Stationary solutions for nonlinear dispersive Schrödinger’s equation

This paper carries out the integration of the nonlinear dispersive Schrödinger’s equation by the aid of Lie group analysis. The stationary solutions are obtained. The two types of nonlinearity that are studied in this paper are power law and dual-power law so that the cases of Kerr law and parabolic law nonlinearity fall out as special cases.     

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,...     

Dynamics of superregular breathers in the quintic nonlinear Schrödinger equation

In this paper, we consider an extended nonlinear Schrödinger equation that includes fifth-order dispersion with matching higher-order nonlinear terms. Via the modified Darboux transformation and Joukowsky transform, we present the superregular breather (SRB), multipeak soliton and hybrid solutions. The latter two modes appear as a result of the higher-order effects and are converted from a SRB one, which cannot exist for the standard NLS equation. These solutions reduce to a small localized perturbation of the background at time zero, which is different from the previous analytical solutions. The corresponding state transition conditions are given analytically. The relationship between modulation instability and state transition is unveiled. Our results will enrich the dynamics of nonlinear waves in a higher-order wave system.     

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...     

Dynamics of solitons in the fourth-order nonlocal nonlinear Schrödinger equation

Nonlinear Dynamics - We consider the fourth-order nonlocal nonlinear Schrödinger equation and generate the Lax pair. We then employ Darboux transformation to generate dark and antidark soliton...     

Soliton-like solutions of a derivative nonlinear Schrödinger equation with variable coefficients in inhomogeneous optical fibers

Under investigation in this paper is a derivative nonlinear Schrödinger equation with variable coefficients, which governs the propagation of the subpicosecond soliton pulses in inhomogeneous optical fibers. Through the nonisospectral Kaup–Newell scheme, the Lax pair is constructed with some constraints on the variable coefficients. Under the integrable conditions, bright one- and multi-soliton-like solutions are derived via the Hirota method. By suitably choosing the dispersion coefficient function, several types of inhomogeneous solitons are obtained in, respectively: (1) exponentially decreasing dispersion profile, (2) linearly decreasing dispersion profile, (3) exponentially increasing dispersion profile, and (4) periodically fluctuating dispersion profile. The intensity of the inhomogeneous soliton can be controlled by means of modifying the loss/gain term. Asymptotic analysis of the two-soliton-like solution is performed, which shows that the changes of the widths, amplitudes, and energies before and after the collision are completely caused by the variable coefficients, but have nothing to do with the collision between two soliton-like envelopes. Through suitable choices of variable coefficients, figures are plotted to illustrate the collision behavior between two inhomogeneous solitons, which has some potential applications in the real optical communication systems.     

On the construction of the exact analytic or parametric closed-form solutions of standing waves concerning the cubic nonlinear Schr?dinger equation

We propose a new approach for the construction of the closed-form solutions of standing waves of the cubic nonlinear Schr?dinger equation (NLS). Through appropriate functional transformations, we reduce the radially symmetric NLS into an Emden–Fowler equation whose solution results to the derivation of the closed forms of the standing waves. We also derive the necessary restrictions under which the derived solutions are admissible.     

The bifurcation and exact travelling wave solutions of (1+2)-dimensional nonlinear Schrödinger equation with dual-power law nonlinearity

By using the method of dynamical systems, this paper researches the bifurcation and the exact traveling wave solutions for a (1+2)-dimensional nonlinear Schrödinger equation with dual-power law nonlinearity. Exact parametric representations of all wave solutions are given.     

The use of Volterra series in the analysis of the nonlinear Schrödinger equation

A Volterra series analysis is used to analyse the dispersive behaviour in the frequency domain for the non-linear Schrödinger equation (NLS). It is shown that the solution of the initial value problem for the nonlinear Schrödinger equation admits a local multi-input Volterra series representation. Higher order spatial frequency responses of the nonlinear Schrödinger equation can therefore be defined in a similar manner as for lumped parameter non-linear systems. A systematic procedure is presented to calculate these higher order spatial frequency response functions analytically. The frequency domain behaviour of the equation, subject to Gaussian initial waves, is then investigated to reveal a variety of non-linear phenomena such as self-phase modulation (SPM), cross-phase modulation (CPM), and Raman effects modelled using the NLS.     

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.     

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-...     

Nonlinear supratransmission of multibreathers in discrete nonlinear Schrödinger equation with saturable nonlinearities

We show that the transport of vibrational energy in protein chains modeled by the Discrete Nonlinear Schrödinger equation (DNSE) with saturable nonlinearities can be done through the nonlinear supratransmission phenomenon: we find numerically and semi-analytically threshold amplitudes beyond which the wave propagation takes place within the molecular chains. Subsequently, it is shown that the saturable higher order nonlinearity parameter reduces the supratransmission threshold amplitude. We also prove that the discrete gap multibreathers can be transmitted or supratransmitted according to the frequency belonging to the lower forbidden band gap. More precisely, the discrete gap multibreathers are supratransmitted close to the edge of the lower forbidden band.