Variable-coefficient extended mapping method for nonlinear evolution equations

In this Letter, a variable-coefficient extended mapping method is proposed to seek new and more general exact solutions of nonlinear evolution equations. Being concise and straightforward, this method is applied to the mKdV equation with variable coefficients and (2+1)-dimensional Nizhnik-Novikov-Veselov equations. As a result, many new and more general exact solutions are obtained including Jacobi elliptic function solutions, hyperbolic function solutions and trigonometric function solutions. It is shown that the proposed method provides a very effective and powerful mathematical tool for solving a great many nonlinear evolution equations in mathematical physics.     

Application of an extended homogeneous balance method to new exact solutions of nonlinear evolution equations

The multiple soliton solutions of the approximate equations for long water waves and soliton-like solutions for the dispersive long-wave equations in 2+1 dimensions are constructed by using an extended homogeneous balance method. Solitary wave solutions are shown to be a special case of the present results. This method is simple and has a wide-ranging practicability, and can solve a lot of nonlinear partial differential equations.     

Differential transform method for solving linear and non-linear systems of partial differential equations

In this Letter, we propose a reliable algorithm to develop exact and approximate solutions for the linear and non-linear systems of partial differential equations. The approach rest mainly on two-dimensional differential transform method which is one of the approximate methods. The method can easily be applied to many linear and non-linear problems and is capable of reducing the size of computational work. Exact solutions can also be achieved by the known forms of the series solutions. Several illustrative examples are given to demonstrate the effectiveness of the present method.     

Some new solitary and travelling wave solutions of certain nonlinear diffusion-reaction equations using auxiliary equation method

Attempts are made to look for the soliton content in the exact solutions of certain types of nonlinear diffusion-reaction (DR) equations with the quadratic and cubic nonlinearities. Such equations may arise in a variety of contexts in physical problems. In this Letter using the auxiliary equation method, some new solitary and travelling wave solutions of such nonlinear DR equations are obtained in a very general form. Several interesting special cases of these general solutions are also discussed.     

An effective method for finding special solutions of nonlinear differential equations with variable coefficients

There are many interesting methods can be utilized to construct special solutions of nonlinear differential equations with constant coefficients. However, most of these methods are not applicable to nonlinear differential equations with variable coefficients. A new method is presented in this Letter, which can be used to find special solutions of nonlinear differential equations with variable coefficients. This method is based on seeking appropriate Bernoulli equation corresponding to the equation studied. Many well-known equations are chosen to illustrate the application of this method.     

A new auxiliary equation method and its application to the Sharma-Tasso-Olver model

A new auxiliary equation method, constructed by a first order nonlinear ordinary differential equation with at most an eighth-degree nonlinear term, is first proposed for exploring more exact solutions to nonlinear evolution equations. Being concise and straightforward, the method, with the aid of symbolic computation, is applied to the Sharma-Tasso-Olver model, and some new exact solitary wave solutions are obtained. The approach is also applicable to searches for exact solutions of other nonlinear evolution equations.     

Exp-function method for N-soliton solutions of nonlinear evolution equations in mathematical physics

In this Letter, the Exp-function method is generalized to construct N-soliton solutions of a KdV equation with variable coefficients. As a result, 1-soliton, 2-soliton and 3-soliton solutions are obtained, from which the uniform formula of N-soliton solutions is derived. It is shown that the Exp-function method may provide us with a straightforward and effective mathematical tool for generating N-soliton solutions of nonlinear evolution equations in mathematical physics.     

A direct method to find solutions of some type of coupled Korteweg-de Vries equations using hyperelliptic functions of genus two

We suggest how one can obtain exact solutions of some type of coupled Korteweg-de Vries equations by means of hyperelliptic functions of genus two.     

A new method to construct the integrable coupling system for discrete soliton equation with the Kronecker product

It is shown that the Kronecker product can be applied to construct a new integrable coupling system of discrete soliton equation hierarchy in this Letter. A direct application to the generalized Toda lattice spectral problem leads to a novel integrable coupling system. It is also indicated that the study of integrable couplings by using of the Kronecker product is an efficient and straightforward method.     

A search for the simplest chaotic partial differential equation

A numerical search for the simplest chaotic partial differential equation (PDE) suggests that the Kuramoto-Sivashinsky equation is the simplest chaotic PDE with a quadratic or cubic nonlinearity and periodic boundary conditions. We define the simplicity of an equation, enumerate all autonomous equations with a single quadratic or cubic nonlinearity that are simpler than the Kuramoto-Sivashinsky equation, and then test those equations for chaos, but none appear to be chaotic. However, the search finds several chaotic, ill-posed PDEs; the simplest of these, in the discrete approximation of finitely many, coupled ordinary differential equations (ODEs), is a strikingly simple, chaotic, circulant ODE system.     

