Self-similar rogue waves in an inhomogeneous generalized nonlinear Schrödinger equation

We present an explicit analytical form of first and second order rogue waves for distributive nonlinear Schrödinger equation (NLSE) by mapping it to standard NLSE through similarity transformation. Upon obtaining the rogue wave solutions, we study the propagation of rogue waves through a periodically distributed system for the two cases when Wronskian of dispersion and nonlinearity is (i) zero, (ii) not equal to zero. For the former case, we discuss a mechanism to control their propagation and for the latter case we depict the interesting features of rogue waves as they propagate through dispersion increasing and decreasing fiber.     

Solitons for a generalized variable-coefficient nonlinear Schrdinger equation

In this paper,we investigate some exact soliton solutions for a generalized variable-coefficients nonlinear Schrdinger equation (NLS) with an arbitrary time-dependent linear potential which describes the dynamics of soliton solutions in quasi-one-dimensional Bose-Einstein condensations. Under some reasonable assumptions,one-soliton and two-soliton solutions are constructed analytically by the Hirota method. From our results,some previous one-and two-soliton solutions for some NLS-type equations can be recovered by some appropriate selection of the various parameters. Some figures are given to demonstrate some properties of the one-and the two-soliton and the discussion about the integrability property and the Hirota method is given finally.     

Modulation of localized solutions in an inhomogeneous saturable nonlinear Schrödinger equation

In this paper we study the modulation of localized solutions by an inhomogeneous saturable nonlinear medium. Throughout an appropriate ansatz we convert the inhomogeneous saturable nonlinear Schrödinger equation in a homogeneous one. Then, via a variational approach we construct localized solutions of the autonomous equation and we present some modulation patterns of this localized structures. We have checked the stability of such solutions through numerically simulations.     

Conservation laws, soliton solutions and modulational instability for the higher-order dispersive nonlinear Schrödinger equation

In this paper, analytically investigated is a higher-order dispersive nonlinear Schrödinger equation. Based on the linear eigenvalue problem associated with this equation, the integrability is identified by admitting an infinite number of conservation laws. By using the Darboux transformation method, the explicit multi-soliton solutions are generated in a recursive manner. The propagation characteristic of solitons and their interactions under the periodic plane wave background are discussed. Finally, the modulational instability of solutions is analyzed in the presence of small perturbation.     

Traveling-waves and solitons in a generalized time-variable coefficients nonlinear Schrödinger equation with higher-order terms

In this work, we study exact solutions of a generalized nonautonomous cubic–quintic nonlinear Schrödinger equation with higher-order terms, and the dispersion and the nonlinear coefficients engendering temporal dependency. Similarity transformations are used to convert the nonautonomous equation into autonomous one and then we present solutions in a general way. These solutions are obtained for the first class by using the F-expansion method and for the second class constituted by most general bright, dark and front by a direct substitution. We also generalize the external potential which traps the system and the nonlinearities. Finally, the stability of the soliton solutions under slight disturbance of the constraint conditions and initial perturbation of white noise is discussed analytically and numerically. The results reveal that solitons can propagate in a stable way under slight disturbance of the constraint conditions and initial perturbation of a 10% white noise.     

Deformed soliton,breather,and rogue wave solutions of an inhomogeneous nonlinear Schrdinger equation

We use the 1-fold Darboux transformation (DT) of an inhomogeneous nonlinear Schro¨dinger equation (INLSE) to construct the deformed-soliton, breather, and rogue wave solutions explicitly. Furthermore, the obtained first-order deformed rogue wave solution, which is derived from the deformed breather solution through the Taylor expansion, is different from the known rogue wave solution of the nonlinear Schro¨dinger equation (NLSE). The effect of inhomogeneity is fully reflected in the variable height of the deformed soliton and the curved background of the deformed breather and rogue wave. By suitably adjusting the physical parameter, we show that a desired shape of the rogue wave can be generated. In particular, the newly constructed rogue wave can be reduced to the corresponding rogue wave of the nonlinear Schro¨dinger equation under a suitable parametric condition.     

Kuznetsov–Ma soliton and Akhmediev breather of higher-order nonlinear Schrdinger equation

In terms of Darboux transformation, we have exactly solved the higher-order nonlinear Schrdinger equation that describes the propagation of ultrashort optical pulses in optical fibers. We discuss the modulation instability(MI) process in detail and find that the higher-order term has no effect on the MI condition. Under different conditions, we obtain Kuznetsov–Ma soliton and Akhmediev breather solutions of higher-order nonlinear Schrdinger equation. The former describes the propagation of a bright pulse on a continuous wave background in the presence of higher-order effects and the soliton's peak position is shifted owing to the presence of a nonvanishing background, while the latter implies the modulation instability process that can be used in practice to produce a train of ultrashort optical soliton pulses.     

