Travelling Wave Solutions of Triple Sine-Gordon Equation

Using a complete discrimination system for polynomials and elementary integral method, we obtain the travelling solutions for triple sine-Gordon equation. This method can be applied to similar problems and has general meaning.     

Projective Synchronization in Time-Delayed Chaotic Systems

For the first time, we report on projective synchronization between two time delay chaotic systems with single time delays. It overcomes some limitations of the previous work, where projective synchronization has been investigated only in finite-dimensional chaotic systems, so we can achieve projective synchronization in infinitedimensional chaotic systems. We give a general method with which we can achieve projective synchronization in time-delayed chaotic systems. The method is illustrated using the famous delay-differential equations related to optical bistability. Numerical simulations fully support the analytical approach.     

Modulational instability in the cubic-quintic nonlinear Schrödinger equation through the variational approach

The dynamics of nonlinear pulse propagation in an average dispersion-managed soliton system is governed by a constant coefficient nonlinear Schrödinger (NLS) equation. For a special set of parameters the constant coefficient NLS equation is completely integrable. The same constant coefficient NLS equation is also applicable to optical fiber systems with phase modulation or pulse compression. We also investigate MI arising in the cubic-quintic nonlinear Schrödinger equation for ultrashort pulse propagation. Within this framework, we derive ordinary differential equations (ODE’s) for the time evolution of the amplitude and phase of modulation perturbations. Analyzing the ensuing ODE’s, we derive the classical modulational instability criterion and identify it numerically. We show that the quintic nonlinearity can be essential for the stability of solutions. The evolutions of modulational instability are numerically investigated and the effects of the quintic nonlinearity on the evolutions are examined. Numerical simulations demonstrate the validity of the analytical predictions.     

Adaptive Control and Function Projective Synchronization in 2D Discrete-Time Chaotic Systems

This study addresses the adaptive control and function projective synchronization problems between 2D Rulkov discrete-time system and Network discrete-time system. Based on backstepping design with three controllers, a systematic, concrete and automatic scheme is developed to investigate the function projective synchronization of discretetime chaotic systems. In addition, the adaptive control function is applied to achieve the state synchronization of two discrete-time systems. Numerical results demonstrate the effectiveness of the proposed control scheme.     

Noncontinuous Solitary Wave Solutions for Double sine-Gordon Equation

Using the tanh method and a variable separated ordinary difference equation method to solve the double sineGordon equation, we derive some new exact travelling wave solutions, especially a new type of noncontinuous solitary wave solutions. These noncontinuous solitary wave solutions are verified by using the conservation law theory.     

He's variational iteration method applied to the solution of the prey and predator problem with variable coefficients

In this Letter, a mathematical model of the problem of prey and predator is presented and He's variational iteration method is employed to compute an approximation to the solution of the system of nonlinear differential equations governing the problem. The results are compared with the results obtained by Adomian decomposition method and homotopy perturbation method. Comparison of the methods show that He's variational iteration method is a powerful method for obtaining approximate solutions to nonlinear equations and their systems.     

Exact and explicit solutions of higher-order nonlinear equations of Schrödinger type

New exact solutions for the higher-order nonlinear Schrödinger equation and coupled higher-order nonlinear Schrödinger equations are obtained by using the generalized Jacobi elliptic function method. Solutions in the limiting cases are also studied.     

Dynamics and interaction of pulses in the modified Manakov model

The two-component vector nonlinear Schrödinger equation, with mixed signs of the nonlinear coefficients, is considered. This equation is integrable by the inverse scattering transform method. The evolution of a single pulse and interaction of pulses are studied. It is shown that the dynamics of a single pulse is reduced to the scalar nonlinear Schrödinger equation of focusing or defocusing type, depending on the initial parameters. It is found that the interaction of pulses results in the appearance of additional solitons and bound states of several solitons. The asymptotic field profile in the non-soliton regime is also obtained.     

The variational iteration method for exact solutions of Laplace equation

In this work, we introduce a framework for analytic treatment of Laplace equation with Dirichlet and Neumann boundary conditions. Exact solutions are developed by using the He's variational iteration method (VIM). The work confirms the power of the method in reducing the size of calculations.     

