Nonlinear Schrodinger Equation Research Articles

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.

Periodic finite-band solutions to the focusing nonlinear Schrödinger equation by the Fokas method: inverse and direct problems

We consider the Riemann–Hilbert (RH) approach to the construction of periodic finite-band solutions to the focusing nonlinear Schrödinger (NLS) equation. An RH problem for the solution of the finite-band problem has been recently derived via the Fokas method (Deconinck et al. 2021 Lett. Math. Phys. 111 , 1–18. ( doi:10.1007/s11005-021-01356-7 ); Fokas & Lenells. 2021 Proc. R. Soc. A 477 , 20200605. ( doi:10.1007/s11005-021-01356-7 )) Building on this method, a finite-band solution to the NLS equation can be given in terms of the solution of an associated RH problem, the jump conditions for which are characterized by specifying the endpoints of the arcs defining the contour of the RH problem and the constants (so-called phases) involved in the jump matrices. In our work, we solve the problem of retrieving the phases given the solution of the NLS equation evaluated at a fixed time. Our findings are corroborated by numerical examples of phases computation, demonstrating the viability of the method proposed.

The quasi-Gramian solution of a non-commutative extension of the higher-order nonlinear Schrödinger equation

The nonlinear Schrödinger (NLS) equation, which incorporates higher-order dispersive terms, is widely employed in the theoretical analysis of various physical phenomena. In this study, we explore the non-commutative extension of the higher-order NLS equation. We treat real or complex-valued functions, such as g 1 = g 1(x, t) and g 2 = g 2(x, t) as non-commutative, and employ the Lax pair associated with the evolution equation, as in the commutation case. We derive the quasi-Gramian solution of the system by employing a binary Darboux transformation. The soliton solutions are presented explicitly within the framework of quasideterminants. To visually understand the dynamics and solutions in the given example, we also provide simulations illustrating the associated profiles. Moreover, the solution can be used to study the stability of plane waves and to understand the generation of periodic patterns within the context of modulational instability.

Frequency downshifting in decaying wavetrains on the ocean surface covered by ice floes

We study analytically and numerically a frequency downshifting due to power-type frequency-dependent decay of surface waves in the ocean covered by ice floes. The downshifting is obtained both within the linear model and within the nonlinear Schrödinger (NLS) equation augmented by viscous terms for the initial condition in the form of an NLS envelope soliton. It is shown that the frequency-dependent dissipation produces a more substantial downshifting when the spectrum is relatively wide. As a result, the nonlinear adiabatic scenario of wavetrain evolution provides a downshifting remarkably smaller in magnitude than in the linear regime. Meanwhile, interactions between nonlinear wavegroups lead to spectral broadening and, thus, result in fast substantial frequency downshifts. Analytic estimates are obtained for an arbitrary power n of the dependence of a dissipation rate on frequency ∼ωn. The developed theory is validated by the numerical modeling of the generalized NLS equation with dissipative terms. Estimates of frequency downshift are given for oceanic waves of realistic parameters.

Propagation of collisional among multi-soliton, multi-singular solition and rogue wave around the critical values in an unmagetized plasma

Progress in understanding the propagation characteristics of (i) collisional acoustic among multi-soliton and multi-singular soliton around the critical values and their corresponding phase shifts and (ii) collision between two rogue waves (RWs) propagating toward each in a plasma environment is presented. The considered plasma environment consists of mobile cold positrons, immobile positive ions and (r, q)-distributed hot positrons, and electrons. To accomplish our goal, the coupled modified Korteweg–de Vries equations (mKdVEs) and nonlinear Schrödinger equations (NLSEs) are derived from the considered plasma environment. Based on the concept of Hirota's bilinear method, the multi-soliton and multi-singular soliton solutions of the coupled mKdVEs are determined directly. In addition, the analytical unstable RWs solutions of the coupled NLSEs are determined. With the impact of physical parameters, (i) the trajectories are described for double, triple, quadruple and quintuple positron acoustic bi-directional multi-soliton and (ii) the variation of collisional RWs profiles are displayed with physical interpretation. The results described by the coupled mKdVEs also show that the scattered double-, triple-, quadruple-, and quintuple-soliton are elastic and preserved their original features oppositely after the collision around the critical value of any specific plasma parameter.

Bright soliton propagation with loss and gain phenomena in transversely connected nonlinear pendulum pairs

In this work, we seek to investigate the dynamics of bright soliton in a chain of coupled pendulum pairs. After deriving the linear dispersion relation from the equation of the model, we find that among the obtained modes, the fast mode is the one on which we are going to be focused. Since the discrete simultaneous equation describing the dynamics of the model has not been extensively studied in the literature, we assume that the two lines of the model are proportional to each other. We use the rotating wave approximation method to derive a NLS equation governing the propagation of waves in the network. Depending on the choice of wave number, we deduce that the system supports bright and hole-soliton solutions. We use the obtained bright soliton as the initial condition for numerical computation, which demonstrates the significant role of the transverse coupling parameter in the system. That is, it affects the behavior of the forward-bright soliton generated in the system. The lattice allows gain and loss phenomena during the propagation of the waves.

