Solutions Of 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.

Influence of NNN interaction and existence of breathers in a quasi-discrete Heisenberg helimagnetic spin system

In this work, we investigate the existence and properties of discrete bright and dark breathers in a one-dimensional Heisenberg helimagnet incorporating bilinear, helical, NNN and anisotropic interactions. The Hamiltonian is bosonized using the Dyson–Maleev transformation and the coherent state representation is selected as the fundamental representation of the system. With the help of multiple scales combined with quasi-discreteness approximation, the dynamical equation is reduced to a nonlinear Schrödinger equation (NLSE). By employing the solution of NLSE derived using inverse scattering technique (IST), the bright and dark type breather solutions are constructed. The effect of helicity and NNN interactions on these breather modes is also analysed.

Ion Acoustic Breathers in Electron-Beam Plasma

The nonlinear excitations of ion acoustic (IA) structures in an electron beam embedded plasma composed of Vasyliunas–Cairns (VC) distributed hot electrons has been studied. The nonlinear Schrödinger equation (NLSE) from the Kadomtsev–Petviashvili (KP) equation with suitable transformation has been derived from rational solutions of NLSE; breathers have been studied. It has been shown that the nonthermality and superthermality of the electrons, the electron beam density, and the beam velocity alter the characteristics of different kinds of breathers. This investigation may be important in interpreting the physics of nonlinear structures in the upper layer of magnetosphere.

Numerical solutions of nonlinear Schrodinger equation with applications in optical fiber communication

Optical soliton is a nonlinear wave with huge information or data which travels over larger distance in optical fibers. This paramount application of optical soliton forced many mathematicians and physicians to focus on it which initiates to study the soliton solutions of mathematical model in optical fiber communication characterized by nonlinear Schrodinger equation (NLSE) along with its important applications. Nonlinear Schrodinger equation represents propagation of light pulse through an optical fiber. In this paper NLSE is solved numerically by using exponential cubic B-spline differential quadrature method. In this method leave-one out cross validation strategy is being used to find optimal value of parameter p which enhances accuracy and efficiency of its results which are near to exact solutions and approximate to available numerical solutions in the literature.

Optical solitons with Kudryashov’s sextic power-law nonlinearity

Objective –The main objective of this paper is to investigate analytic soliton solutions of a nonlinear Schrödinger equation (NLSE), including Kudryashov’s sextic power-law nonlinearity by introducing different approaches of two efficient analytical methods. The considered equation has been recently introduced by N. A. Kudryashov to describe pulse propagation in optical fibers. The refractive index in the equation comprises six terms, each of the terms containing a power-law component. Methods –Applying a wave transformation to the considered NLSE and splitting up the real and imaginary parts, the NLSE is converted to the nonlinear ordinary differential equations (NLODEs). Then, the solutions of NLODEs are considered as suggested in the proposed method and suggested solutions that include some unknown parameters are substituted into the NLODE. An algebraic equations system is acquired by collecting the same power of the unknown function and equating all coefficients to zero. The unknown parameters in the system, and so the solutions of the NLSE, are found by solving the system. In the proposed first method, the modified extended tanh method is enhanced by proposing more solutions. The proposed second method, a different version of auxiliary methods, can remarkably reduce calculations to easily get solutions for the NLSEs with higher-order or power-law nonlinearity. Results –The two proposed methods are successfully applied to the considered NLSE and the abundant solutions of the NLSE are attained. Besides, 2D, 3D and contour graphs are demonstrated in figures for the physical illustrations of the gained solutions. Conclusion –Obtaining the solutions of NLSEs with higher-order or power-law nonlinearity has crucial importance but still challenging work. So, we propose different approaches of two efficient analytical methods, namely, enhanced modified extended tanh expansion method and an auxiliary function method. The derived results imply that the used methods are very efficient, reliable and powerful such that they can be easily implemented to many nonlinear NLSEs with higher-orders or higher power-law nonlinearities that describe real-life phenomena.

New optical soliton solutions for the variable coefficients nonlinear Schrödinger equation

This paper is devoted to seek new optical soliton solutions of nonlinear Schrödinger equation (NLSE) with time-dependent coefficients which describes the dispersion decreasing fiber. To achieve optical soliton solutions of NLSE, the basic idea of homogenous balance approach has been used to propose Bernoulli $$(G'/G)$$ -expansion method, where $$G =G(\zeta )$$ satisfies Bernoulli equation, which is easier to solve than previous studies. By applying some transformations and using this method, some periodic wave, bright and dark soliton solutions are successfully obtained. Moreover, 3D surfaces, standard deviation line plots and contour maps graphs of the obtained results under effect of different values of coefficients are illustrated to have acceptable image of dynamic structures and to find the relation between the parameters and wave behaviors.

