Conservation laws and Darboux transformation for the coupled cubic–quintic nonlinear Schrödinger equations with variable coefficients in nonlinear optics

In this paper, by Darboux transformation and symbolic computation we investigate the coupled cubic–quintic nonlinear Schrödinger equations with variable coefficients, which come from twin-core nonlinear optical fibers and waveguides, describing the effects of quintic nonlinearity on the ultrashort optical pulse propagation in the non-Kerr media. Lax pair of the equations is obtained, and the corresponding Darboux transformation is constructed. One-soliton solutions are derived; some physical quantities such as the amplitude, velocity, width, initial phases, and energy are, respectively, analyzed; and finally an infinite number of conservation laws are also derived. These results might be of some value for the ultrashort optical pulse propagation in the non-Kerr media.     

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.     

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.     

Spatiotemporal vector and scalar solitons of the coupled nonlinear Schrödinger equation with spatially modulated cubic–quintic–septimal nonlinearities

Nonlinear Dynamics - The spatially modulated cubic–quintic–septimal nonlinearities and transverse modulation are introduced to study the impact on a...     

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.     

Modulation instability and rogue waves for the sixth-order nonlinear Schrödinger equation with variable coefficients on a periodic background

Nonlinear Dynamics - In this paper, rogue wave solutions of a sixth-order focusing nonlinear Schrödinger (NLS) equation with variable coefficients are investigated on a periodic background. To...     

Dynamics of various waves in nonlinear Schrödinger equation with stimulated Raman scattering and quintic nonlinearity

Nonlinear Dynamics - In this work, we consider a nonlinear Schrödinger equation with stimulated Raman scattering and quintic nonlinearity, which works as a model for the propagation of...     

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.     

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.     

Vortex solitons of the (3+1)-dimensional spatially modulated cubic–quintic nonlinear Schrödinger equation with the transverse modulation

Vortex solitons in the spatially modulated cubic–quintic nonlinear media are governed by a (3+1)-dimensional cubic–quintic nonlinear Schrödinger equation with spatially modulated nonlinearity and transverse modulation. Via the variable separation principle with the similarity transformation, we derive two families of vortex soliton solutions in the spatially modulated cubic–quintic nonlinear media. For the disappearing and parabolic transverse modulation, vortex solitons with different configurations are constructed. The similar configurations of vortex solitons exist for the same value of \(l-k\) with the topological charge k and degree number l. Moreover, the number of the inner layer structure of vortex solitons getting rid of the package covering layer is related to \((n-1)/2+1\) with the soliton order number n. For the disappearing transverse modulation, there exist phase azimuthal jumps around their cores of vortex solitons with \(2\pi \) phase change in every jump, and any two jumps one after another realize the change in \(\pi \). For the parabolic transverse modulation, all phases of vortex soliton exist k-jump, and every jump realizes the change in \(2\pi /k\); thus, k-jumps totally realize the azimuthal change in \(2\pi \) around their cores.     

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-...