Lie Algebraic Structures and Integrability of Long-Short Wave Equation in （2＋1）Dimensions

The hidden symmetry and integrability of the long-short wave equation in (2+1) dimensions are considered using the prolongation approach. The internal algebraic structures and their linear spectra are derived in detail which show that the equation is integrable.

Supersymmetric Sawada-Kotera-Ramani Equation: Bilinear Approach

In this paper, using the Hirota＇s bilineax method, we consider the N = 1 supersymmetric Sawada-Kotera- Ramani equation and obtain the Bazcklund transformation of it. Its one- and two-supersoliton solutions axe obtained and N-supersoliton solutions for N ≥ 3 are given under the condition kiξj = kjξi.

Multiple Periodic-Soliton Solutions for(3+1)-Dimensional Jimbo-Miwa Equation

2N line-soliton solutions of the (3+1)-dimensional Jimbo-Miwa equation can be presented by resorting tothe Hirota bilinear method.In this paper,N periodic-soliton solutions of the (3+1)-dimensional Jimbo-Miwa equationare obtained from the 2N line-soliton solutions by selecting the parameters into conjugated complex parameters in pairs.

Higher-Dimensional KdV Equations and Their Soliton Solutions

A （2＋1）-dimensional KdV equation is obtained by use of Hirota method, which possesses N-soliton solution, specially its exact two-soliton solution is presented. By employing a proper algebraic transformation and the Riccati equation, a type of hell-shape soliton solutions are produced via regarding the variable in the Riccati equation as the independent variable. Finally, we extend the above （2＋1）-dimensional KdV equation into （3＋1）-dimensional equation, the two-soliton solutions are given.

Soliton Solutions for Nonisospectral AKNS Equation by Hirota＇s Method

Bilinear form of the nonisospectral AKNS equation is given. The N-soliton solutions are obtained through Hirota＇s method.

Solving the KdV Equation Under Bargmann Constraint via Bilinear Approach

In this paper we consider exact solutions to the KdV equation under the Bargmann constraint. Solutions expressed through exponential polynomials and Wronskians are derived by bilinear approach through solving the Lax pair under the Bargmann constraint. It is also shown that the potential u in the stationary Sehrodinger equation can be a summation of squares of wave functions from bilinear point of view.

Quasi-Periodic Waves and Asymptotic Property for Boiti-Leon-Manna-Pempinelli Equation

In this paper,multi-periodic (quasi-periodic) wave solutions are constructed for the Boiti-Leon-Manna-Pempinelli(BLMP) equation by using Hirota bilinear method and Riemann theta function.At the same time,weanalyze in details asymptotic properties of the multi-periodic wave solutions and give their asymptotic relations betweenthe periodic wave solutions and the soliton solutions.

Double-Parameter Solutions of Projective Riccati Equations and Their Applications

The purpose of the present paper is twofold. First, the projective Riccati equations （PREs for short） are resolved by means of a linearized theorem, which was known in the literature. Based on the signs and values of coeffcients of PREs, the solutions with two arbitrary parameters of PREs can be expressed by the hyperbolic functions, the trigonometric functions, and the rational functions respectively, at the same time the relation between the components of each solution to PREs is also implemented. Second, more new travelling wave solutions for some nonlinear PDEs, such as the Burgers equation, the mKdV equation, the NLS^＋ equation, new Hamilton amplitude equation, and so on, are obtained by using Sub-ODE method, in which PREs are taken as the Sub-ODEs. The key idea of this method is that the travelling wave solutions of nonlinear PDE can be expressed by a polynomial in two variables, which are the components of each solution to PREs, provided that the homogeneous balance between the higher order derivatives and nonlinear terms in the equation is considered.

Periodic Wave Solutions for(2+1)-Dimensional Boussinesq Equation and(3+1)-Dimensional Kadomtsev-Petviashvili Equation

For describing various complex nonlinear phenomena in the realistic world,the higher-dimensional nonlinearevolution equations appear more attractive in many fields of physical and engineering sciences.In this paper,by virtueof the Hirota bilinear method and Riemann theta functions,the periodic wave solutions for the(2+1)-dimensionalBoussinesq equation and(3+1)-dimensional Kadomtsev-Petviashvili(KP)equation are obtained.Furthermore,it isshown that the known soliton solutions for the two equations can be reduced from the periodic wave solutions.

Painlevé Property and Complexiton Solutions of a Special Coupled KdV Equation

A special coupled KdV equation is proved to be the Painleve property by the Kruskal＇s simplification of WTC method. In order to search new exact solutions of the coupled KdV equation, Hirota＇s bilinear direct method and the conjugate complex number method of exponential functions are applied to this system. As a result, new analytical eomplexiton and soliton solutions are obtained synchronously in a physical field. Then their structures, time evolution and interaction properties are further discussed graphically.

Extended Complex tanh-Function Method and Exact Solutions to （2＋1）-Dimensional Hirota Equation

In this paper, a new extended complex tanh-function method is presented for constructing traveling wave, non-traveling wave, and coefficient functions＇ soliton-like solutions of nonlinear equations. This method is more powerful than the complex tanh-function method [Chaos, Solitons and Fractals 20 （2004） 1037]. Abundant new solutions o[ （2q-1）-dimensional Hirota equation are obtained by using this method and symbolic computation system Maple.

Heteroclinic Breather-Wave Solutions for Davey-Stewartson Equation

Exact heteroclinic breather-wave solutions for Davey-Stewartson （DSI, DSII） system with periodic boundary condition are constructed using Hirota＇s bilinear form method and generalized ansatz method. The heteroclinic structure of wave is investigated.