Dynamical plane wave solutions for the Heisenberg model of ferromagnetic spin chains with beta derivative evolution and obliqueness

The oblique plane waves with their dynamical behaviors for a (2+1)-dimensional nonlinear Schrödinger equation (NLSE) having beta derivative spatial-temporal evolution are investigated. In order to study such phenomena, NLSE is converted to a nonlinear ordinary differential equation with a planar dynamical system by considering the variable wave transform with obliqueness and the properties of the beta derivative. Some more new general forms of analytical solutions, like bright, dark, singular, and pure periodic solutions of NLSE are constructed by employing the auxiliary ordinary differential equation method and the extended simplest equation method. The effect of obliqueness and beta derivative parameter on several types of wave structures along with the phase portrait diagrams are reported by considering some special values of parameters for the existence of attained solutions. It is found that the planar dynamical system is not supported by any type of orbit for Θ=450. It is also confirmed from the obtained solutions that no plane waves are generated for Θ=450. The presented studies on bifurcation analysis and analytical solutions for (2+1)-dimensional NLSE would be very useful to understand the physical scenarios of nonlinear spin dynamics in magnetic materials for Heisenberg models of ferromagnetic spin chains.

Fast sixth-order algorithm based on the generalized Cayley transform for the Zakharov-Shabat system associated with nonlinear Schrodinger equation

The nonlinear Schrödinger equation (NLSE) is widely used in telecommunication applications, since it allows one to describe the propagation of pulses in an optical fiber. Recently some new approaches based on the nonlinear Fourier transform (NFT) have been actively explored to compensate for fiber nonlinearity and to exceed the limitations of nonlinearity-imposed limits of linear transmission methods. Despite the fact that the numerical solution of NLSE is a general problem, nevertheless, the optical community has been focusing on this issue. Improving the accuracy of the NFT algorithms remains an urgent problem in optics. In particular, it is important to increase the approximation order of the methods, especially in problems where it is necessary to analyze the structure of complex waveforms. To correctly describe them and their spectral parameters, more accurate and fast numerical methods are needed.We propose a novel general approach for constructing sixth-order (with respect to an integration step) finite-difference schemes for first-order linear differential systems. These schemes are based on the generalized Cayley transform and include exponential integrators as a special case. If the system has a time-dependent skew-hermitian matrix then the schemes conserve the quadratic first integral automatically. Then we apply our method to solve the direct spectral problem for the Zakharov-Shabat system. New schemes with fractional rational transition matrix allow the use of fast algorithms to solve the initial problem for a large number of values of the spectral parameter.

Plane Wave Solutions With Dynamical Behaviors for Heisenberg Model of Ferromagnetic Spin Chain With Beta Derivative Evolution and Obliqueness

The plane waves with their dynamical behaviors of (2+1)-dimensional nonlinear Schrodinger equation (NLSE) having beta derivative spatial-temporal evolution are investigated. In order to study such phenomena, NLSE is converted to a nonlinear ordinary differential equation with a planar dynamical system (PDS) by considering the variable wave transform and the properties of the beta derivative. By employing two distinct solution methods, namely, the auxiliary ordinary differential equation method and the extended simplest equation method, some more new general form of analytical solutions of NLSE are constructed. The effect of beta derivative parameter and obliqueness on several types of wave structures along with the phase portrait diagrams are reported. It is found that the PDS is not supported by any type of orbits for Θ = 45°. It is also confirmed from the obtained solutions that no plane waves are generated for Θ = 45°. The presented studies on bifurcation analysis and analytical solutions for (2+1)-dimensional NLSE would be very useful to understand the physical scenarios for Heisenberg models of ferromagnetic spin chain with magnetic interactions.

Head-on collision of ion-acoustic multi-solitons and study of rogue waves in electron-beam superthermal plasma

In this investigation, head-on collision of ion-acoustic (IA) multi-solitons and study of characteristics of ion acoustic rogue waves in an unmagnetized plasma which includes ion fluid, superthermal hot electrons, and penetrated by electron beam have been presented. The Korteweg-de Vries equation have been derived by employing the extended Poincaré-Lighthill-Kuo (PLK) method. Further, multi-solitons of different KdV equations have also been determined by following the Hirota bilinear method. The combined effects of electron beam parameters and variation of different physical parameters on the characteristics of IA solitons and phase shifts of multi-solitons have been analyzed. It is remarked that electron beam components and other plasma parameters significantly influence the phase shifts and other properties of ion-acoustic solitons (IASs). Furthermore, the nonlinear Schrodinger equation (NLSE) has been derived from the KdV equation by using appropriate transformation. The characteristics of different orders rogue waves have also been studied from the different solutions of NLSE. The results of this study may be of great importance to understand the characteristics properties of various kinds of nonlinear solitary structures in different kinds of plasma systems.

