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This paper applies the Improved Order of Convergent of the Cubic B-Spline collocation method (IOCuBSM) to obtain the numerical solution of the nonlinear Schrödinger Equation with a single soliton as the test problem. The approximate solution is based on the Crank-Nicolson scheme with the finite-difference method applied to time discretization and the improved order of convergent cubic B-Splines as the interpolation function in the space dimension. The error between numerical and analytical solutions is determined by discrete maximum norm and Euclidean norm and the method is proved to be unconditionally stable. Furthermore, the computational time for the methods is also studied. [ABSTRACT FROM AUTHOR]

Researchers are currently investigating the impact of collisions between solitons and external potentials, particularly the Nonlinear Schrödinger Equation (NLSE), which governs the soliton interaction with the potential in the presence of nonlinearity. In the case of discrete systems of one-dimensional NLSE, the behaviour and propagation of discrete solitons through a series of discrete sites or points are described. In this paper, we investigated the interaction of a discrete soliton with a localized impurity, namely the Gaussian external potential, using two methods: the variational approximation (VA) method and the direct numerical method. Discrete Cubic-Quintic NLSE is the main equation used to describe the phenomenon. This equation is non-integrable and does not have analytical solutions, but it can be reduced to ordinary differential equations using the variational approach. The VA method is applied by taking into account the Gaussian profile as the ansatz, which then produces evolution equations for soliton solutions of width, linear and quadratic phase-front correction and center-of-mass position. These equations play typical roles in describing the scattering process of the discrete soliton with the localized impurity. The direct numerical method of discrete NLSE is then performed to compare and validate the accuracy of the approximation results from the VA method. Considering different values of potential strengths and a fixed soliton initial velocity, it is found that the soliton can be transmitted, trapped or reflected after the interaction with external Gaussian potential. The findings imply that the VA method is a successful and practical way for examining the scattering of soliton in discrete NLSE on external potential. [ABSTRACT FROM AUTHOR]

This paper aims to study and determine the necessary condition of obtaining soliton solutions in multi-component generalizations of the non-local reduction for nonlinear Schrödinger equation with PT-symmetry. We consider particularly square barrier initial conditions. This work includes: evaluating the spectral problem, introducing the Jost solutions and scattering matrix of the Zakharov-Shabat system, and determining discrete eigenvalues. As the main example, we use the Zakharov-Shabat system which corresponds to the Manakov system of nonlinear Schrödinger equation. In addition, three cases of square barrier potential corresponding to symmetric and asymmetric potentials are shown and studied. A multi-soliton configuration is shown to be allowed for multi-component non-local nonlinear Schrödinger equations of the Manakov type. Numerical simulations are used to obtain and illustrate the results that depend on solving a system of equations. [ABSTRACT FROM AUTHOR]

This paper deals with three classes of nonlinear Schrödinger equations (NLSE) having general polynomial nonlinearities. We find travelling wave solutions of soliton type (smooth solitons, peakons and cuspons), periodic ones (smooth, peakons and cuspons) and kinks. In the case of some special polynomials the solutions are expressed up to the inverse map by Gauss and Appell's hypergeometric functions, by some elliptic functions of Jacobi and Weierstrass types etc. In each case the above mentioned solutions of NLSE are obtained into integral form. [ABSTRACT FROM AUTHOR]

In the present paper, a nonlinear fractional Schrödinger equation is investigated with a numerical approach. First and second orders of accuracy difference schemes are constructed for a mixed problem for a nonlinear fractional Schrödinger equation with Neumann boundary condition. Results of numerical experiments support the convergence of solutions of constructed first and second order of accuracy difference schemes to exact solution of the problem. [ABSTRACT FROM AUTHOR]

The nonlinear Schrödinger equation is a well-known partial differential equation, which provides a successful model in nonlinear optic theory, as well as other applications. The conservation law plays an important role in the study of nonlinear evolutionary equations. In this paper, we construct infinitely many conservation laws for the two-dimensional nonlinear Schrödinger equation with the Lax pair. [ABSTRACT FROM AUTHOR]

