFD, are classified. These obtained ESRs include periodic wave pulse, bright-dark periodic wave pulse, multiple bright-dark soliton pulse, and so many types. We provide 2-D, 3-D and contour diagrams, Hamiltonian function (HF) for phase plane dynamics analysis and bifurcation analysis diagrams to examine the nonlinear effects and the impact of fractional and time parameters on these obtained ESRs. The obtained ESRs illustration the novelty and prominence of the dynamical construction and promulgation performance of the resultant equation and also have practical effects for real world problems. The ESRs obtained with MG′/GES show that this scheme is very humble, well-organized and can be implemented to other the NLSE with ACFD.

On the proximity between the wave dynamics of the integrable focusing nonlinear Schrödinger equation and its non-integrable generalizations

The question of whether features and behaviors that are characteristic to completely integrable systems persist in the transition to non-integrable settings is a central one in the field of nonlinear dispersive equations. In this work, we investigate this topic in the context of focusing nonlinear Schrödinger (NLS) equations. In particular, we consider non-integrable counterparts of the (integrable) focusing cubic NLS equation, which are distinct generalizations of cubic NLS and involve a broad class of nonlinearities, with the cases of power and saturable nonlinearities serving as illustrative examples. This is a notably different direction from the one explored in other works, where the non-integrable models considered are only small perturbations of the integrable one. We study the Cauchy problem on the real line for both vanishing and non-vanishing boundary conditions at infinity and quantify the proximity of solutions between the integrable and non-integrable models via estimates in appropriate metrics as well as pointwise. These results establish that the distance of solutions grows at most linearly with respect to time, while the growth rate of each solution is chiefly controlled by the size of the initial data and the nonlinearity parameters. A major implication of these closeness estimates is that integrable dynamics emerging from small initial conditions may persist in the non-integrable setting for significantly long times. In the case of zero boundary conditions at infinity, this persistence includes soliton and soliton collision dynamics, while in the case of nonzero boundary conditions at infinity, it establishes the nonlinear behavior of the non-integrable models at the early stages of the ubiquitous phenomenon of modulational instability. For this latter and more challenging type of boundary conditions, the closeness estimates are proved with the aid of new results concerning the local existence of solutions to the non-integrable models. In addition to the infinite line, we also consider the cubic NLS equation and its non-integrable generalizations in the context of initial-boundary value problems on a finite interval. Apart from their own independent interest and features such as global existence of solutions (which does not occur in the infinite domain setting), such problems are naturally used to numerically simulate the Cauchy problem on the real line, thereby justifying the excellent agreement between the numerical findings and the theoretical results of this work.

Spectral structure and even-order soliton solutions of a defocusing shifted nonlocal NLS equation via Riemann-Hilbert approach

Spectral structure and even-order soliton solutions of a defocusing shifted nonlocal NLS equation via Riemann-Hilbert approach

Optical solitons and stability analysis for NLSE with nonlocal nonlinearity, nonlinear chromatic dispersion and Kudryashov’s generalized quintuple-power nonlinearity

In this paper, we consider the nonlinear Schrödinger equation (NLSE) with nonlocal nonlinearity having nonlinear chromatic dispersion and Kudryashov’s generalized quintuple-power. For the suggested model, dark solitons, and singular solitons are investigated using the modified extended mapping approach. Furthermore, rational solutions, exponential solutions, hyperbolic wave solutions, singular periodic solutions, and Jacobi elliptic function solutions are shown. We impose certain limits on the parameters in order to guarantee the existence of the obtained soliton solutions. Furthermore, a few chosen solutions are also displayed graphically to illustrate the physical characteristics of the precise solutions. Furthermore, we use the idea of modulation instability analysis to discuss the stability of the derived wave solutions.

SH waves in a weakly inhomogeneous half space with a nonlinear thin layer coating

We investigate the self-modulation of Love waves propagating in a nonlinear half-space covered by a nonlinear layer. We assume that the constituent material of the layer is nonlinear, homogeneous, isotropic, compressible, and hyperelastic, whereas for the half-space, it is nonlinear, heterogeneous, compressible and a different hyperelastic material. By employing the nonlinear thin layer approximation, the problem of wave propagation in a layered half-space is reduced to the one for a nonlinear heterogeneous half-space with a modified nonlinear homogeneous boundary condition on the top surface. This new problem is analyzed by a relevant perturbation method, and a nonlinear Schrödinger (NLS) equation defining the self-modulation of waves asymptotically is obtained. The dispersion relation is derived for different heterogeneous properties of the half-space and the thin layer. Then the results of the thin layer approximation are compared with the ones for the finite layer obtained in Teymur et al. (Int J Eng Sci 85:150–162, 2014). The solitary solutions of the derived NLS equation are obtained for selected real material models. It has been discussed how these solutions are influenced by the heterogeneity of the semi-infinite space.