Symmetrical localized states in three-layered structure consisting of linear layer between defocusing media separated by interfaces with nonlinear response

The three-layered symmetric structure consisting of a linear layer sandwiched between defocusing nonlinear media was considered. The layers are separated by interfaces with nonlinear properties. The nonlinear symmetrical localized states arising in such a structure were described theoretically. The proposed model is based on the nonlinear Schrodinger equation with negative Kerr-type nonlinear term and nonlinear self-consistent potential describing the interaction of the waves and interfaces. Even and odd solutions of nonlinear Schrodinger equation correspond to the localized states of two types existing in different energy ranges. The stationary state energies as the system parameter functions are calculated in explicit form. The conditions of their existence were analyze depending on the combination of signs of nonlinear interface parameters. The localized states of the special kinds existing only for the case of interfaces, which are characterized by a strong nonlinear response, were found. The dependencies of localization energies on the amplitude of the interface oscillations were studied analytically.

Effects of adiabaticity of electrons and negative ions on the modulational instability of ion-acoustic waves in electronegative plasma

In the present work, we investigate modulational instability (MI) in an unmagnetised collisionless electronegative plasma consisting of inertialess adiabatic electrons, inertialess adiabatic negative ions and mobile positive ions. For this purpose, a nonlinear Schrodinger equation (NLSE) is derived to study envelope evolution of the ion-acoustic carrier waves for our system by using reductive perturbation technique. In examining the stability/instability of the ion-acoustic envelope wavepackets, the critical wavenumber kc at which MI sets in is found to be shifted to higher values as adiabaticity γ increases. The effects of electron-to-negative ion temperature ratio σn and negative ion density fraction f on the region for propagation of stable wave pulses are also discussed. Within the instability region, the rational solution of NLSE provides the rogue wave (RW) profiles which is localised in space and time. With the increase of adiabaticity, the possibility for the formation of RW decreases at the lower wavenumber which is in good agreement with experimental observations. The change in adiabaticity significantly influences the rogue wave amplitudes. Maximum pulse height for RW is found to be increasing with the increase of adiabaticity. Our theoretical findings may be helpful for basic characteristic study of rogue waves in the laboratory as well as space plasmas.

Normalized solutions for nonlinear Schrödinger equations

Due to its very important applications in many physical problems, the normalized solution of nonlinear Schrodinger equations has gradually attracted the attention of a large number of researchers in recent years:beginalign*begincases-Δ u+λ u=g(u), x∈ mathbbR^N, u∈ H¹(mathbbR^N), displaystyle∫mathbbR^Nu²=c,endcasesendalign*where the normalization condition $c\\in~\\mathbb{R}^+$ is given, but the Lagrange multiplier $\\lambda$ is unknown. We first introduce the existence, multiplicity, and other properties of the normalized solution of a single Schrodinger equation under different conditions. Then we introduce some new results related to the normalized solution of the nonlinear Schrodinger systems. Finally we list some open problems related to the normalized solution.

An Efficient Meshless Method for Solving Multi-dimensional Nonlinear Schrödinger Equation

In this article, we study an efficient combination of the meshless local Petrov–Galerkin and time-splitting methods for the numerical solution of nonlinear Schrodinger equation in two and three dimensions. The Strang splitting technique is used to separate the original equation in two parts, linear and nonlinear. The linear part is approximated with the meshless local Petrov–Galerkin method in the space variable and the Crank–Nicolson method in time. Also, the nonlinear part can be solved analytically. We use the moving Kriging interpolation instated of the moving least squares approximation to make the shape functions of the meshless local Petrov–Galerkin method which have the Kronecker delta property, so the Dirichlet boundary condition is imposed directly and easily. In the meshless local Petrov–Galerkin method, the Heaviside step function is chosen as the test function in each sub-domain. Several test problems for two and three dimensions are presented, and the results are compared to their analytical and other numerical methods to illustrate the accuracy and capability of this technique.

Periodic and rogue waves for Heisenberg models of ferromagnetic spin chains with fractional beta derivative evolution and obliqueness

ABSTRACT The nonlinear Schrodinger equation (NLSE) in (2 + 1) dimensions with beta derivative evolution is considered here to study nonlinear coherent structures for Heisenberg models of ferromagnetic spin chain with magnetic exchanges. Such structures are studied by determining the analytical solutions of NLSE having beta derivative evolution via two different mathematical techniques. The dynamical behaviors of equilibrium points are also studied by deriving the planar dynamical system from the considered equation. Some of obtained analytical solutions are described with graphical representation by varying beta derivative parameter (BDP) and obliqueness. It is revealed that the obliqueness is extensively affected both on the plane wave dynamics as well as equilibrium points of the system, whereas the equilibrium points are independent of BDP.

