Exact Solutions to Degasperis-Procesi Equation

In this paper, dependent and independent variable transformations are introduced to solve the Degasperis- Procesi equation. It is shown that different kinds of solutions can be obtained to the Degasperis-Procesi equation.     

Exact Solutions to Degasperis-Procesi Equation

In this paper, dependent and independent variable transformations are introduced to solve the Degasperis-Procesi equation. It is shown that different kinds of solutions can be obtained to the Degasperis-Procesi equation.     

Exact Solutions to Short Pulse Equation

In this paper, dependent and independent variable transformations are introduced to solve the short pulse equation. It is shown that different kinds of solutions can be obtained to the short pulse equation.     

Exact Solutions to Triple sine-Gordon Equation

In this paper, a new transformation is introduced to solve triple sine-Gordon equation. It is shown that this intermediate transformation method is powerful to solve complex special type nonlinear evolution equation.     

Exact Periodic-Wave Solutions to Nizhnik-Novikov-Veselov Equation

Exact periodic-wave solutions to the generalized Nizhnik-Novikov-Veselov (NNV) equation are obtained by using the extended Jacobi elliptic-function method, and in the limit case, the solitary wave solution to NNV equation are also obtained.     

New Exact Solutions of Time Fractional Gardner Equation by Using New Version of F-Expansion Method

In this article, we consider analytical solutions of the time fractional derivative Gardner equation by using the new version of F-expansion method. With this proposed method multiple Jacobi elliptic functions are situated in the solution function. As a result, various exact analytical solutions consisting of single and combined Jacobi elliptic functions solutions are obtained.     

Exact Solutions of the Harry-Dym Equation

The aim of this paper is to generate exact travelling wave solutions of the Harry-Dym equation through the methods of Adomian decomposition, He's variational iteration, direct integration, and power series. We show that the two later methods are more successful than the two former to obtain more solutions of the equation.     

Exact Solutions to a Combined sinh-cosh-Gordon Equation

Based on a transformed Painlevé property and the variable separated ODE method, a function transformation 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 Painlevé property and reduce the given PDEs to a variable-coefficient ordinary differential equations, then we seek for solutions to the resulting equations by some methods. As an application, exact solutions for the combined sinh-cosh-Gordon equation are formally derived.     

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     

Exact Solutions to Nonlinear Schroedinger Equation and Higher-Order Nonlinear Schroedinger Equation

We study solutions of the nonlinear Schroedinger equation （NLSE） and higher-order nonlinear Sehroedinger equation （HONLSE） with variable coefficients. By considering all the higher-order effect of HONLSE as a new dependent variable, the NLSE and HONLSE can be changed into one equation. Using the generalized Lie group reduction method （CLGRM）, the abundant solutions of NLSE and HONLSE are obtained.     

New Exact Solutions to the Combined KdV and mKdV Equation

The modified mapping method is developed to obtain new exact solutions to the combined KdV and mKdV equation. The method is applicable to a large variety of nonlinear evolution equations, as long as odd- and even-order derivative terms do not coexist in the equation under consideration.     

Reduction and Exact Solutions of the Monge-Ampere Equation

Abstract

On the basis of a subgroup structure of the Poincaré group P(1, 3) the ansatzes which reduce the Monge–Ampere equation to differential equations with fewer independent variables have been constructed. The corresponding symmetry reduction has been done. By means of the solutions of the reduced equations some classes of exact solutions of the investigated equation have been found.     

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 Solutions of the Matrix KP Equation

The matrix KP equation is a many component extension of the ordinary KP (Kodomtsev-Petvjshvilli) equation. Although the matrix KP equation is very important, however,its explicit exact solutions have not been reported up to now. In this letter we give a method to construct the matrix KP hierarchy and its exact solutions. Then we give here the explicit expressions of the exact solutions of the matrix KP equation with arbitrary soliton number.     

Exact Eternal Solutions of the Boltzmann Equation

We construct two families of self-similar solutions of the Boltzmann equation in an explicit form. They turn out to be eternal and positive. They do not possess finite energy. Asymptotic properties of the solutions are also studied.     

Exact Jacobian Elliptic Function Solutions to sinh-Gordon Equation

In this paper, two transformations are introduced to solve sinh-Gordon equation by using the knowledge of elliptic equation and Jacobian elliptic functions. It is shown that different transformations are required in order to obtain more kinds of solutions to the sinh-Gordon equation.     

Exact Solutions and Invariant Sets to General Reaction-Diffusion Equation

In this paper, we introduce new invariant sets, and the invariant sets and exact solutions to general reactiondiffusion equation are discussed. It is shown that there exist a class of exact solutions to the equations that belong to the invariant sets.     

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 Jacobian Elliptic Function Solutions to sine-Gordon Equation

In this paper, four transformations are introduced to solve single sine-Gordon equation by using the knowledge of elliptic equation and Jacobian elliptic functions. It is shown that different transformations are required in order to obtain more kinds of solutions to the single sine-Gordon equation.     

Exact Jacobian Elliptic Function Solutions to sine-Gordon Equation

In this paper, four transformations are introduced to solve single sine-Gordon equation by using the knowledge of elliptic equation and Jacobian elliptic functions. It is shown that different transformations are required in order to obtain more kinds of solutions to the single sine-Gordon equation.