The Cubic-Quintic Nonlinear Schrödinger Equation (CQNLSE) is one of the universal mathematical models constituting many interesting problems in physics such as plasma physics, condensed matter physics, Bose-Einstein condensates, nonlinear optics, etc. This paper studies the scattering of the soliton of the CQNLSE on the localized external potential namely Gaussian potential. The approximate analytical method, also known as variational method has been applied in order to derive the equations for soliton parameters evolution during the scattering process. The validity of approximations was tested by direct numerical simulations of CQNLSE with soliton initially located far from potential. It was shown, in case of the potential in the form of Gaussian function, that depending on initial velocity of the soliton, the soliton may be reflected by potential or transmitted through it. The critical values of the velocity separating these two scenarios have been identified. [ABSTRACT FROM AUTHOR]

We construct a discrete fundamental solution for the parabolic Dirac operator which factorizes the non-stationary Schrödinger operator. With such fundamental solution we construct a discrete counterpart for the Teodorescu and Cauchy-Bitsadze operators and the Bergman projectors. We finalize this paper with convergence results regarding the operators and a concrete numerical example. [ABSTRACT FROM AUTHOR]

In this paper we study the use of energy-conserving methods, in the class of Hamiltonian Boundary Value Methods, for the numerical solution of the nonlinear Schrodinger equation. [ABSTRACT FROM AUTHOR]

The stability of the rational solutions of the nonlinear Schr odinger (NLS) equation has only recently began to be addressed. In this paper we develop a Chebyshev pseudo-spectral method for the NLS equation to study the stability of the rational solutions. Using the map x = cot θ and the Fast Fourier Transform (FFT) to approximate uxx, the Chebyshev scheme effectively handles the infinite line boundary conditions. An extensive numerical study, involving large ensembles of perturbed initial data for the Peregrine solution (the lowest order rational solution), indicates it is linearly unstable. Working with unstable solutions is numerically challenging. In the current literature, numerical experiments related to the Peregrine solution frequently employ standard Fourier methods without a discussion of the related numerical issues. We examine the performance of a Fourier pseudo-spectral method (FPS4) using Peregrine initial data. Applying FPS4, tiny Gibbs oscillations occur in the first few steps of the numerical solution. These oscillations grow to O(1), providing further evidence of the instability of the Peregrine solution. We modify the FPS4 method using a spectral-splitting technique which resolves the Gibbs oscillations and significantly improves the numerical solution. [ABSTRACT FROM AUTHOR]

In this paper, average vector field method (AVF) is derived for strongly coupled Schrödinger equation (SCNLS). The SCNLS equation is discretized in space by finite differences and is solved in time by structure preserving AVF method. Numerical results for different paremeter compare with the Lobatto IIIA-IIIB method. The results indicate that AVF method are effective to preserve global energy and momentum. [ABSTRACT FROM AUTHOR]

In this paper, a unified formula of a series of breather soliton solutions for the standard (1+1) dimension nonlinear Schrödinger equation is obtained through Exp-function method. Further, by using appropriate transformation, based on solutions obtained, breather soliton solutions of the variable coefficient nonlinear Schrödinger equation are also obtained. One free function of time t and several arbitrary parameters are involved to generate abundant wave structures. [ABSTRACT FROM AUTHOR]

Coupled Nonlinear Schrödinger (CNLS) equation is a second order nonlinear partial differential equation commonly related to nonlinear optical fiber. In this paper, CNLS equation is solved using Finite Difference Method (FDM) and Hybrid Cubic B-Spline collocation method (HCBM) with appropriate initial and boundary conditions. Theta-weighted scheme is applied to the equations and the nonlinear terms are linearized using Taylor series expansion. The temporal space is discretized by forward difference and for the spatial dimensions, central difference is applied for FDM while B-Spline functions are applied for HCBM. The HCBM is shown to be unconditionally stable using von Neumann stability analysis. To test the accuracy, a numerical example is discussed and the error norms are computed. The results obtained show that FDM and HCBM are reliable and easy to implement. [ABSTRACT FROM AUTHOR]

The article surveys the recent results on integrable systems arising from quadratic pencil of Lax operator L, with values in a Hermitian symmetric space. The counterpart operator M in the Lax pair defines positive, negative and rational flows. The results are illustrated with examples from the A.III symmetric space. The modeling aspect of the arising higher order nonlinear Schrödinger equations is briefly discussed. [ABSTRACT FROM AUTHOR]

