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.     

On the Darboux transformation of a generalized inhomogeneous higher-order nonlinear Schrödinger equation

Recently, a paper about the Nth-order rogue waves for an inhomogeneous higher-order nonlinear Schrödinger equation using the generalized Darboux transformation is published. Song et al. (Nonlinear Dyn 82(1):489–500. doi: 10.1007/s11071-015-2170-6, 2015). However, the inhomogeneous equation which admits a nonisospectral linear eigenvalue problem is mistaken for having a constant spectral parameter by the authors. This basic error causes the results to be wrong, especially regarding the Darboux transformation (DT) in Sect. 2 when the inhomogeneous terms are dependent of spatial variable x. In fact, the DT for inhomogeneous equation has an essential difference from the isospectral case, and their results are correct only in the absence of inhomogeneity which was already discussed in detail before. Consequently, we firstly modify the DT based on corresponding nonisospectral linear eigenvalue problem. Then, the nonautonomous solitons are obtained from zero seed solutions. Properties of these solutions in the inhomogeneous media are discussed graphically to illustrate the influences of the variable coefficients. Finally, the failure of finding breather and rogue wave solutions from this modified DT is also discussed.     

Darboux transformation for an integrable generalization of the nonlinear Schr?dinger equation

A Darboux transformation for an integrable generalization of the coupled nonlinear Schr?dinger equation is derived with the help of the gauge transformation between the Lax pair. As a reduction, a Darboux transformation for an integrable generalization of the nonlinear Schr?dinger equation is obtained, from which some new solutions for the integrable generalization of the nonlinear Schr?dinger equation are given.     

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,...     

Evolution of optical solitary waves in a generalized nonlinear Schr?dinger equation with variable coefficients

We derive two types of exact analytical solutions in terms of rational-like functions for a generalized nonlinear Schr?dinger equation with variable coefficients via the methods of similarity transformation and direct ansatz. Based on these solutions, several novel optical solitary waves are constructed by selecting appropriate functions, and the main evolution features of these waves are shown by some interesting figures with computer simulation.     

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.     

Breathers,solitons and rogue waves of the quintic nonlinear Schrödinger equation on various backgrounds

Nonlinear Dynamics - We investigate the generation of breathers, solitons, and rogue waves of the quintic nonlinear Schrödinger equation (QNLSE) on uniform and elliptical backgrounds. The...     

Solitons and rogue waves of the quartic nonlinear Schrödinger equation by Riemann–Hilbert approach

Nonlinear Dynamics - We systematically develop a Riemann–Hilbert approach for the quartic nonlinear Schrödinger equation on the line with both zero boundary condition and nonzero...     

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.     

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...     

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...     

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.     

General high-order rogue waves to nonlinear Schrödinger–Boussinesq equation with the dynamical analysis

Nonlinear Dynamics - General high-order rogue waves of the nonlinear Schrödinger–Boussinesq equation are obtained by the KP-hierarchy reduction theory, and the N-order rogue waves are...