Spectral structures and soliton dynamical behaviors of two shifted nonlocal NLS equations via a novel Riemann–Hilbert approach: A reverse-time NLS equation and a reverse-spacetime NLS equation

By extending the traditional Riemann–Hilbert (RH) approach of soliton equations to a novel version, this paper is devoted to studying two shifted nonlocal NLS equations: a reverse-time NLS equation with a temporal parameter t0, and a reverse-spacetime NLS equation with the spatial and temporal parameters x0,t0. It is shown that the defocusing cases of the two shifted nonlocal NLS equations are equivalent to their focusing cases under linear transformations, only the focusing cases of the two shifted nonlocal NLS equations need to be treated in this work which differs from the classical local NLS equation whose focusing and defocusing cases should be treated separately. Firstly, the spectral analysis of the Lax pair of the two shifted nonlocal NLS equations are performed, from which spectral structures of the two shifted equations are revealed in detail, respectively. Secondly, based on the obtained symmetry relations of the scattering data, the soliton solutions of the two shifted nonlocal NLS equations are rigorously obtained by introducing a novel reduction-proof technique which is different but more direct than the traditional RH approach. It is demonstrated that, due to the existence of the shifted parameters, the spectral analysis for deriving the symmetry relations of the scattering data and thus the calculations of the soliton solutions for the two shifted nonlocal NLS equations involve more tedious and ingenious computations than those unshifted ones without shifted parameters. Finally, the dynamical behaviors underlying the obtained soliton solutions are theoretically explored and graphically illustrated by highlighting the roles that the shifted parameters play, which manifest the peculiar soliton characteristics that may have potential applications of the shifted nonlocal reverse-time and reverse-spacetime NLS equations.

Optimal error bounds on time-splitting methods for the nonlinear Schrödinger equation with low regularity potential and nonlinearity

We establish optimal error bounds on time-splitting methods for the nonlinear Schrödinger equation with low regularity potential and typical power-type nonlinearity [Formula: see text], where [Formula: see text] is the density with [Formula: see text] the wave function and [Formula: see text] the exponent of the nonlinearity. For the first-order Lie–Trotter time-splitting method, optimal [Formula: see text]-norm error bound is proved for [Formula: see text]-potential and [Formula: see text], and optimal [Formula: see text]-norm error bound is obtained for [Formula: see text]-potential and [Formula: see text]. For the second-order Strang time-splitting method, optimal [Formula: see text]-norm error bound is established for [Formula: see text]-potential and [Formula: see text], and optimal [Formula: see text]-norm error bound is proved for [Formula: see text]-potential and [Formula: see text] (or [Formula: see text]). Compared to those error estimates of time-splitting methods in the literature, our optimal error bounds either improve the convergence rates under the same regularity assumptions or significantly relax the regularity requirements on potential and nonlinearity for optimal convergence orders. A key ingredient in our proof is to adopt a new technique called regularity compensation oscillation (RCO), where low frequency modes are analyzed by phase cancellation, and high frequency modes are estimated by regularity of the solution. Extensive numerical results are reported to confirm our error estimates and to demonstrate that they are sharp.

New asymptotic lower bound for the radius of analyticity of solutions to nonlinear Schrödinger equation

In this paper, we show that the radius of analyticity [Formula: see text] of solutions to the one-dimensional nonlinear Schrödinger (NLS) equation [Formula: see text] is bounded from below by [Formula: see text] when [Formula: see text] and by [Formula: see text] when [Formula: see text] as [Formula: see text], given initial data that is analytic with fixed radius. This improves results obtained by Tesfahun [On the radius of spatial analyticity for cubic nonlinear Schrödinger equations, J. Differential Equations 263(11) (2017) 7496–7512] for [Formula: see text] and Ahn et al. [On the radius of spatial analyticity for defocusing nonlinear Schrödinger equations, Discrete Contin. Dyn. Syst. 40(1) (2020) 423–439] for any odd integers [Formula: see text], where they obtained a decay rate [Formula: see text] for larger [Formula: see text]. The proof of our main theorems is based on a modified Gevrey space introduced in [T. T. Dufera, S. Mebrate and A. Tesfahun, On the persistence of spatial analyticity for the beam equation, J. Math. Anal. Appl. 509(2) (2022) 126001], the local smoothing effect, maximal function estimate of the Schrödinger propagator, a method of almost conservation law, Schrödinger admissibility and one-dimensional Sobolev embedding.

A Semiclassical Approach to the Nonlocal Nonlinear Schrödinger Equation with a Non-Hermitian Term

The nonlinear Schrödinger equation (NLSE) with a non-Hermitian term is the model for various phenomena in nonlinear open quantum systems. We deal with the Cauchy problem for the nonlocal generalization of multidimensional NLSE with a non-Hermitian term. Using the ideas of the Maslov method, we propose the method of constructing asymptotic solutions to this equation within the framework of semiclassically concentrated states. The semiclassical nonlinear evolution operator and symmetry operators for the leading term of asymptotics are derived. Our approach is based on the solutions of the auxiliary dynamical system that effectively linearizes the problem under certain algebraic conditions. The formalism proposed is illustrated with the specific example of the NLSE with a non-Hermitian term that is the model of an atom laser. The analytical asymptotic solution to the Cauchy problem is obtained explicitly for this example.