In this paper the notion of a group of equivalence transformations of the family of differential equations is considered. Moreover invariants of Lie groups are investigated. By using these tools some problems from mathematical physics such as: transformations of potential in the nonlinear Schrödinger equation, neutron diffusion problem and relations in Maxwell and material equations have been studied. [ABSTRACT FROM AUTHOR]

In this paper, Nonlinear Schrödinger (NLS) equation with Neumann boundary conditions is solved using finite difference method (FDM) and cubic B-spline interpolation method (CuBSIM). First, the approach is based on the FDM applied on the time and space discretization with the help of theta-weighted method. However, our main interest is the second approach, whereby FDM is applied on the time discretization and cubic B-spline is utilized as an interpolation function in the space dimension with the same help of theta-weighted method. The CuBSIM is shown to be stable by using von Neumann stability analysis. The proposed method is tested on a test problem with single soliton motion of the NLS equation. The accuracy of the numerical results is measured by the Euclidean-norm and infinity-norm. CuBSIM is found to produce more accurate results than the FDM. [ABSTRACT FROM AUTHOR]

Weakly nonlinear propagation of plane progressive pressure waves of a short wavelength in an initially quiescent liquid uniformly containing many spherical gas bubbles is theoretically studied. We focus on an initial polydispersity of bubble radius (i.e., an initial nonuniform distribution of bubble size); noting that an initial number density is uniform. Previously examined the long-wave case [Ishitsuka & Kanagawa, Proc. Mtgs. Acoust., 39, 045002 (2019)] is then extended to a short-wave case. By using the method of multiple scales, the nonlinear Schrödinger type equation including an advection term with the variable coefficient as the polydisperse effect, is derived from the basic equations. [ABSTRACT FROM AUTHOR]

In this study we apply the Simple Equations Method (SEsM) to obtain exact solution of equations which are connected to the nonlinear Equation of Schrödinger. We use specific case of SEsM which is based on the use of 2 simple equations. By means of the first simple equation we construct the envelope of the solution of the nonlinear Equation of Schrödinger. The second simple equation is connecet to the relationship for the oscillating part of the solution. [ABSTRACT FROM AUTHOR]

We study the Cauchy problem for the Schrödinger equation with a mixed type nonlinear term. We analyze the local and global properties of solutions in dependence of the nonlinear power, the repulsive or attractive character of nonlinear interaction and the size of the initial data. [ABSTRACT FROM AUTHOR]

As is well known, for PDEs that enjoy conservation properties, numerical schemes that inherit the properties are advantageous in that the schemes give qualitatively better solutions in practice. Lately Furihata and Matsuo have developed “the discrete variational method” that automatically constructs conservative finite difference schemes on uniform meshes for a class of PDEs with certain variational structures. We extend this method to mixed meshes and derive a numerical scheme that conserves the energy and the density for the nonlinear Schrödinger equation on such meshes. [ABSTRACT FROM AUTHOR]

Solitons in photorefractive crystals have been known to exist and currently become a trend of research in the modern era. Study on soliton in photorefractive crystals has involved a variety of disciplines, which work complementary among theories, models and experiments. Physics provides a theoretical basis that can be used for experimentation. In compiling this theory, it is very often faced with complicated mathematical problems, especially when solving the equation models of soliton dynamics. In this study, the numerical split-step Fourier method has been successfully applied to the time-dependent model of bright soliton dynamics in photorefractive crystals having both linear and quadratic electro-optical effects. From the numerical calculations, the bright soliton is formed in a steady-state and undergoes evolution until finally stable. Photorefractive crystal are found to have a response time (τ ≈ 2) in forming a soliton. Finally, the electro-optical effect was found to affect the stability of bright solitons. [ABSTRACT FROM AUTHOR]

We discuss the Simple Equations Method (SEsM) for obtaining exact solutions of nonlinear partial differential equations. We show that the Hirota method is particular case of SEsM for specific form of the function from the Step. 2 of SEsM and for simple equations from the kind of differential equation for exponential function. We illustrate the methodology by obtaining the three-soliton solution of the Korteweg - de Vries equation, two soliton solution of the nonlinear Schrödinger equation, and the soliton solution of the Ishimori equation for the spin dynamics of ferromagnetic materials. [ABSTRACT FROM AUTHOR]

