By using the sine-cosine method proposed recently, we give the exact periodic and soliton solutions of the (2 + 1)-dimensional soliton equation in this paper. Many new families of exact traveling wave solutions of the (2 + 1)-dimensional soliton equation are successfully obtained. The computation for the method appears to be easier and faster by general mathematical software.

By using the sine-cosine method proposed recently, we give the exact periodic and soliton solutions of the (2 + 1)-dimensional Bogoyavlenskii equation in this paper. Many new families of exact traveling wave solutions of the (2 + 1)-dimensional Bogoyavlenskii equation are successfully obtained. The computation for the method appears to be easier and faster by general mathematical software.

Two distinct and novel formalisms for deriving exact closed solutions of a class of variable-coefficient differential-difference equations arising from a plate solidification problem are introduced. Thereupon, exact closed traveling wave and similarity solutions to the plate solidification problem are obtained for some special cases of time-varying plate surface temperature.

In this paper, using modified FF-expansion method, we present some explicit formulas of exact traveling wave solutions for the foam drainage equation. A modified FF-expansion method is proposed by taking full advantages of FF-expansion method and Riccati equation in seeking exact solutions of non-linear partial differential equations.

In this paper, the modified exp-function method is used to seek generalized wave solutions of Kudryashov-Sinelshchikov equation. As a result, some new types of exact traveling wave solutions for arbitrary α, β are obtained which include exponential function, hyperbolic function and trigonometric function. The related results are extend. Obtained results clearly indicate the reliability and effi...

The KdV equation arises in the framework of the Boussinesq scaling as a model equation for waves at the surface of an inviscid fluid. Encoded in the KdV model are relations that may be used to reconstruct the velocity field in the fluid below a given surface wave. In this paper, velocity fields associated to exact solutions of the KdV equation are found, and particle trajectories are computed n...

Liu [Liu Rui, Coexistence of multifarious exact nonlinear wave solutions for generalized b-equation, Inter. J. Bifurcation and Chaos, 20, pp. 3193-3208, 2010] investigated the coexistence of multifarious exact traveling wave solutions of the generalized b-equation, and also presented three conjectures and two questions. In this note, by using the method of complete discrimination system for pol...

case II diffusion of a penetrant through a polymer matrix is characterized by constant front speed. Hence, a traveling-wave analysis is appropriate for the model equations. For the previously validated model analyzed here, conditions on the molecular and stress diffusion coefficients are obtained which guarantee the existence of a traveling wave. Conditions are derived under which an interior m...

in this article, the modified exp-function method is used to construct many exact solutions to the nonlineargeneralized k(n,n) and bbm equations with variable coefficients. under different parameter conditions, explicitformulas for some new exact solutions are successfully obtained. the proposed solutions are found to beimportant for the explanation of some practical physical problems.