Nonlinear Schrodinger Equation Research Articles

Multistep Reduced Differential Transform Method in Solving Nonlinear Schrodinger Equations

This paper obtains semi-analytical solutions for the nonlinear Schrodinger equations (NLSEs) using the multistep reduced differential transform method (MsRDTM). The implemented method yields an analytical approximate solution over a longer time frame, in which the method applied is treated as an algorithm in a sequence of small sub-division of intervals of identical length compared to the traditional reduced differential transform method (RDTM). Excluding the need of perturbation, linearization, or discretization, this method offers the benefit and reliability of the multistep algorithm. The outcomes show that the MsRDTM generated highly accurate solutions of NLSEs than the RDTM. In addition, the results show that the suggested method is straightforward to use, saves a significant amount of computing work when solving NLSEs, and has potential for broad application in other complex partial differential equations (PDEs) in the fields of engineering and science. The accuracy of the method is shown through the tables and graphical illustrations provided.

Predicting positon solutions of a family of nonlinear Schrödinger equations through deep learning algorithm

We consider a hierarchy of nonlinear Schrödinger equations (NLSEs) and forecast the evolution of positon solutions using a deep learning approach called Physics Informed Neural Networks (PINN). Notably, the PINN algorithm accurately predicts positon solutions not only in the standard NLSE but also in other higher order versions, including cubic, quartic and quintic NLSEs. The PINN approach also effectively handles two coupled NLSEs and two coupled Hirota equations. In addition to the above, we report exact second-order positon solutions of the sextic NLSE and coupled generalized NLSE. These solutions are not available in the existing literature and we construct them through generalized Darboux transformation method. Further, we utilize PINNs to forecast their behaviour as well. To validate PINN's accuracy, we compare the predicted solutions with exact solutions obtained from analytical methods. The results show high fidelity and low mean squared error in the predictions generated by our PINN model.

Modified failproof physics-informed neural network framework for fast and accurate optical fiber transmission link modeling

Physics-informed neural networks (PINNs) have recently emerged as an important and ground-breaking technique in scientific machine learning for numerous applications including in optical fiber communications. However, the vanilla/baseline version of PINNs is prone to fail under certain conditions because of the nature of the physics-based regularization term in its loss function. The use of this unique regularization technique results in a highly complex non-convex loss landscape when visualized. This leads to failure modes in PINN-based modeling. The baseline PINN works very well as an optical fiber model with relatively simple fiber parameters and for uncomplicated transmission tasks. Yet, it struggles when the modeling task becomes relatively complex, reaching very high error, for example, numerous modeling tasks/scenarios in soliton communication and soliton pulse development in special fibers such as erbium-doped dispersion compensating fibers. We implement two methods to circumvent the limitations caused by the physics-based regularization term to solve this problem, namely, the so-called scaffolding technique for PINN modeling and the progressive block learning PINN modeling strategy to solve the nonlinear Schrödinger equation (NLSE), which models pulse propagation in an optical fiber. This helps PINN learn more accurately the dynamics of pulse evolution and increases accuracy by two to three orders of magnitude. We show in addition that this error is not due to the depth or architecture of the neural network but a fundamental issue inherent to PINN by design. The results achieved indicate a considerable reduction in PINN error for complex modeling problems, with accuracy increasing by up to two orders of magnitude.

Spatial optical soliton cluster solutions in strongly nonlocal nonlinear media

A (2+1)-dimensional variable coefficient nonlinear Schrödinger equation (NLSE) is described under strongly nonlocal nonlinear media. The soliton cluster solutions are obtained by using the improved self-similar reduction and F-expansion method. The regulation of nonlinear coefficients on the dynamics of spatial optical solitons and the self-similar propagation characteristics of the optical soliton clusters are investigated. It is observed that dark solitons transmitted in nonlinear media show two different spatial distributions. It is also found that the peak intensity of the soliton cluster only fluctuates in a small range during the propagation process, but the soliton cluster after harmonic potential modulation has some loss during the propagation process. But the loss is not as dramatic as when bright solitons are modulated. A new kind of soliton cluster is formed after the modulation of dark solitons, which has great practical significance such as application in logic gates, all-optical devices.