We review the derivation as well as the free particle solution of nonlinear master Schrödinger equation. Even though this equation is founded by the Hamilton-Jacobi and continuity equations, the plane-wave solution does still exist for the free particle case. In addition, we will also formulate the gauge invariance form, which describes the electromagnetic interaction. In this case, the Hamilton-Jacobi equation should be modified while the continuity equation remains unchanged. [ABSTRACT FROM AUTHOR]

The split-step modified cubic B-splines collocation (SSM3BC) method is constructed by combining the split-step approach with the modified cubic B-splines collocation method for the two-dimensional (2D) nonlinear Schrödinger (NLS) equation. Numerical tests are carried out, and the SSM3BC schemes are efficient both for the one-dimensional (1D) and 2D NLS equations. It is also numerically verified that the proposed 1D and 2D schemes are second-order convergence both in time and space. [ABSTRACT FROM AUTHOR]

We investigate spatially localized voltage oscillations in a discrete nonlinear electrical transmission line (NETL) considered in Physics Letters A 380 (2016) 2017-2020. According to an appropriate substitution of the judiciously chosen parameters of the system, the modified extended nonlinear Schrodinger (MENLS) obtained leads to a nonlinear equation of particular class with eight-parameter group and three singular straight lines. By using bifurcation theory of planar dynamical systems and investigating the dynamical behavior, we obtain bifurcations of the phase portraits of the system under different parameter conditions. Additionally to some special level curves and the well-known traveling wave solutions, rich varieties of exotic solutions are determined and their implicit expressions are studied. [ABSTRACT FROM AUTHOR]

The present study theoretically deals with a finite amplitude propagation of pressure waves in an initially quiescent liquid containing many small spherical gas bubbles. A fast propagation mode induced by the incorporation of compressibility in the liquid phase is focused. The wave frequency is larger than the eigenfrequency of single bubble oscillations and the wavelength is comparable to the initial bubble radius. Then, a nonlinear Schrödinger (NLS) equation with some correction terms for envelope wave of the carrier wave is derived from a set of averaged equations in a two-fluid model (Egashira et al., Fluid Dyn. Res., 34, 2004) via a general and systematic way based on the asymptotic expansion method of multiple scales (Kanagawa, J. Acoust. Soc. Am., 137, 2015). The NLS equation describes propagation properties in a far field. [ABSTRACT FROM AUTHOR]

The near-conservation of energy on long time intervals in numerical discretizations of Hamiltonian partial differential equations is discussed using the cubic nonlinear Schrödinger equation and its discretization by the split-step Fourier method as a model problem. [ABSTRACT FROM AUTHOR]

The damping effect of the interaction of high-frequency amide-I vibrations with the low-frequency acoustic vibrations of the protein is investigated. The phenomena studied phenomonologically by extension of the nonlinear Schrodinger equation. By introducing a local approximation, the damping factor can be expressed as a new term iγφ in the nonlinear Schrodinger equation. The result show that the soliton with damping propagate slower than original one. By introducing a periodic external force, the equation of motion is described by the force-damped nonlinear Schrodinger equation. Solution based on the variational methods show that the Davidov's soliton will be accelerated by a periodic external force. [ABSTRACT FROM AUTHOR]

We investigate complexes of localized states in two dynamical systems: (i) directly driven nonlinear Schrödinger equation (NLS) and (ii) double sine-Gordon equation (2SG). Our numerical approach is based on the numerical continuation with respect to the control parameters of the stationary solutions and stability analysis by means of the linearized eigenvalue problem. We show that in the weak damping case the directly driven NLS equation holds stable and unstable multi-soliton complexes. We also show that the second harmonic changes properties and increases the complexity of coexisting static fluxons of 2SG equation. [ABSTRACT FROM AUTHOR]

We study the non-integrability of a Hamiltonian system describing the stationary solutions in Bose-Fermi mixtures in one dimensional optical lattices. The proof is based on the Differential Galois approach and Morales-Ramis method. [ABSTRACT FROM AUTHOR]

The Gross-Pitaevskii equation, or more generally the nonlinear Schrödinger equation, models the Bose-Einstein condensates in a macroscopic gaseous superfluid wave-matter state in ultra-cold temperature. We provide analytical study of the NLS with L2 initial data in order to understand propagation of the defocusing and focusing waves for the BEC mechanism in the presence of electromagnetic fields. Numerical simulations are performed for the two-dimensional GPE with anisotropic quadratic potentials. [ABSTRACT FROM AUTHOR]

