Soliton solutions and a Bäcklund transformation for a generalized nonlinear Schrödinger equation with variable coefficients from optical fiber communications

Under investigation in this paper is a generalized nonlinear Schrödinger model with variable dispersion, nonlinearity and gain/loss, which could describe the propagation of optical pulse in inhomogeneous fiber systems. By employing the Hirota method, one- and two-soliton solutions are obtained with the aid of symbolic computation. Furthermore, a general formula which denotes multi-soliton solutions is given. Some main properties of the solutions are discussed simultaneously. As one important property of nonlinear evolution equation, the Bäcklund transformation in bilinear form is also constructed, which is helpful on future research and as far as we know is firstly proposed in this paper.     

Exact solutions for fractional DDEs via auxiliary equation method coupled with the fractional complex transform

Dynamical behavior of many nonlinear systems can be described by fractional‐order equations. This study is devoted to fractional differential–difference equations of rational type. Our focus is on the construction of exact solutions by means of the (G'/G)‐expansion method coupled with the so‐called fractional complex transform. The solution procedure is elucidated through two generalized time‐fractional differential–difference equations of rational type. As a result, three types of discrete solutions emerged: hyperbolic, trigonometric, and rational. Copyright © 2016 John Wiley & Sons, Ltd.     

Infinitely many stationary solutions for a simple climate model via a shooting method

In this paper, we study the number of steady solutions of a non‐linear model arising in Climatology. By applying a shooting method we show the existence of infinitely many steady solutions for some values of a parameter (the solar constant). This method allows us to determine how many times a solution attains the critical temperature (?10°C) at which the coalbedo is assumed to be discontinuous. Copyright © 2002 John Wiley & Sons, Ltd.     

A new hyperbolic auxiliary function method and exact solutions of the mBBM equation

We propose a new hyperbolic auxiliary function method in this communication. Applying this, exact traveling wave solutions for the modified Benjamin–Bona–Mahoney are constructed.     

Solving nonlinearly constrained global optimization problem via an auxiliary function method

This paper considers the nonlinearly constrained continuous global minimization problem. Based on the idea of the penalty function method, an auxiliary function, which has approximately the same global minimizers as the original problem, is constructed. An algorithm is developed to minimize the auxiliary function to find an approximate constrained global minimizer of the constrained global minimization problem. The algorithm can escape from the previously converged local minimizers, and can converge to an approximate global minimizer of the problem asymptotically with probability one. Numerical experiments show that it is better than some other well known recent methods for constrained global minimization problems.     

Some new Jacobi elliptic function solutions for the short-pulse equation via a direct symbolic computation method

Applying a direct symbolic computation method combined with variable transformations, some new Jacobi elliptic function solutions are obtained to the short-pulse equation in nonlinear optics. When Jacobi elliptic function modulus m??1 or?0, the travelling wave solutions degenerate to two types of solutions, namely, the loop-like soliton solution and the trigonometric function solution.     

Soliton solutions for the Fitzhugh–Nagumo equation with the homotopy analysis method

An analytic technique, the homotopy analysis method (HAM), is applied to obtain the soliton solution of the Fitzhugh–Nagumo equation. The homotopy analysis method (HAM) is one of the most effective method to obtain the exact solution and provides us with a new way to obtain series solutions of such problems. HAM contains the auxiliary parameter ?, which provides us with a simple way to adjust and control the convergence region of series solution.     

Exact stationary-wave solutions in the standard model of the Kerr-Nonlinear optical fiber with the Bragg grating

By using dynamical system method to the standard model of the Kerr-Nonlinear optical fiber with the Bragg grating, under fixed parameter conditions, all possible exact parametric representations of the bounded stationary wave solutions are obtained from the double sine-Gordon equation.     

Soliton solutions for a negative order non-isospectral AKNS equation

In this paper, the bilinear form of a negative order non-isospectral AKNS equation is given. The soliton solutions are obtained through Hirota’s direct method and the Wronskian technique, respectively.     

A new constructing auxiliary function method for global optimization

A new auxiliary function method based on the idea which executes a two-stage deterministic search for global optimization is proposed. Specifically, a local minimum of the original function is first obtained, and then a stretching function technique is used to modify the objective function with respect to the obtained local minimum. The transformed function stretches the function values higher than the obtained minimum upward while it keeps the ones with lower values unchanged. Next, an auxiliary function is constructed on the stretched function, which always descends in the region where the function values are higher than the obtained minimum, and it has a stationary point in the lower area. We optimize the auxiliary function and use the found stationary point as the starting point to turn to the first step to restart the search. Repeat the procedure until termination. A theoretical analysis is also made. The main feature of the new method is that it relaxes significantly the requirements for the parameters. Numerical experiments on benchmark functions with different dimensions (up to 50) demonstrate that the new algorithm has a more rapid convergence and a higher success rate, and can find the solutions with higher quality, compared with some other existing similar algorithms, which is consistent with the analysis in theory.     

Exact solutions for nonlinear partial differential equations via Exp‐function method

In this article, the Exp‐function method is applied to nonlinear Burgers equation and special fifth‐order partial differential equation. Using this method, we obtain exact solutions for these equations. The method is straightforward and concise, and its applications are promising. This method can be used as an alternative to obtain analytical and approximate solutions of different types of nonlinear differential equations. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

New exact solutions for mCH and mDP equations by auxiliary equation method

With the aid of symbolic computation, auxiliary equation method is introduced to investigate modified forms of Camassa-Holm and Degasperis-Procesi equations. A series of new exact traveling wave solutions, including smooth solitary wave solution, peakons, singular solution, periodic wave solution, Jacobi elliptic solution, are obtained in general form. These new exact solutions will enrich previous results and help us further understand the physical structures of these two nonlinear equations.     

Soliton solutions for a generalized KdV and BBM equations with time-dependent coefficients

A generalized KdV equation with time-dependent coefficients will be studied. The BBM equation with time-dependent coefficients and linear damping term will also be examined. The wave soliton ansatz will be used to obtain soliton solutions for both equations. The conditions of existence of solitons are presented.     

A new rational auxiliary equation method and exact solutions of a generalized Zakharov system

A new rational auxiliary equation method for obtaining exact traveling wave solutions of constant coefficient nonlinear partial differential equations of evolution is proposed. Its effectiveness is evinced by obtaining exact solutions of a generalized Zakharov system, some of which are new. It is shown that the G^′/G and the generalized projective Ricatti expansion methods are special cases of the auxiliary equation method. Further, due the solutions obtained, four other new and practicable rational methods are deduced.     

More solutions of the auxiliary equation to get the solutions for a class of nonlinear partial differential equations

In this paper, a new auxiliary function method is presented for constructing exact travelling wave solutions of nonlinear evolution equations. By the relationships of Jacobi elliptic functions, we get more solutions of the auxiliary equation compared with El-Wakila and Abdou (2006) [22]. So, more new exact travelling wave solutions are obtained for a class of nonlinear partial differential equations.