Analysis of performance limits in optical communications due to fiber nonlinearity and third order dispersion

In this study, an analysis is conducted to treasure the expressions of the pulsation expansion factor, in addition to the standardized output, and solve the nonlinear Schrödinger equation (NLSE), reflecting the impact of XPM on third order dispersion. Using large effective area fiber (LEAF) and standard single-mode fiber (SSMF), the effects of transmission distances and varying input powers are assessed at various transmission speeds. The first and second order GVD XPM effects are the only factors influencing the pulse’s propagation. The second-order effects of GVD are not noticeable at short distances and low bit rates, but they become noticeable and impact system performance as the bit rate increases. The study discovered that input dominance has less of an impact on pulse width than data rate and fiber length. Methodical derivation and numerical simulation using the split-phase Fourier method at the same data rate and input power yield the SSMF and LEAF consequences. In comparison to LEAF fibers, XPM has a greater beneficial impact on second and third order dispersion in SSMF fibers.

Envelope solitary waves in two-negative ions with stationary dust grains

Abstract Using a multi-fluid model, we look at how modulated electrostatic dust-ion-acoustic wave
packets move nonlinearly through a plasma made up of a three-ion fluid with Maxwellian
electrons and stationary dust grains. A nonlinear Schr¨odinger (NLS) equation describes
the electric potential envelope wave packet. The analysis reveals the existence of different
types of localized modes, namely bright, dark, and grey solitons. We numerically analyse
the coefficients of the NLS equation to identify stable or unstable regions for wave packet
propagation. It is found that higher relative density ratios increase the group velocity of the
wave packets. Stable pulses can become unstable when plasma parameters exceed certain
relative density ratio values. Stable pulses can exist within a crucial window of the relative
dust density ratio. Controlling the dust grain density ratio outside the zone can cause unstable
wave packets or bright envelope solitons to propagate.

Optical solitons retrieval for an extension of novel dual-mode of a dispersive non-linear Schrödinger equation

In this study, we rearrange the non-linear Schrödinger equation (NLSE) into a dual-mode structure, which expands it, and then we investigate the geometrical evaluation and representation of this novel model. Explicit, exact solutions are obtained by implementing the improved modified extended tanh-function (IMETF) approach. The study’s findings have important ramifications for how solitons propagate in nonlinear optics. The resultant solutions can be found in several forms: singular, bright, rational, exponential, Jacobi elliptic function (JEF), Weierstrass elliptic doubly periodic solution, and singular periodic solutions. By comparing our obtained traveling wave solutions with existing literature, we demonstrate their novelty and substantive contribution to current research. The success of our approach suggests its potential applicability to address various nonlinear challenges in different fields, particularly within soliton theory, since the analyzed model appears in a multitude of applications. Besides, in order to better comprehend some of these found solutions behaviors, we showed their outlines in 3D and 2D graphs.

Random numerical stability of attractors for nonlinear Schrodinger equations on infinite lattices

Random numerical stability of attractors for nonlinear Schrodinger equations on infinite lattices

Soliton molecules, breather molecules and stability analysis of the three-coupled nonlinear Schrödinger equation

We investigate the three-coupled nonlinear Schrödinger (NLS) equation, which holds a significant physical implications in optical communications and biophysics. The multi-soliton solutions are derived utilizing the Hirota bilinear method. Multi-breather solutions are presented by assuming certain relationship for the parameters of the multi-solitons. By introducing the velocity resonance mechanism to multiple soliton/breather solutions, it is found that the three-coupled NLS model possesses abundant soliton molecule, breather molecule and breather-soliton molecule structures. The interactions among different types of soliton molecules which can be observed in marine and oceanic waters are investigated through numerical simulation. Ultimately, baseband modulation instability of the three-coupled NLS equation is analyzed. It is highly significance for the study of soliton formation and rogue waves in the three-coupled NLS equation.

A combined Liouville integrable hierarchy associated with a fourth-order matrix spectral problem

Abstract This paper aims to discuss a fourth-order matrix spectral problem involving four potentials and to generate an associated Liouville integrable hierarchy via the zero curvature formulation. A bi-Hamiltonian formulation is furnished by applying the trace identity and a recursion operator is explicitly worked out, which exhibits the Liouville integrability of each model in the resulting hierarchy. Two specific examples, consisting of novel generalized combined nonlinear Schroedinger equations and modified Korteweg-de Vries equations, are given. 