We developed a method that combines a similarity transformation with a mapping that solves a generalized nonautonomous nonlinear Schrödinger (NLS) equation containing cubic, quintic and higher order terms in the conjunction with spatially and temporary varying dispersion, containing higher nonlinearities, and external potential. We have studied various classes of solutions in closed form representing front, bright and dark solitary-like waves. We introduced a transformation that solves the related transformed NLS in the constant coefficients. As an application of this technique, we have analyzed the dynamical behavior of several specific classes of solutions such as moving, breathing, resonant both for periodic and quasi-periodic solitary-like waves. The stability of the obtained solitary-like waves is examined using analytical and numerical methods. [ABSTRACT FROM AUTHOR]

We analyze a certain class of integral equations associated with Marchenko equations and Gel’fand-Levitan equations. Such integral equations arise through a Fourier transformation on various ordinary differential equations involving a spectral parameter. When the integral operator is perturbed by a finite-rank perturbation, we explicitly evaluate the change in the solution in terms of the unperturbed quantities and the finite-rank perturbation. We show that this result provides a fundamental approach to derive Darboux transformations for various systems of ordinary differential operators. We illustrate our theory by providing the explicit Darboux transformation for the Zakharov-Shabat system and show how the potential and wave function change when a simple discrete eigenvalue is added to the spectrum, and thus we also provide a one-parameter family of Darboux transformations for the nonlinear Schrödinger equation. [ABSTRACT FROM AUTHOR]

Two discrete nonlinear Schrödinger chains are investigated for both linear and nonlinear coupling. The scattering of solitons from point defects in these systems is studied numerically. The soliton dynamics depends on the interchain coupling, the soliton parameters (velocity and width) and the defect strength. It is obtained that solitons which are excited in one of the two chains can be perfectly switched and at the same time transmitted, trapped or reflected by the impurity. [ABSTRACT FROM AUTHOR]

We propose a method for the construction of potentials and nonlinearities, for which the Schrödinger equation is invariant with respect to a group transformations with discrete parameters. This is a generalization of the known Lie symmetry of differential equations. [ABSTRACT FROM AUTHOR]

The results of recently developed investigations, that have been carried out within the framework of the controlling potential method (CPM), are reviewed. This method allows one to decompose a three dimensional (3D) Gross-Pitaevskii equation (GPE) into the pair of coupled Schrödinger-type equations. Under suitable mathematical conditions, the solutions of the 3D controlled GPE can be constructed from the solutions of a 2D linear Schrödinger equation (the transverse component of the GPE) coupled with a 1D nonlinear Schrödinger equation (the longitudinal component of the GPE). Such decomposition allows one to cast the solutions in the form of the product of the solutions of the transverse and the longitudinal components of the GPE. The coupling between these two equations is the functional of both the transverse and the longitudinal profiles. It is shown that the CPM can be used to obtain a new class of three-dimensional solitary waves solutions of the GPE, which governs the dynamics of Bose-Einstein condensates. By imposing an external controlling potential, the desired time-dependent shape of the localized BECs is obtained. The stability of the exact solutions was checked with direct simulations of the time -dependent, three-dimensional GPE. Our simulations show that the localized condensates are stable with respect to perturbed initial conditions. [ABSTRACT FROM AUTHOR]

Weak-nonlinear acoustic pulse propagation in a waveguide channel with longitudinal inhomogeneity is investigated. Euler equations are reduced to a nonlinear wave equation with quadratic nonlinearity, and it is stated that the medium is self-focusing if the adiabatic exponent is greater than 3/2 and defocusing otherwise. The propagation is shown to be characterized with three scales. Nonlinear Schrödinger equation with variable coefficients is obtained for the pulse envelope and a “dark” soliton is presented for a special type of longitudinal inhomogeneity. [ABSTRACT FROM AUTHOR]

The long distance dynamics of modulational instability in optical fibers is studied. A perturbed nonlinear Schrödinger equation which includes higher order dispersive and nonlinear terms as well as damping is investigated. When damping is significant, amplifiers are used to compensate for the fiber losses. Periodically modulated cw waves are shown to evolve chaotically for a variety of physical parameters. A nonlinear spectral analysis of the data shows that the chaotic evolution is characterized by homoclinic transitions in the spectrum. © 2005 American Institute of Physics [ABSTRACT FROM AUTHOR]