A differential quadrature algorithm for nonlinear Schrödinger equation

Numerical solutions of a nonlinear Schrödinger equation is obtained using the differential quadrature method based on polynomials for space discretization and Runge–Kutta of order four for time discretization. Five well-known test problems are studied to test the efficiency of the method. For the first two test problems, namely motion of single soliton and interaction of two solitons, numerical results are compared with earlier works. It is shown that results of other test problems agrees the theoretical results. The lowest two conserved quantities and their relative changes are computed for all test examples. In all cases, the differential quadrature Runge–Kutta combination generates numerical results with high accuracy.     

Complex excitations for the derivative nonlinear Schrödinger equation

Nonlinear Dynamics - The Darboux transformation (DT) formulae for the derivative nonlinear Schrödinger (DNLS) equation are expressed in concise forms, from which the multi-solitons, n-periodic...     

Spatial frequency range analysis for the nonlinear Schrödinger equation

An analysis of the spatial frequency ranges for the nonlinear Schrödinger equation (NLS), subject to initial conditions with Gaussian and band-limited spatial frequency spectra, is presented in this paper. The analysis is based on a Volterra series representation of the NLS equation. This study reveals the relationship between the spatial frequency ranges of the solution, along with the evolution of the system, and the spatial frequency ranges of the initial conditions, and extends previous results in linear and nonlinear finite dimensional systems. The analysis also reveals a variety of nonlinear phenomena including self-phase modulation, cross-phase modulation and Raman effects modelled using the NLS equation.     

Some soliton-type analytical solutions and numerical simulation of nonlinear Schrödinger equation

Nonlinear Dynamics - In this article, we study some soliton-type analytical solutions of Schrödinger equation, with their numerical treatment by Galerkin finite element method. First of all,...     

Exact solutions for a perturbed nonlinear Schrödinger equation by using Bäcklund transformations

The Bäcklund transformation from the Riccati form of inverse method is presented for the Perturbed Nonlinear Schrödinger Equation. Consequently, the exact solutions for Perturbed Nonlinear Schrödinger equation can be obtained by the AKNS class. The technique developed relies on the construction of the wave functions which are solutions of the associated AKNS; that is, a linear eigenvalues problem in the form of a system of PDE. Moreover, we construct a new soliton solution from the old one and its wave function.     

Multi-rogue wave solutions for a generalized integrable discrete nonlinear Schrödinger equation with higher-order excitations

Nonlinear Dynamics - In this paper, a variable-coefficient nonlinear Schrödinger equation that describes the optical soliton propagation in dispersion management fiber systems is studied. Two-...     

Localized traveling wave solution for a logarithmic nonlinear Schrödinger equation

We investigate localized traveling wave solutions for a Schrödinger equation with two logarithmic nonlinear terms under no external potential. It is shown that it can have solitary wave type solutions whose envelope profile depends on the two types of nonlinearity. Remarkably, the profile has cutoffs in the coordinate of propagation. We argue also some fundamental properties that discriminate it from power law type nonlinear Schrödinger equations.     

The bound-state soliton solutions of a higher-order nonlinear Schrödinger equation for inhomogeneous Heisenberg ferromagnetic system

Nonlinear Dynamics - The inverse scattering of a higher-order nonlinear Schrödinger equation for inhomogeneous Heisenberg ferromagnetic system with zero boundary condition is calculated by an...     

Modulation instability,conservation laws and soliton solutions for an inhomogeneous discrete nonlinear Schrödinger equation

In this paper, an inhomogeneous discrete nonlinear Schrödinger equation is analytically investigated. The modulation instability condition and conservation laws are derived. By virtue of the discrete Darboux transformation, two types of explicit solutions on the vanishing and non-vanishing backgrounds are generated. Those results might be useful in the study of solitons propagation in discrete optical fibers.     

Two-breather solutions for the class I infinitely extended nonlinear Schrödinger equation and their special cases

Nonlinear Dynamics - We derive the two-breather solution of the class I infinitely extended nonlinear Schrödinger equation. We present a general form of this multi-parameter solution that...     

Stability of exact solutions of the cubic-quintic nonlinear Schrödinger equation with periodic potential

The nonlinear Schr?dinger equation with attractive quintic nonlinearity in periodic potential in 1D, modeling a dilute-gas Bose–Einstein condensate in a lattice potential, is considered and one family of exact stationary solutions is discussed. Some of these solutions have an analog neither in the linear Schr?dinger equation nor in the integrable nonlinear Schr?dinger equation. Their stability is examined analytically and numerically.     

Soliton interaction control through dispersion and nonlinear effects for the fifth-order nonlinear Schrödinger equation

Nonlinear Dynamics - Optical fiber communication has developed rapidly because of the needs of the information age. Here, the variable coefficients fifth-order nonlinear Schrödinger equation...     

Dynamic behaviors of general N-solitons for the nonlocal generalized nonlinear Schrödinger equation

Nonlinear Dynamics - The general N-solitons of nonlocal generalized nonlinear Schrödinger equations with third-order, fourth-order and fifth-order dispersion terms and nonlinear terms (NGNLS)...     

Bright hump solitons for the higher-order nonlinear Schrödinger equation in optical fibers

Under investigation is the higher-order nonlinear Schrödinger equation with the third-order dispersion (TOD), self-steepening (SS) and self-frequency shift, which can be used to describe the propagation and interaction of ultrashort pulses in the subpicosecond or femtosecond regime. Through the introduction of an auxiliary function, bilinear form is derived. Bright one- and two-soliton solutions are obtained with the Hirota method and symbolic computation. From the one-soliton solutions, we present the parametric regions for the existence of single- and double-hump solitons, and find that they are affected by the coefficients of the group velocity dispersion (GVD) and TOD. Besides, propagation of the one single- or double-hump soliton is observed. We analytically obtain the amplitudes for the single- and double-hump solitons, and calculate the interval between the two peaks for the double-hump soliton. Moreover, soliton amplitudes are related to the coefficients of the GVD, TOD and SS, while the interval between the two peaks for the double-hump soliton is dependent on the coefficients of the GVD and TOD. Interactions are seen between the (i) two single-hump solitons, (ii) two double-hump solitons, and (iii) single- and double-hump solitons. Those interactions are proved to be elastic via the asymptotic analysis.