Some exact solutions to the inhomogeneous higher-order nonlinear Schrdinger equation by a direct method

By symbolic computation and a direct method, this paper presents some exact analytical solutions of the one-dimensional generalized inhomogeneous higher-order nonlinear Schr?dinger equation with variable coefficients, which include bright solitons, dark solitons, combined solitary wave solutions, dromions, dispersion-managed solitons, etc. The abundant structure of these solutions are shown by some interesting figures with computer simulation.     

Collapse in the nonlinear Schrödinger equation

A new class of exact solutions with a singularity at finite time (collapse) is obtained for the nonlinear Schrödinger equation.     

Optical soliton solutions of the generalized higher-order nonlinear Schrödinger equations and their applications

The propagation of the optical solitons is usually governed by the higher order nonlinear Schrödinger equations (NLSE). In optics, the NLSE modelizes light-wave propagation in an optical fiber. In this article, modified extended direct algebraic method with add of symbolic computation is employed to construct bright soliton, dark soliton, periodic solitary wave and elliptic function solutions of two higher order NLSEs such as the resonant NLSE and NLSE with the dual-power law nonlinearity. Realizing the properties of static and dynamic for these kinds of solutions are very important in various many aspects and have important applications. The obtaining results confirm that the current method is powerful and effectiveness which can be employed to other complex problems that arising in mathematical physics.     

Inverse scattering approach to coupled higher-order nonlinear Schrödinger equation and N-soliton solutions

A generalized inverse scattering method has been developed and applied to the linear problem associated with the coupled higher-order nonlinear Schrödinger equation to obtain it's N-soliton solution. An infinite number of conserved quantities have been obtained by solving a set of coupled Riccati equations. It has been shown that the coupled system admits two different class of solutions, characterized by the number of local maxima of amplitude of the soliton.     

The modify unstable nonlinear Schrödinger dynamical equation and its optical soliton solutions

In this research, we work on a specific class of nonlinear evolution equation which is the modify unstable nonlinear Schrödinger equation. This equation is used to describe a time evolution of disturbances in unstable media. Various solutions have been obtained. The results deduced are of varied types and include bright solution, dark solution, rational dark-bright solution, as well as cnoidal solutions. These solutions might be useful in engineering fields. Some conditions for the stability of these solutions are presented. The method used here is understandable and very powerful for solving the nonlinear problems.     

Bifurcations and solutions for the generalized nonlinear Schrödinger equation

The theory of bifurcations for dynamical system is employed to construct new exact solutions of the generalized nonlinear Schrödinger equation. Firstly, the generalized nonlinear Schrödinger equation was converted into ordinary differential equation system by using traveling wave transform. Then, the system's Hamiltonian, orbits phases diagrams are found. Finally, six families of solutions are constructed by integrating along difference orbits, which consist of Jacobi elliptic function solutions, hyperbolic function solutions, trigonometric function solutions, solitary wave solutions, breaking wave solutions, and kink wave solutions.     

Novel multi-band quantum soliton states for a derivative nonlinear Schrödinger model

We show that localized N-body soliton states exist for a quantum integrable derivative nonlinear Schrödinger model for several nonoverlapping ranges (called bands) of the coupling constant η. The number of such distinct bands is given by Euler's φ-function which appears in the context of number theory. The ranges of η within each band can also be determined completely using concepts from number theory such as Farey sequences and continued fractions. We observe that N-body soliton states appearing within each band can have both positive and negative momentum. Moreover, for all bands lying in the region η>0, soliton states with positive momentum have positive binding energy (called bound states), while the states with negative momentum have negative binding energy (anti-bound states).     

Periodic nonlinear Schrödinger equation and invariant measures

In this paper we continue some investigations on the periodic NLSEiu _u +iu _xx +u|u| ^p-2 (p6) started in [LRS]. We prove that the equation is globally wellposed for a set of data of full normalized Gibbs measrue (after suitableL ²-truncation). The set and the measure are invariant under the flow. The proof of a similar result for the KdV and modified KdV equations is outlined. The main ingredients used are some estimates from [B1] on periodic NLS and KdV type equations.     

Dark two-soliton solutions for nonlinear Schrödinger equations in inhomogeneous optical fibers

In this paper, the variable coefficient nonlinear Schrödinger equation is investigated analytically. With the bilinear method, the bilinear forms and analytic soliton solutions are obtained. Based on the obtained analytic solutions, the effect of free parameters on the control of soliton transmission is studied. Influences of second-order and third-order dispersion coefficients on dark solitons are discussed. Results in this paper would be of great significance in the generation of dark solitons.