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.     

Generalized differential transform method to differential-difference equation

In this Letter, we generalize the differential transform method to solve differential-difference equation for the first time. Two simple but typical examples are applied to illustrate the validity and the great potential of the generalized differential transform method in solving differential-difference equation. A Padé technique is also introduced and combined with GDTM in aim of extending the convergence area of presented series solutions. Comparisons are made between the results of the proposed method and exact solutions. Then we apply the differential transform method to the discrete KdV equation and the discrete mKdV equation, and successfully obtain solitary wave solutions. The results reveal that the proposed method is very effective and simple. We should point out that generalized differential transform method is also easy to be applied to other nonlinear differential-difference equation.     

Application of Exp-function method to Riccati equation and new exact solutions with three arbitrary functions of Broer-Kaup-Kupershmidt equations

In this Letter, the Exp-function method is used to seek generalized solitonary solutions of Riccati equation. Based on the Riccati equation and its generalized solitonary solutions, new exact solutions with three arbitrary functions of the (2+1)-dimensional Broer-Kaup-Kupershmidt equations are obtained. It is shown that the Exp-function method provides a straightforward and important mathematical tool for nonlinear evolution equations in mathematical physics.     

A general approach to specific second order ordinary differential equations using homotopy perturbation method

Homotopy perturbation method is used to solve specific second order ordinary differential equations and tested for different examples. The results obtained demonstrate efficiency of the proposed method.     

A new similarity solution of the Burgers equation with linear damping

The direct method of Clarkson and Kruskal is used to study the symmetry reductions of the Burgers equation with linear damping. The classical similarity reductions reported previously is recovered. The new, nonclassical similarity reduction is obtained. The new similarity solution is given, and it is also obtained by means of the singular manifold method. The nonclassical method is used to demonstrate that the new exact solution is indeed a nonclassical similarity solution. This work has been supported by the Postdoctoral Science Foundation of China.     

Some quasi-periodic solutions to the Kadometsev-Petviashvili and modified Kadometsev-Petviashvili equations

The Kadometsev-Petviashvili (KP) and modified KP (mKP) equations are retrieved from the first two soliton equations of coupled Korteweg-de Vries (cKdV) hierarchy. Based on the nonlinearization of Lax pairs, the KP and mKP equations are ultimately reduced to integrable finite-dimensional Hamiltonian systems in view of the r-matrix theory. Finally, the resulting Hamiltonian flows are linearized in Abel-Jacobi coordinates, such that some specially explicit quasi-periodic solutions to the KP and mKP equations are synchronously given in terms of theta functions through the Jacobi inversion.     

Initial value problem of the Whitham equations for the Camassa-Holm equation

We study the Whitham equations for the Camassa-Holm equation. The equations are neither strictly hyperbolic nor genuinely nonlinear. We are interested in the initial value problem of the Whitham equations. When the initial values are given by a step function, the Whitham solution is self-similar. When the initial values are given by a smooth function, the Whitham solution exists within a cusp in the x-t plane. On the boundary of the cusp, the Whitham solution matches the Burgers solution, which exists outside the cusp.     

The improved sub-ODE method for a generalized KdV-mKdV equation with nonlinear terms of any order

In this Letter, Li and Wang's sub-ODE method [X.Z. Li, M.L. Wang, Phys. Lett. A 361 (2007) 115] is improved and applied to the generalized KdV-mKdV equation with nonlinear terms of any order. As a result, more travelling wave solutions are obtained including not only all the known solutions found by Li and Wang but also other formal solutions. This improved sub-ODE method can be used for solving other nonlinear partial differential equations with nonlinear terms of any order in mathematical physics.     

He's homotopy perturbation method for solving heat-like and wave-like equations with variable coefficients

In this Letter, He's homotopy perturbation method is applied to heat-like and wave-like equations with variable coefficients. The solutions are introduced in this Letter are in recursive sequence forms which can be used to obtain the closed form of the solutions if they are required. The method is tested on various examples which are revealing the effectiveness and the simplicity of the method.     

A simple and direct method for generating travelling wave solutions for nonlinear equations

We propose a simple and direct method for generating travelling wave solutions for nonlinear integrable equations. We illustrate how nontrivial solutions for the KdV, the mKdV and the Boussinesq equations can be obtained from simple solutions of linear equations. We describe how using this method, a soliton solution of the KdV equation can yield soliton solutions for the mKdV as well as the Boussinesq equations. Similarly, starting with cnoidal solutions of the KdV equation, we can obtain the corresponding solutions for the mKdV as well as the Boussinesq equations. Simple solutions of linear equations can also lead to cnoidal solutions of nonlinear systems. Finally, we propose and solve some new families of KdV equations and show how soliton solutions are also obtained for the higher order equations of the KdV hierarchy using this method.