Two Types of New Solutions to KdV Equation

It is common knowledge that the soliton solutions u（x, t） defined by the bell-shape form is required to satisfy the following condition lira u（x, t） = u（±∞, t） = 0. However, we think that the above condition can be modified as lim u（x, t） = u（±∞, t）^x→ = c, where c is a constant, which is called as a stationary height of u（x, t） in the present paper.^x→∞ If u（x, t） is a bell-shape solitary solution, then the stationary height of each solitary wave is just c. Under the constraint c = 0, all the solitary waves coming from the N-bell-shape-sollton solutions of the KdV equation are the same-oriented travelling. A new type of N-soliton solution with the bell shape is obtained in the paper, whose stationary height is an arbitrary constant c. Taking c ≥ 0, the resulting solitary wave is bound to be the same-oriented travelling. Otherwise, the resulting solitary wave may travel at the same orientation, and also at the opposite orientation. In addition, another type of singular rational travelling solution to the KdV equation is worked out.

A Modified Homogeneous Balance Method and Its Applications

A modified homogeneous balance method is proposed by improving some key steps in the homogeneousbalance method.Bilinear equations of some nonlinear evolution equations are derived by using the modified homogeneousbalance method.Generalized Boussinesq equation,KP equation,and mKdV equation are chosen as examples to illustrateour method.This approach is also applicable to a large variety of nonlinear evolution equations.

Exact Solutions to a Combined sinh-cosh-Gordon Equation

Based on a transformed Painlev~ property and the variable separated ODE method, a function transfor- mation method is proposed to search for exact solutions of some partial differential equations （PDEs） with hyperbolic or exponential functions. This approach provides a more systematical and convenient handling of the solution process of this kind of nonlinear equations. Its key point is to eradicate the hyperbolic or exponential terms by a transformed Painleve property and reduce the given PDEs to a variable-coefficient the resulting equations by some methods. As an application, are formally derived. ordinary differential equations, then we seek for solutions to exact solutions for the combined sinh-cosh-Gordon equation

Free Surface Flow Generated by Submerged Twin-cylinders in Forward Motion Using a Fully Nonlinear Method

The free surface flow generated by twin-cylinders in forced motion submerged beneath the free surface is studied based on the boundary element method. Two relative locations, namely, horizontal and vertical, are examined for the twin cylinders. In both cases, the twin cylinders are starting from rest and ultimately moving with the same constant speed through an accelerating process. Assuming that the fluid is inviscid and incompressible and the flow to be irrotational, the integral Laplace equation can be discretized based on the boundary element method. Fully-nonlinear boundary conditions are satisfied on the unknown free surface and the moving body surface. The free surface is traced by a Lagrangian technique. Regriding and remeshing are applied, which is crucial to quality of the numerical results. Single circular cylinder and elliptical cylinder are calculated by linear method and fully nonlinear method for accuracy checking and then fully nonlinear method is conducted on the twin cylinder cases, respectively. The generated wave elevation and the resultant force are analysed to discuss the influence of the gap between the two cylinders as well as the water depth. It is found that no matter the kind of distribution, when the moving cylinders are close to each other, they suffer hydrodynamic force with large absolute value in the direction of motion. The trend of force varying with the increase of gap can be clearly seen from numerical analysis. The vertically distributed twin cylinders seem to attract with each other while the horizontally distributed twin cylinders are opposite when they are close to each other.

On Differential Equations Describing 3-Dimensional Hyperbolic Spaces

In this paper, we introduce the notion of a （2＋1）-dimenslonal differential equation describing three-dimensional hyperbolic spaces （3-h.s.）. The （2＋1）-dimensional coupled nonlinear Schrodinger equation and its sister equation, the （2＋1）-dimensional coupled derivative nonlinear Schrodinger equation, are shown to describe 3-h.s, The （2 ＋ 1 ）-dimensional generalized HF model：St=（1/2i[S,Sy]＋2iσS）x,σx=-1/4i tr（SSxSy）, in which S ∈ GLc（2）/GLc（1）×GLc（1）,provides another example of （2＋1）-dimensional differential equations describing 3-h.s. As a direct con-sequence, the geometric construction of an infinire number of conservation lairs of such equations is illustrated. Furthermore we display a new infinite number of conservation lairs of the （2＋1）-dimensional nonlinear Schrodinger equation and the （2＋1）-dimensional derivative nonlinear Schrodinger equation by a geometric way.

Two-grid method for characteristic mixed finite-element solutions of nonlinear convection-diffusion equations

A two-grid method for solving nonlinear convection-dominated diffusion equations is presented. The method use discretizations based on a characteristic mixed finite-element method and give the linearization for nonlinear systems by two steps. The error analysis shows that the two-grid scheme combined with the characteristic mixed finite-element method can decrease numerical oscillation caused by dominated convections and solve nonlinear advection-dominated diffusion problems efficiently.

Analysis to Some Solutions Obtained by Modified Extended tanh-Function Method

First, two tanh-coth type solutions of a class of nonlinear wave equation are derived by using a simplified modified extended tanh-function method. Then, further analysis to some tanh-coth type solutions of nonlinear evolution equations are given. The results show that when balance number m is one or two, the tanh-eoth type solutions obtained by the modified extended tanh-funetion method can be obtained by using the hyperbolic-function method.

HBFTrans2: A Maple Package to Construct Hirota Bilinear Form for Nonlinear Equations

An improved algorithm for symbolic computation of Hirota bilinear form of nonlinear equations by a logarithm transformation is presented. The improved algorithm is more efficient by using the property of Hirota-D operator. The software package HBFTrans2 is written in Maple and its running efficiency is tested by a variety of soliton equations.