Similarity Reductions and Similarity Solutions of the （3＋1）-Dimensional Kadomtsev-Petviashvili Equation

Employing the compatibility method, we obtain the symmetries of the （3＋1）-dimensional Kadomtsev Petviashvili （KP） equation. Four types of similarity reductions of the KP equation are obtained by solving the corresponding characteristic equations associated with symmetry equations. In addition, a lot of similarity solutions to the KP equation are obtadned.     

Interferometric generation of dark spatial solitons in a photorefractive Bi12TiO20 crystal

We present an interferometric way to generate dark spatial solitons by propagating two coherent beams in a photorefractive Bi₁₂TiO₂₀ crystal under application of an external electric field. Our results shown that the initial width, initial separation, and relative phase between the beams allow to control the type of dark soliton generated.     

A Bilinear Backlund Transformation and Explicit Solutions for a （3＋1）-Dimensional Soliton Equation

Considering the bilinear form of a （3＋1）-dimensional soliton equation, we obtain a bilinear Backlund transformation for the equation. As an application, soliton solution and stationary rational solution for the （3＋1）- dimensional soliton equation are presented.     

Dispersive optical solitons by Kudryashov's method

This paper studies the dynamics of dispersive optical solitions that is modeled by the fourth order nonlinear Schrödinger's equation and Schrödinger–Hirota equation, the latter of which is considered with power law nonlinearity. Kudryashov's method is applied to obtain soliton solutions to the model equations. These results and the solution methodology makes a profound impact in the study of optical solitons.     

Localized Waves in Nonlinear Systems with Spatially Inhomogeneous Nonlinearity

A type of （2＋1）-dimensional nonlinear Schrǒdinger equation with spatially inhomogeneous nonlinearity and an external potential is studied. It is found that special external potentials and spatially nonlinearities can support nonlinear localized waves.     

Eigenvalues of the Zakharov-Shabat scattering problem for two separated sech-shaped pulses

It is shown that the initial condition of two separated sech-shaped in-phase pulses in the nonlinear Schrödinger equation, may give rise to not only stationary solitons, but also symmetrically separating solitons, provided the initial distance of separation is large enough. The critical distance between the pulses for which a separating soliton pair can be found for certain amplitudes is derived using a variational approach.     

Soliton-like solutions of certain types of nonlinear diffusion-reaction equations with variable coefficient

With a view to exploring new soliton-like solutions of certain types of nonlinear diffusion-reaction (DR) equations with a variable coefficient, we demonstrate the viability of a method which is the combination of both the symbolic computation technique of Gao and Tian [Y.T. Gao, B. Tian, Comput. Phys. Commun. 133 (2001) 158] and auxiliary equation method of Sirendaoreji [Sirendaoreji, Phys. Lett. A 356 (2006) 124] and used recently for the KdV equation. In particular, the DR equations with quadratic and cubic nonlinearities with a time-dependent velocity in the convective flux term are studied and the existence of soliton-like solutions is shown.     

A Darboux Transformation for the Coupled Kadomtsev--Petviashvili Equation

The coupled Kadomtsev-Petviashvili equation is considered. It is shown that a Darboux transformation can be constructed by means of an elementary approach.     

N-Soliton Solutions of Non-Isospectral Derivative Nonlinear Schrödinger Equation

Bilinear forms of the non-isospectral derivative nonlinear Schrǒdinger equation are derived. The N-soliton solutions of this equation are obtained by Hirota＇s method.     

Bäcklund transformation, nonlinear superposition formula and solutions of the Calogero equation

In this Letter, the Bäcklund transformation for the (2+1)-Calogero equation is presented in the bilinear form. Furthermore, a nonlinear superposition formula related to the transformation is proved rigorously. By the way, the Wronskian determinant solution is also derived and verified completely.     

Chirped and chirpfree soliton solutions of generalized nonlinear Schrödinger equation with distributed coefficients

Using the F-expansion method, we systematically present exact solutions of the generalized nonlinear nonlinear Schrödinger equation with varying intermodal dispersion and nonlinear gain or loss. This approach allows us to obtain large variety of solutions in terms of Jacobi-elliptical and Weierstrass-elliptical functions. The chirped and unchirped spatiotemporal soliton solutions and trigonometric-function solutions have been also obtained as limiting cases. The dynamics of these spatiotemporal soliton is discussed in context of optical fiber communication. To visualize the propagation characteristics of chirp and unchirped dark-bright soliton solutions, few numerical simulations are given. It is found that wave profile of solitons depend on the group velocity dispersion and the gain or loss functions.