Stability theory for two-lobe states on the tadpole graph for the NLS equation

The aim of this work is to present new spectral tools for studying the orbital stability of standing waves solutions for the nonlinear Schrödinger equation (NLS) with power nonlinearity on a tadpole graph, namely, a graph consisting of a circle with a half-line attached at a single vertex. By considering δ-type boundary conditions at the junction and bound states with a positive two-lobe profile, the main novelty of this paper is at least twofold. Via a splitting eigenvalue method developed by the author, we identify the Morse index and the nullity index of a specific linearized operator around of an a priori positive two-lobe state profile for every positive power; and we also obtain new results about the existence and the orbital stability of positive two-lobe states at least in the cubic NLS case. To our knowledge, the results contained in this paper are the first in studying positive bound states for the NLS on a tadpole graph by non-variational techniques. In particular, our approach has prospect of being extended to study stability properties of other bound states for the NLS on a tadpole graph or on other non-compact metric graph such as a looping edge graph, as well as, for other nonlinear evolution models on a tadpole graph.

Numerical Simulation of Soliton Propagation Behavior for the Fractional-in-Space NLSE with Variable Coefficients on Unbounded Domain

The soliton propagation of the fractional-in-space nonlinear Schrodinger equation (NLSE) is much more complicated than that of the corresponding integer NLSE. The aim of this paper is to discover some novel fractal soliton propagation behaviors (FSPBs) of this fractional-in-space NLSE. Firstly, the exact solution is compared with the present numerical solution, and the validity and accuracy of the present numerical method are verified. Secondly, the effect of fractional derivatives on soliton propagation is explored through the present numerical simulation results. At the same time, the present method is extended to the three-dimensional fractional-order NLSE. Finally, some novel FSPBs of the fractional-in-space NLSE are given.

Two-stage initial-value iterative physics-informed neural networks for simulating solitary waves of nonlinear wave equations

We propose a new two-stage initial-value iterative neural network (IINN) algorithm for solitary wave computations of nonlinear wave equations based on traditional numerical iterative methods and physics-informed neural networks (PINNs). Specifically, the IINN framework consists of two subnetworks, one of which is used to fit a given initial value, and the other incorporates physical information and continues training on the basis of the first subnetwork. Importantly, the IINN method does not require any additional data information including boundary conditions, apart from the given initial value. Corresponding theoretical guarantees are provided to demonstrate the effectiveness of our IINN method. The proposed IINN method is efficiently applied to learn some types of solutions in different nonlinear wave equations, including the one-dimensional (1D) nonlinear Schrödinger equations (NLS) equation (with and without potentials), the 1D saturable NLS equation with PT-symmetric optical lattices, the 1D focusing-defocusing coupled NLS equations, the KdV equation, the two-dimensional (2D) NLS equation with potentials, the 2D amended GP equation with a potential, the (2+1)-dimensional KP equation, and the 3D NLS equation with a potential. These applications serve as evidence for the efficacy of our method. Finally, by comparing with the traditional methods, we demonstrate the advantages of the proposed IINN method.

Soliton solutions of the improved perturbed nonlinear Schrödinger equation having parabolic law with non-local nonlinearity in the presence of chromatic and spatio-temporal dispersion terms

This paper was organized to examine the analytical solutions of the improved perturbed parabolic-law nonlinear Schrodinger equation including non-local nonlinearity in the presence of chromatic and spatio-temporal dispersion. This model mostly makes use of studying the propagation of optical pulses in fiber optic communication systems. We performed the Sinh-Gordon equation expansion method so that we produce the analytical solutions of the model under consideration. It was confirmed that the acquired solutions satisfy the main model. Therefore, bright and dark soliton solutions were retrieved; besides, various 3D and 2D graphical illustrations of the solitons were demonstrated via appropriate values of the parameters. Furthermore, this manuscript focused on the parameters’ effect on the acquired solitons behavior.

Dynamical stricture of optical soliton solutions and variational principle of nonlinear Schrödinger equation with Kerr law nonlinearity

The aim of this paper is to discover some new exact solutions for two kinds of nonlinear Schrödinger equation (NLSE), including the (2+1)-dimensional NLSE and the derivative NLSE, through the use of the variational principle method and amplitude ansatz method. We will find the solutions to these equations by selecting a trial function with a single nontrivial variational parameter, and it is continuous at all intervals in three cases from a region of a rectangular box, then using this trial function to obtain the functional integral and the Lagrangian of the system without any loss. After that, we approximate this trial function by quadratic polynomials with two free parameters rather than a piecewise linear ansatz function. Similarly, this trial function will be approximated by the tanh function. Additionally, we accept several new types of soliton solutions, including bright solitons, dark solitons, bright–dark solitary wave solutions, rational dark–bright solutions, and periodic solitary wave solutions. Also, conditions for the stability of the solutions will be proposed. These explanations are of great importance in the fields of applied science and engineering. The results will be shown through different types of graphs, including 2D, 3D, and contour plots.

