Zhenya Yan

New explicit solitary wave solutions and periodic wave solutions for Whitham–Broer–Kaup equation in shallow water

Abstract Based upon the well-known Riccati equation, a new generalized transformation is presented and applied to solve Whitham–Broer–Kaup (WBK) equation in shallow water. As a result, many explicit exact solutions, which contain new solitary wave solutions, periodic wave solutions and the combined formal solitary wave solutions and periodic wave solutions, are obtained. And variant Boussinesq equation and the system of approximate equation for long water waves, as the special cases of WBK equation, can also obtain the corresponding solitary wave solutions and periodic wave solutions. In addition, with the aid of Mathematica and Wu elimination method to solve a large system of algebraic equations, the course of solving equations can be carried out in computer.

New explicit travelling wave solutions for two new integrable coupled nonlinear evolution equations

Abstract In this Letter, a more powerful method to seek exact travelling wave solutions of nonlinear partial differential equations is presented, which uses the good ideas of the extended-tanh function method and our previous method. The two new integrable coupled potential KdV equation and modified KdV-type equations which were firstly presented by Foursov (J. Math. Phys. 41 (2000) 6173) are chosen to illustrate the method by using symbolic computation such that multiple travelling wave solutions are obtained which contain new kink-like soliton solutions, kink-shaped solitons, bell-shaped solitons, rational solutions and singular solitons that may be useful to explain the “blow-up” phenomena.

New explicit and exact travelling wave solutions for a system of variant Boussinesq equations in mathematical physics

Abstract With the aid of Mathematica, we obtain several types of explicit and exact travelling wave solutions to a system of variant Boussinesq equations by using an improved sine-cosine method and the Wu elimination method. These solutions contain Wangs results and other types of solitary wave solutions and new solutions. The method can also be applied to solve more systems of nonlinear partial differential equations, such as the coupled KdV equations.

New explicit solitary wave solutions for (2 + 1)-dimensional Boussinesq equation and (3 + 1)-dimensional KP equation

Abstract In this Letter, we study (2+1)-dimensional Boussinesq equation and (3+1)-dimensional KP equation by using the new generalized transformation in Homogeneous Balance Method (HBM). As a result, many explicit exact solutions, which contain new solitary wave solutions, periodic wave solutions and the combined formal solitary wave solutions and periodic wave solutions, are obtained.

Explicit and exact traveling wave solutions of Whitham–Broer–Kaup shallow water equations

Abstract In this Letter, four pairs solutions of Whitham–Broer–Kaup (WBK) equations, which contain blow-up solutions and periodic solutions, are obtained by using of hyperbolic function method, Mathematica and Wu elimination method. The method can also be applied to solve more nonlinear partial differential equation or equations.

Exact Solutions for a Family of Variable-Coefficient “Reaction–Duffing” Equations via the Bäcklund Transformation

The homogeneous balance method is extended and applied to a class of variable-coefficient “reaction–duffing” equations, and a Bäcklund transformation (BT) is obtained. Based on the BT, a nonlocal symmetry and several families of exact solutions of this equation are obtained, including soliton solutions that have important physical significance. The Fitzhugh–Nagumo and Chaffee–Infante equations are also considered as special cases.

Weierstrass semi-rational expansion method and new doubly periodic solutions of the generalized Hirota-Satsuma coupled KdV system

In the paper, with the aid of symbolic computation, we investigate the generalized Hirota-Satsuma coupled KdV system via our Weierstrass semi-rational expansion method presented recently using the rational expansion of Weierstrass elliptic function and its first-order derivative. As a consequence, three families of new Weierstrass elliptic function solutions via Weierstrass elliptic function P(Ξ;g2,g3) and its first-order derivative P(Ξ;g2, g3). Moreover, the corresponding new Jacobi elliptic function solutions and solitary wave solutions are also presented, and when Ξ; → ∞, these solitary wave solutions approach to some constants.

Applications of Fractional Exterior Differential in Three-Dimensional Space

A brief survey of fractional calculus and fractional differential forms was firstly given. The fractional exterior transition to curvilinear coordinate at the origin were discussed and the two coordinate transformations for the fractional differentials for three-dimensional Cartesian coordinates to spherical and cylindrical coordinates are obtained, respectively. In particular, for v=m=1, the usual exterior transformations, between the spherical coordinate and Cartesian coordinate, as well as the cylindrical coordinate and Cartesian coordinate, are found respectively, from fractional exterior transformation.