Extraction of optical wave structures to the coupled fractional system in magneto-optic waveguides

In this article, truncated M-fractional coupled nonlinear Schrodinger equation (NLSE) with quadratic–cubic nonlinearity is under observation. The studied model is composed of chromatic dispersion, magneto-optic parameter and inter-modal dispersion. The NLSE is the most significant physical model to explain the fluctuations of optical soliton proliferation. The NLSEs have become more popular because of the clarity with which they explain a wide range of complex physical phenomena and the depth with which they display dynamical patterns via localized wave solutions. Optical soliton propagation in magneto-optic is currently a subject of great interest due to the multiple prospects for ultrafast signal routing systems and short light pulses in communications. The optical solitons are secured in the forms of bright, dark, singular and combo solitons. In addition, hyperbolic, periodic and exponential function solutions have been recovered. The modified Sardar subequation and enhanced modified extended tanh-expansion approaches recently developed integration tools are adopted in this study for securing the solutions. In nonlinear dispersive media, optical solitons are stretched electromagnetic waves that maintain their intensity due to a balance between the effects of dispersion and nonlinearity. The effect of parameters have been observed by allotting suitable values and sketching the different shapes of the graphs.

Formation and propagation of fundamental and vortex soliton families in 1D and 2D dual-Lévy-index fractional nonlinear Schrödinger equations with cubic–quintic nonlinearity

We address effects of interplay of two different fractional-diffraction terms, corresponding to different Lé vy indices (LIs), in the framework of nonlinear Schrödinger equation (NLSE) with cubic–quintic nonlinearity (dual-LI fractional NLSE ), in one- and two-dimensional (1D and 2D) settings. The critical (in 1D) and supercritical (in 2D) wave collapses are suppressed in the presence of a defocusing quintic term, making it possible to produce stable localized modes, including fundamental and vortical ones. In 2D, families of fundamental and vortex solitons, with topological charges S=0,1,2,3, are produced in a numerical form and by dint of the variational approximation (VA). In particular, the threshold power necessary for the existence of 2D fundamental solitons is predicted by the VA, being close to the numerical results. In the case of fractional-diffraction terms with unequal LIs acting in two transverse directions, anisotropic 2D fundamental and vortex solitons are constructed. Stability of the solitons is investigated for small perturbations governed by the linearized equations, and results are corroborated by direct simulations of the perturbed evolution. Those vortex solitons which are unstable are split by azimuthal perturbations into fragments, whose number is determined by the shape of the perturbation eigenmode with the largest growth rate. These results extend the concept of the 2D fractional diffraction and solitons to media with the anisotropic fractionality.

Fast nonlinear Fourier transform algorithms for optical data processing.

The nonlinear Fourier transform (NFT) is an approach that is similar to a conventional Fourier transform. In particular, NFT allows to analyze the structure of a signal governed by the nonlinear Schrödinger equation (NLSE). Recently, NFT applied to NLSE has attracted special attention in applications of fiber-optic communication. Improving the speed and accuracy of the NFT algorithms remains an urgent problem in optics. We present an approach that allows to find all variants of symmetric exponential splitting schemes suitable for the fast NFT (FNFT) algorithms with low complexity. One of the obtained schemes showed good numerical results in computing the continuous spectrum compared with other fast fourth-order NFT schemes.

The application of the "inverse problem" method for constructing confining potentials that make N-soliton waveforms exact solutions in the Gross-Pitaevskii equation.

In this work, we discuss an application of the "inverse problem" method to find the external trapping potential, which has particular N trapped soliton-like solutions of the Gross-Pitaevskii equation (GPE) also known as the cubic nonlinear Schrödinger equation (NLSE). This inverse method assumes particular forms for the trapped soliton wave function, which then determines the (unique) external (confining) potential. The latter renders these assumed waveforms exact solutions of the GPE (NLSE) for both attractive (g<0) and repulsive (g>0) self-interactions. For both signs of g, we discuss the stability with respect to self-similar deformations and translations. For g<0, a critical mass Mc or equivalently the number of particles for instabilities to arise can often be found analytically. On the other hand, for the case with g>0 corresponding to repulsive self-interactions which is often discussed in the atomic physics realm of Bose-Einstein condensates, the bound solutions are found to be always stable. For g<0, we also determine the critical mass numerically by using linear stability or Bogoliubov-de Gennes analysis, and compare these results with our analytic estimates. Various analytic forms for the trapped N-soliton solutions in one, two, and three spatial dimensions are discussed, including sums of Gaussians or higher-order eigenfunctions of the harmonic oscillator Hamiltonian.

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.&#x0D; 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.