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.     

Dispersive bright,dark and singular optical soliton solutions in conformable fractional optical fiber Schrödinger models and its applications

In this paper, we study three different space-time fractional models of the Schrödinger equation. By using the properties of conformable derivative and fractional complex transform, the bright, dark and singular optical solitons for conformable space–time fractional nonlinear \((1+1)\)-dimensional Schrödinger models are determined.     

Exact explicit solitary wave and periodic wave solutions and their dynamical behaviors for the Schamel–Korteweg–de Vries equation

The Schamel–Korteweg–de Vries equation is investigated by the approach of dynamics.The existences of solitary wave including ω-shape solitary wave and periodic wave are proved via investigating the dynamical behaviors with phase space analyses.The sufficient conditions to guarantee the existences of the above solutions in different regions of the parametric space are given.All possible exact explicit parametric representations of the waves are also presented.Along with the details of the analyses,the analytical results are numerically simulated lastly.     

Auto-Bäcklund transformations and solitary wave solutions for the nonlinear evolution equation

In the present work, according to the concept of extended homogeneous balance method and with help of Maple, we get auto-Bäcklund transformations for a (2 + 1)-dimensional nonlinear evolution equation. Subsequently, by using these auto-Bäcklund transformation, exact explicit solutions of this equation are obtained.     

Periodic and solitary wave solutions of cubic–quintic nonlinear reaction-diffusion equation with variable convection coefficients

Attempts have been made to explore the exact periodic and solitary wave solutions of nonlinear reaction diffusion (RD) equation involving cubic–quintic nonlinearity along with time-dependent convection coefficients. Effect of varying model coefficients on the physical parameters of solitary wave solutions is demonstrated. Depending upon the parametric condition, the periodic, double-kink, bell and antikink-type solutions for cubic–quintic nonlinear reaction-diffusion equation are extracted. Such solutions can be used to explain various biological and physical phenomena.     

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.     

Optical dark and dark-singular soliton solutions of (1+2)-dimensional chiral nonlinear Schrodinger’s equation

In this study, the dynamical analysis of optical dark and singular solitons is carried out for chiral (1+2)-dimensional nonlinear Schrödinger’s equation with the implementation of extended direct algebraic and extended trial equation method independently. The constraint conditions guarantee the perseverance of these soliton solutions. Along with optical dark and singular solitons, these integration techniques yield other wave solutions such as Jacobi elliptic function, rational function, and hyperbolic function solutions as outgrowth.     

On solitary wave solutions of the compound Burgers–Korteweg–de Vries equation

The goal of this note is to construct a class of traveling solitary wave solutions for the compound Burgers–Korteweg–de Vries equation by means of a hyperbolic ansatz. A computational error in a previous work has been clarified.     

Explicit,periodic and dispersive optical soliton solutions to the generalized nonlinear Schrödinger dynamical equation with higher order dispersion and cubic-quintic nonlinear terms

In this article, the nonlinear Schrödinger equation with higher order dispersion and nonlinear terms have been discussed analytically using extended Fan sub-equation method. The results hold numerous traveling wave solutions like optical, bright, dark, explicit, periodic and combined wave solutions with the aid of five parameters that are of key importance in elucidating some physical circumstance.     

Mixed lump-stripe,bright rogue wave-stripe,dark rogue wave-stripe and dark rogue wave solutions of a generalized Kadomtsev–Petviashvili equation in fluid mechanics

Under investigation in this paper is a (3+1)-dimensional generalized Kadomtsev–Petviashvili equation in fluid mechanics. We derive the mixed lump-stripe waves, bright mixed rogue wave-stripe, dark mixed rogue wave-stripe and dark rogue waves solutions by virtue of the symbolic computation. We observe the fission and fusion phenomena between the lump and one-stripe wave through the mixed-stripe wave solutions. Then, we observe that the influence of l₁, l₂, l₃, l₄, l₅, l₆, l₇ and l₈ on the mixed lump-stripe waves, where l₁ and l₂ represent the dispersion and nonlinear effects, l₃, l₆, l₇ and l₈ are the perturbed effects, while l₄ and l₅ stand for the disturbed wave velocities along the transverse spatial coordinates y and z, respectively. We graphically present the interaction between a rogue wave and a pair of stripe waves through the mixed rogue wave-stripe solutions. We derive a dark mixed rogue wave-stripe when l₁ < 0. We study the influence of l₁, l₂, l₃, l₄, l₅, l₆, l₇ and l₈ on the rogue wave and a pair of stripe waves. We present the dark rogue wave with certain parameters and observe that two stripe waves merge into one stripe wave.     

Modified KdV–Zakharov–Kuznetsov dynamical equation in a homogeneous magnetised electron–positron–ion plasma and its dispersive solitary wave solutions

This paper presents a new compound synchronisation scheme among four hyperchaotic memristor system with incommensurate fractional-order derivatives. First a new controller was designed based on adaptive technique to minimise the errors and guarantee compound synchronisation of four fractional-order memristor chaotic systems. According to the suitability of compound synchronisation as a reliable solution for secure communication, we then examined the application of the proposed adaptive compound synchronisation scheme in the presence of noise for secure communication. In addition, the unpredictability and complexity of the drive systems enhance the security of secure communication. The corresponding theoretical analysis and results of simulation validated the effectiveness of the proposed synchronisation scheme using MATLAB.     

Bright and dark soliton solutions of the (3 + 1)-dimensional generalized Kadomtsev–Petviashvili equation and generalized Benjamin equation

In this paper, we obtain the 1-soliton solutions of the (3?+?1)-dimensional generalized Kadomtsev–Petviashvili (gKP) equation and the generalized Benjamin equation. By using two solitary wave ansatz in terms of sech ^p and $\tanh^{p}$ functions, we obtain exact analytical bright and dark soliton solutions for the considered model. These solutions may be useful and desirable for explaining some nonlinear physical phenomena in genuinely nonlinear dynamical systems.     

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.     

Self-similar cnoidal and solitary wave solutions of the (1+1)-dimensional generalized nonlinear Schrödinger equation

With the help of the similarity transformation and the solvable stationary nonlinear Schrödinger equation (NLSE),we obtain exact chirped and chirp-free self-similar cnoidal wave and solitary wave solutions of the generalized NLSE exhibitingspatial inhomogeneity, inhomogeneous nonlinearity and gain or loss at the same time. As an example, we investigate their propagationdynamics in a nonlinear optical system, and present a series of interesting properties of optical waves.     

Dispersive traveling wave solutions to the space–time fractional equal-width dynamical equation and its applications

In this article, some new traveling wave solutions to the space–time fractional equal-width equation are constructed with the help of the extended Fan sub-equation method. A simple transformation is introduced to convert the fractional order partial differential equation into an ordinary differential equation. As a result, the bright, dark, singular and combined wave solitons are observed for different values of two parameters. Moreover, the graphical representations are also depicted.     

Soliton and rogue wave solutions of two-component nonlinear Schr?dinger equation coupled to the Boussinesq equation

The nonlinear Schr?dinger(NLS) equation and Boussinesq equation are two very important integrable equations.They have widely physical applications. In this paper, we investigate a nonlinear system, which is the two-component NLS equation coupled to the Boussinesq equation. We obtain the bright–bright, bright–dark, and dark–dark soliton solutions to the nonlinear system. We discuss the collision between two solitons. We observe that the collision of bright–bright soliton is inelastic and two solitons oscillating periodically can happen in the two parallel-traveling bright–bright or bright–dark soliton solution. The general breather and rogue wave solutions are also given. Our results show again that there are more abundant dynamical properties for multi-component nonlinear systems.