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 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.     

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.     

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 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 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.     

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 Solutions for a Nonisospectral and Variable-Coefficient KdV Equation

The bilinear form for a nonisospectral and variable-coefficient KdV equation is obtained and some exact soliton solutions are derived through Hirota method and Wronskian technique. We also derive the bilinear transformation from its Lax pairs and find solutions with the help of the obtained bilinear transformation.     

Exact Travelling Wave Solutions to a Coupled Nonlinear Evolution Equation

By using an improved hyperbola function method, several types of exact travelling wave solutions to a coupled nonlinear evolution equation are obtained, which include kink-shaped soliton solutions, bell-shaped soliton solutions, envelop solitary wave solutions, and new solitary waves. The method can be applied to other nonlinear evolution equations in mathematical physics.     

Exact Travelling Wave Solutions to a Coupled Nonlinear Evolution Equation

By using an improved hyperbola function method, several types of exact travelling wave solutions to a coupled nonlinear evolution equation are obtained, which include kink-shaped soliton solutions, bell-shaped soliton solutions, envelop solitary wave solutions, and new solitary waves. The method can be applied to other nonlinear evolution equations in mathematical physics.     

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.     

Exact Solutions of Generalized Burgers-Fisher Equation with Variable Coefficients

The generalized Riccati equation vational expansion method is extended in this paper. Several exact solutions for the generalized Burgers-Fisher equation with variable coefficients are obtained by this method, and some of which are derived for the first time. It is concluded from the results that this approach is simple and efficient even in solving partial differential equations with variable coefficients.     

Exact Solutions of a Fractional-Type Differential-Difference Equation Related to Discrete MKdV Equation

The extended simplest equation method is used to solve exactly a new differential-difference equation of fractional-type, proposed by Narita [J. Math. Anal. Appl. 381 （2011） 963] quite recently, related to the discrete MKdV equation. It is shown that the model supports three types of exact solutions with arbitrary parameters： hyperbolic, trigonometric and rational, which have not been reported before.     

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.     

New Exact Solutions of Multi-component mKdV Equation

In this paper, new Jacobi elliptic function solutions of multi-component mKdV equation are obtained directly in a unified way. When the modulus m→1, those periodic solutions degenerate as the corresponding hyperbolic function solutions. Then, to the three-component mKdV equation, five types of effective solution are presented in detail.     

Solving mKdV-sinh-Gordon equation by a modified variable separated ordinary differential equation method

By introducing a more general auxiliary ordinary differential equation (ODE), a modified variable separated ODE method is developed for solving the mKdV--sinh-Gordon equation. As a result, many explicit and exact solutions including some new formal solutions are successfully picked up for the mKdV--sinh-Gordon equation by this approach.     

Two-Dimensional Rossby Waves: Exact Solutions to Petviashvili Equation

The two-dimensional (2D) nonlinear Rossby waves described by the Petviashvili equation, which has been invoked as an ageostrophic extension of the barotropic quasi-geostrophic potential vorticity equation, can be investigated through the exact periodic-wave solutions for the Petviashvili equation, while the exact analytical periodic-wave solutions to the Petviashvili equation are obtained by using the Jacobi elliptic function expansion method. It is shown that periodic-wave 2D Rossby solutions can be obtained by this method, and in the limit cases, the 2D Rossby soliton solutions are also obtained.