Simple high-order boundary conditions for computing rogue waves in the nonlinear Schrödinger equation

This paper proposes some simple and accurate artificial boundary conditions in order to develop numerical methods to compute the rational solution of the nonlinear Schrödinger equation (NLSE) modeling the dynamics of rogue waves. To overcome the nonzero background condition and the algebraically slow decaying ratio at spatial infinity, we design the boundary conditions by means of a far field asymptotic expansion formulation of the rational solution and some transformations for resting the oscillation in time. Compared to the existing periodic boundary condition, the proposed boundary conditions can achieve fourth-order accuracy at most, i.e., O(L−4) with the interval length L, which means that for getting the presupposed accuracy, a smaller computational interval can be chosen. Meanwhile, their forms are simple, so there is hardly any additional computational cost to implement the numerical methods. Then we compare the proposed boundary conditions and numerical methods in terms of efficiency and accuracy, and identify the most efficient and accurate one for computing the rational solution of NLSE and numerically study their stability and interaction.

Analytical solutions of the time-fractional non-linear Schrodinger equation with zero and non zero trapping potential through the Sumudu decomposition method

Sumudu decomposition method is used to construct the approximate analytical solutions of time-fractional nonlinear Schrodinger equations with zero and nonzero trapping potential. The Sumudu decomposition method is a combined form of the Sumudu transform and the Adomian decomposition method. The fractional derivatives are defined in the Caputo sense. The exact solutions of some nonlinear Schrodinger equations are given as a special case of our approximate analytical solutions. The computations show that the described method is easy to apply, and it needs smaller size of computation as compared to the other existing methods. Further, the solutions are derived in a convergent series form which shows the effectiveness of the method for solving a wide variety of nonlinear fractional differential equations.

Field confinement effect induced by nonlinear interface between nonlinear focusing and defocusing media

We describe the non-symmetrical spatial-distributed electric field near the thin optic layer with nonlinear properties that separated two focusing and defocusing media with Kerr-type nonlinearity differing by values of refractive index. The nonlinear Schrödinger equation (NLSE) with nonlinear potential containing two parameters describes the distribution of electric field strength in adiabatic approximation. The problem is reduced to the solution of NLSEs at the half-spaces with the nonlinear boundary conditions at the interface plane. We obtain five new types of nonlinear stationary states describing the confinement effect of electric field strength across the interface. The field confinement effect is the localization of nonlinear spatially periodic wave during the transition from a one half-space to another one where the field distribution is monotonically damping from the interface. We derive and analyze the frequencies of field confinement effect possibility in dependence of media and interface characteristics.

The evolution of rogue wave triplets and super rogue waves in superthermal polarized space dusty plasma

This present investigation has been instigated to examine the impact of polarization force on modulational instability of dust acoustic (DA) waves and transition of rogue wave triplets to super rogue waves in dusty plasma composed of negative dust as fluid, Boltzmannian electrons, and superthermal ions. The presence of superthermal ions has remarkably altered the impact of polarization force. An increment in ion superthermality index restricts the polarization parameter toward smaller values. By adopting the reductive perturbation technique, the nonlinear Schrödinger equation (NLSE) is procured that determines the modulational instability of the dust acoustic waves. It is observed that the effect of polarization force is constricted by the wavenumber domain in the advent of the instability region. The rational solution of NLSE describes the evolution of dust acoustic rogue wave triplets, which further transform into the super rogue waves by means of superposition of triplets. It is remarked that the amalgamation of polarization force and superthermal ions have an explicit impact on the characteristics of different kinds of dust acoustic rogue waves. It is intensified that our present theoretical pronouncements may shed light on the salient features of different kinds of DA rogue waves in laboratory experiments and space/astrophysical regions, especially in Saturn's magnetosphere, planetary rings, and comet tails, etc.

Existence of positive solution for a fractional elliptic equation in exterior domain

In this paper, we show the existence of positive solutions for nonlinear Schrodinger equation with fractional Laplacian(−Δ)su+λu=|u|p−2uinΩ, where Ω⊂RN is an unbounded domain, ∂Ω≠∅ is bounded, λ∈R+, s∈(0,1), N>2s and 2<p<2s⁎. Precisely, we show that the problem has at least one positive solutions. We achieved our results by using variational method together with Brouwer theory of degree. Moreover, we also prove a version to the Fractional operator in unbounded domain of the Global Compactness result due to Struwe (see [24]).