Solitary Wave Solutions of Triple Sine-Gordon Equation

Through a mapping relation between the triple sine-Gordon and Ф5 model, we discuss the equation in detail. A series of solitary wave solutions for the triple sine-Gordon equation is obtained by the existing ones of the Ф5 model. Some detailed solitary wave solutions are given for some special coefficients which have significance in physics.     

New Exact Travelling Wave Solutions to Kundu Equation

Based on a first-order nonlinear ordinary differential equation with Six-degree nonlinear term, we first present a new auxiliary equation expansion method and its algorithm. Being concise and straightforward, the method is applied to the Kundu equation. As a result, some new exact travelling wave solutions are obtained, which include bright and dark solitary wave solutions, triangular periodic wave solutions, and singular solutions. This algorithm can also be applied to other nonlinear evolution equations in mathematical physics.     

Exact Travelling Solutions of Discrete sine-Gordon Equation via Extended Tanh-Function Approach

In this paper, we generalize the extended tanh-function approach, which was used to find new exact travelling wave solutions of nonlinear partial differential equations or coupled nonlinear partial differential equations, to nonlinear differential-difference equations. As illustration, two series of exact travelling wave solutions of the discrete sine-Gordon equation are obtained by means of the extended tanh-function approach.     

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.     

New Types of Travelling Wave Solutions From （2＋l）-Dimensional Davey-Stewartson Equation

In this paper, based on new auxiliary nonlinear ordinary differential equation with a sixtb-aegree nonnneal term, we study the （2＋l ）-dimensional Davey-Stewartson equation and new types of travelling wave solutions are obtained, which include new bell and kink profile solitary wave solutions, triangular periodic wave solutions, and singular solutions. The method used here can be also extended to many other nonlinear partial differential equations.     

Elimination of Spiral Waves and Competition between Travelling Wave Impulses and Spiral Waves

The interaction between travelling wave impulses and spiral waves is studied and the results of their competition are related to the exciting period. From the results, it is known that the formation and development of spiral waves in cardiac tissue depend on the period by which the travelling wave impulses are excited. A method is proposed to eliminate spiral waves, which is easily operated.     

Radio-Frequency Characteristics of the Coaxial Step-Disk-Loaded Slow-Wave Structure for Relativistic Travelling Wave Tubes

We present a new periodic all-metal slow wave structure, a coaxial step-disc-loaded system and the dispersion characteristics of the structure. By using the field-matching method, the dispersion equation and the coupling impedance of this structure are obtained. The coaxial structure makes the bandwidth broader than that of the non-coaxial one. Compared with the coaxial disc-loaded and ridged-disc-loaded structures, the pass-band of coaxial step-disc-loaded structure is the broadest. The calculation results show that increasing the step width and decreasing the step thickness can improve the bandwidth.     

New Exact Envelope Traveling Wave Solutions of High-Order Dispersive Cubic-Quintic Nonlinear SchrSdinger Equation

Using trial equation method, abundant exact envelope traveling wave solutions of high-order dispersive cubic-quintic nonlinear Schr6dinger equation, which include envelope soliton solutions, triangular function envelope solutions, and Jacobian elliptic function envelope solutions, are obtained. To our knowledge, all of these results are new. In particular, our proposed method is very simple and can be applied to a lot of similar equations.     

New Variable Separated Solutions and Ghoston Structure for the （2＋1）-Dimensional Sine-Gordon System

The Baecklund transformation and variable separation approach are developed for sine-Gordon systems. Three new types of variable separated solutions with some arbitrary functions have been obtained. A new kind of ghoston structure, which is invisible at most of time and can only be detected when it meets with a foldon, is found. This new structure shows a novel interesting and mysterious phenomenon.     

Nonsingular Travelling Complexiton Solutions to a Coupled Korteweg--de Vries Equation

A class of novel nonsingular travelling complexiton solutions to a coupled Korteweg-de Vries （KdV） equation is presented via the first step Darboux transformation of the complex KdV equation with nonzero seed solution. Furthermore, the properties of the nonsingular solutions are discussed.     

Adaptive Function Projective Synchronization of Discrete-Time Chaotic Systems

By backstepping control law and the active control method, adaptive function projective synchronization of 2D and 3D discrete-time chaotic systems with Uncertain parameters are investigated. To illustrate the effectiveness of the new scheme, some numerical examples are given.     

Grammian Solutions to a Variable-Coefficient KP Equation

Solutions in the Grammian form for a variable-coefficient Kadomtsev-Petviashvili （KP） equation which has the Wronskian solutions are derived by means of Pfaffian derivative formulae.     

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.     

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.     

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.     

Periodic Homoclinic Wave of （1＋1）-Dimensional Long-Short Wave Equation

The exact periodic homoclinic wave of （1＋1）D long-short wave equation is obtained using an extended homoclinic test technique. This result shows complexity and variety of dynamical behaviour for a （1＋1）-dimensional long-short wave equation.     

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.     

Multi-Soliton Solutions of the Levi Equations

The multisoliton solutions of the Levi equations are derived with the Hirota method and Wronskian technique respectively.