Analyzing optical soliton propagation in perturbed nonlinear Schrödinger equation: A multi-technique study

In this study, the extended hyperbolic function method and unified solver method are applied to examine the propagation of optical solitons in perturbed nonlinear Schrodinger equation. The proposed techniques are characterized as simpler, more concise, and straightforward. They allow for the extension of the class of solutions and the retrieval of a diverse range of optical solitons, including singular, periodic singular, bright, and dark ones. This highlights the significance of the results. The authors emphasize the novelty of these techniques, as they have not been previously applied to recover optical solitons in this model These out comes may be beneficial for further understanding of numerous phenomena that arises in various physical systems and can be castoff for the optical communication resolution. The existence measures for these solutions are also unveiled in the form of constraint parameters. In addition, the pictorial representations of certain solutions depicted in figures undoubtedly play a key role in understanding the behavior and apprehending some of the physical features of the the discovery of a plethora of novel soliton solutions and their accompanying behaviors.

Analysis of modulational instability of waves in the fluid-filled elastic tube

In this paper, we investigate the modulational instability of waves within a fluid-filled elastic tube considering the fluid viscosity. Through the reductive perturbation method, we derived a modified Nonlinear Schrödinger Equation (NLSE) with damping term for the system. Based on the modified NLSE, we found that modulational instability exists in this system though the fluid viscosity is considered. Both the instability and the stability region are given. Furthermore, the growth rate of the instability is obtained. The dependence of the growth rate on the system parameters such as perturbation wave length, the background wave length, the viscosity of the fluid and the elasticity of the tube is given in the present paper. It indicates that the fluid viscosity refrain the modulational instability. The results may be used to predict the production of the rogue wave in this system, as well as to the other similar system.

Numerical investigation of a wideband supercontinuum source based on a large-mode-area photonic crystal fiber pumped at 1.3 μm

A numerical investigation was conducted to obtain a supercontinuum spanning about two octaves using a large mode area photonic crystal fiber (PCF) pumped at 1.3 μm. In our study, a 1.3 μm femtosecond laser and a silica-based large mode area PCF were selected as the pump source and nonlinear medium, respectively. The nonlinear Schrodinger equation was solved with split-step Fourier method to simulate the evolution of pulse and the broadening of spectrum. The effect of several parameters including the length of PCF, the pulse width, and the average pump power on characteristics of the output spectrum was studied. The simulation results revealed that the supercontinuum extended from near 600 nm to over 2450 nm at 20 dB with length of 30 cm, pulse width of 100 fs and average power of 12 W, respectively. This work proved this large mode area PCF a potentially excellent medium for supercontinuum source and provided some theoretical guidance for future experiments.

Cnoidal waves and their soliton limits in single mode fiber lasers

Cnoidal waves are a type of nonlinear periodic wave solutions of the nonlinear dynamic equations. They are well known in fluid dynamics, but it is not the case in optics. In this paper we show both experimentally and numerically that cnoidal waves could be formed in a fiber laser either in the net normal or net anomalous cavity dispersion regime, especially because, as the pump power is increased, the formed cnoidal waves could eventually evolve into a train of bright (in the net anomalous cavity dispersion regime) or dark (in the net normal cavity dispersion regime) solitons. Numerical simulations of the laser operation based on the extended nonlinear Schrödinger equation (NLSE) have well reproduced the experimental observations. The result not only explains why solitons can still be formed in a fiber laser even without mode locking but also suggests a new effective way of automatic stable periodic pulse train generation in lasers with a nonlinear cavity.

Modeling Wave Packet Dynamics and Exploring Applications: A Comprehensive Guide to the Nonlinear Schrödinger Equation

The nonlinear Schrödinger (NLS) equation stands as a cornerstone model for exploring the intricate behavior of weakly nonlinear, quasi-monochromatic wave packets in dispersive media. Its reach extends across diverse physical domains, from surface gravity waves to the captivating realm of Bose–Einstein condensates. This article delves into the dual facets of the NLS equation: its capacity for modeling wave packet dynamics and its remarkable breadth of applications. We illuminate the derivation of the NLS equation through both heuristic and multiple-scale approaches, underscoring how distinct interpretations of physical variables and governing equations give rise to varied wave packet dynamics and tailored values for dispersive and nonlinear coefficients. To showcase its versatility, we present an overview of the NLS equation’s compelling applications in four research frontiers: nonlinear optics, surface gravity waves, superconductivity, and Bose–Einstein condensates. This exploration reveals the NLS equation as a powerful tool for unifying and understanding a vast spectrum of physical phenomena.