A novel approach for solving the Fisher equation using Exp-function method

In this Letter, Exp-function method is employed to obtain traveling wave solutions of the Fisher equation. It is shown that, on this example, the Exp-function method is easy to implement and concise method for nonlinear evolution equations in mathematical physics.     

Solitons and periodic solutions for the fifth-order KdV equation with the Exp-function method

In this Letter the Exp-function method is applied to obtain new generalized solitonary solutions and periodic solutions of the fifth-order KdV equation. It is shown that the Exp-function method, with the help of symbolic computation, provides a powerful mathematical tool for solving nonlinear equations arising 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.     

The variational iteration method for exact solutions of Laplace equation

In this work, we introduce a framework for analytic treatment of Laplace equation with Dirichlet and Neumann boundary conditions. Exact solutions are developed by using the He's variational iteration method (VIM). The work confirms the power of the method in reducing the size of calculations.     

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.     

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.     

Integrability of a general Gross-Pitaevskii equation and exact solitonic solutions of a Bose-Einstein condensate in a periodic potential

We consider a general form of the Gross-Pitaevskii equation with time- and space-dependent effective mass, external potential and strength of interatomic interaction. Using the inverse-scattering method, we derive the integrability condition of this equation within a general scheme that can be used to find exact solutions of a wide range of linear and nonlinear partial differential equations. We use this condition to derive exact solitonic solutions of the one-dimensional time-dependent Gross-Pitaevskii equation corresponding to a Bose-Einstein condensate trapped by a periodic potential. Both attractive and repulsive interatomic interactions are considered. The values of the parameters of the potential can be chosen such that the periodic potential becomes almost identical to that of an optical lattice.     

New interaction solutions to the KdV equation

In this Letter, a few new types of interaction solutions to the KdV equation are obtained through a constructed Wronskian form expansion method. The method takes advantage of the forms and structures of Wronskian solutions to the KdV equation, and the functions used in the Wronskian determinants don't satisfy the systems of linear partial differential equations.     

Painlevé analysis and exact solutions of the resonant Davey-Stewartson system

It is shown that the resonant Davey-Stewartson (RDS) system can pass the Painlev test. By truncating the Laurent series to a constant level term, a dependent variable transformation is naturally derived, which leads to the bilinear forms of the RDS system. From the bilinear equations, through making suitable assumptions, some new soliton solutions are obtained. Some representative profiles of the solitary waves are graphically displayed including the two-line soliton solution, “Y” soliton solution, “V” soliton solution, solitoff, etc. The solutions might be useful to describe the nonlinear phenomena in Madelung fluids, capillarity fluids, and so on.     

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.     

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.     

Decomposition of the generalized KP, cKP and mKP and their exact solutions

The generalized (2+1)-dimensional KP, cKP and mKP are decomposed into the known (1+1)-dimensional soliton equations. Then, we show that the (1+1)-dimensional soliton equations give rise to the explicit soliton solutions of the generalized KP, cKP and mKP.     

Chirped and chirpfree soliton solutions of generalized nonlinear Schrödinger equation with distributed coefficients

Using the F-expansion method, we systematically present exact solutions of the generalized nonlinear nonlinear Schrödinger equation with varying intermodal dispersion and nonlinear gain or loss. This approach allows us to obtain large variety of solutions in terms of Jacobi-elliptical and Weierstrass-elliptical functions. The chirped and unchirped spatiotemporal soliton solutions and trigonometric-function solutions have been also obtained as limiting cases. The dynamics of these spatiotemporal soliton is discussed in context of optical fiber communication. To visualize the propagation characteristics of chirp and unchirped dark-bright soliton solutions, few numerical simulations are given. It is found that wave profile of solitons depend on the group velocity dispersion and the gain or loss functions.     

Deltons, peakons and other traveling-wave solutions of a Camassa-Holm hierarchy

In this letter, we study an integrable Camassa-Holm hierarchy whose high-frequency limit is the Camassa-Holm equation. Phase plane analysis is employed to investigate bounded traveling wave solutions. An important feature is that there exists a singular line on the phase plane. By considering the properties of the equilibrium points and the relative position of the singular line, we find that there are in total three types of phase planes. Those paths in phase planes which represented bounded solutions are discussed one-by-one. Besides solitary, peaked and periodic waves, the equations are shown to admit a new type of traveling waves, which concentrate all their energy in one point, and we name them deltons as they can be expressed as some constant multiplied by a delta function. There also exists a type of traveling waves we name periodic deltons, which concentrate their energy in periodic points. The explicit expressions for them and all the other traveling waves are given.     

New positon, negaton and complexiton solutions for the Bogoyavlensky-Konoplechenko equation

New positon, negaton and complexiton solutions for the Bogoyavlensky-Konoplechenko equation are constructed by means of the Darboux transformation with constant seed solution. The new positon, negaton and complexiton solutions are analytical or singular and given out both analytically and graphically.     

Multi-order exact solutions to the Drinfel'd-Sokolov-Wilson equations

In this Letter, based on the Lamé function and Jacobi elliptic function, the perturbation method is applied to the classical Drinfel'd-Sokolov-Wilson (hereafter DSW for short) equations, and many multi-order solutions are derived. It is shown that different Lamé functions can exist in the first order solutions of DSW system.     

Bäcklund transformation, nonlinear superposition formula and solutions of the Calogero equation

In this Letter, the Bäcklund transformation for the (2+1)-Calogero equation is presented in the bilinear form. Furthermore, a nonlinear superposition formula related to the transformation is proved rigorously. By the way, the Wronskian determinant solution is also derived and verified completely.