Wenjun Yuan

New exact traveling wave solutions for the Klein-Gordon-Zakharov equations

Based on the extended hyperbolic functions method, we obtain the multiple exact explicit solutions of the Klein-Gordon-Zakharov equations. The solutions obtained in this paper include (a) the solitary wave solutions of bell-type for u and n, (b) the solitary wave solutions of kink-type for u and bell-type for n, (c) the solitary wave solutions of a compound of the bell-type and the kink-type for u and n, (d) the singular traveling wave solutions, (e) periodic traveling wave solutions of triangle function types, and solitary wave solutions of rational function types. We not only rederive all known solutions of the Klein-Gordon-Zakharov equations in a systematic way but also obtain several entirely new and more general solutions. The variety of structures of the exact solutions of the Klein-Gordon-Zakharov equations is illustrated.

The extended hyperbolic functions method and new exact solutions to the Zakharov equations

Abstract The multiple exact solutions for the nonlinear evolution equations describing the interaction of laser–plasma are developed. The extended hyperbolic function method are employed to reveal these new solutions. The solutions include that of the solitary wave solutions of bell-type for n and E , the solitary wave solutions of kink-type for E and bell-type for n , the solitary wave solutions of a compound of the bell-type and the kink-type for n and E , the singular traveling wave solutions, periodic traveling wave solutions of triangle function types, and solitary wave solutions of rational function types. In addition to re-deriving all known solutions in a systematic way, several new and more general solutions can be obtained by using our method.

Abundant exact and explicit solitary wave and periodic wave solutions to the Sharma-Tasso-Olver equation

Abstract This paper is concerned with a double nonlinear dispersive equation—the Sharma–Tasso–Olver equation. The extended hyperbolic function method is employed to investigate the solitary and periodic travelling waves in this equation. With the aid of Mathematica and Wu-elimination Method, the abundant exact explicit solutions of the nonlinear Sharma–Tasso–Olver equation are derived. The solutions obtained in this paper include (a) solitary wave solutions, (b) the singular travelling wave solutions, and (c) periodic travelling wave solutions of triangle function types. Several entirely new exact solutions to the equation are explicitly obtained, in addition to deriving all known solutions in a systematic way. This work can be regard as an extension to the recent work by Wazwaz.

Bäcklund transformations and abundant exact explicit solutions of the Sharma–Tasso–Olver equation

Abstract By using an extension of the homogeneous balance method and Maple, the Backlund transformations for the Sharma–Tasso–Olver equation are derived. The connections between the Sharma–Tasso–Olver equation and some linear partial differential equations are found. With the aid of the transformations given here and the computer program Maple 12, abundant exact explicit special solutions to the Sharma–Tasso–Olver equation are constructed. In addition to all known solutions re-deriving in a systematic way, several entirely new and more general exact explicit solitary wave solutions can also be obtained. These solutions include (a) the algebraic solitary wave solution of rational function, (b) single-soliton solutions, (c) double-soliton solutions, (d) N-soliton solutions, (e) singular traveling solutions, (f) the periodic wave solutions of trigonometric function type, and (g) many non-traveling solutions. By using the Airy’s function and the Backlund transformations obtained here, the exact explicit solution of the initial value problem for the STO equation is presented. The variety of the structure of the solutions for the Sharma–Tasso–Olver equation is illustrated.

All traveling wave exact solutions of the variant Boussinesq equations

In this article, we employ the complex method to obtain all meromorphic solutions of complex variant Boussinesq equations (1), then find out related traveling wave exact solutions of System (vB). The idea introduced in this paper can be applied to other nonlinear evolution equations. Our results show that all rational and simply periodic solutions w r , 1 ( k x - λ t ) , w r , 2 ( k x - λ t ) , w s , 1 ( k x - λ t ) and w s , 2 ( k x - λ t ) of System (vB) are solitary wave solutions, and there exist some rational solutions wr,2(z) and simply periodic solutions ws,2(z) which are not only new but also not degenerated successively by the elliptic function solutions. We believe that this method should play an important role for finding exact solutions in the mathematical physics. We also give some computer simulations to illustrate our main results.

All exact traveling wave solutions of the combined KdV-mKdV equation

In this article, we employ the complex method to obtain all meromorphic solutions of complex combined Korteweg-de Vries-modified Korteweg-de Vries equation (KdV-mKdV equation) at first, then we find all exact traveling wave solutions of the combined KdV-mKdV equation. The idea introduced in this paper can be applied to other nonlinear evolution equations. Our results show that all rational and simply periodic exact traveling wave solutions of the combined KdV-mKdV equation are solitary wave solutions, the complex method is simpler than other methods, and there exist some rational solutions wr,2(z) and simply periodic solutions ws,2(z) such that they are not only new but also not degenerated successively by the elliptic function solutions. We believe that this method should play an important role in finding exact solutions in mathematical physics. We also give some computer simulations to illustrate our main results.MSC: Primary 30D35; secondary 34A05.PACS Codes: 02.30.Jr; 02.70.Wz; 02.30.-f.

The general traveling wave solutions of the Fisher type equations and some related problems

In this article, we introduce two recent results with respect to the integrality and exact solutions of the Fisher type equations and their applications. We obtain the sufficient and necessary conditions of integrable and general meromorphic solutions of these equations by the complex method. Our results are of the corresponding improvements obtained by many authors. All traveling wave exact solutions of many nonlinear partial differential equations are obtained by making use of our results. Our results show that the complex method provides a powerful mathematical tool for solving a great number of nonlinear partial differential equations in mathematical physics. We will propose four analogue problems and expect that the answer is positive, at last.MSC:30D35, 34A05.

All meromorphic solutions for two forms of odd order algebraic differential equations and its applications

Abstract In this article, we employ the Nevanlinna’s value distribution theory to investigate the existence of meromorphic solutions of algebraic differential equations. We obtain the representations of all meromorphic solutions for two classes of odd order algebraic differential equations with the weak 〈 p , q 〉 and dominant conditions. Moreover, we give the complex method to find all traveling wave exact solutions of corresponding partial differential equations. As an example, we obtain all meromorphic solutions of some generalized Bretherton equations by using our complex method. Our results show that the complex method provides a powerful mathematical tool for solving great many nonlinear partial differential equations in mathematical physics, and using the traveling wave nobody can find other new exact solutions for many nonlinear partial differential equations by any method.

Normal Families and Growth of Meromorphic Functions with Their th Derivatives

Relying on the normal family theory, we mainly study uniqueness problems of meromorphic functions and their th derivatives and estimate sharply the growth order of their meromorphic functions. Our theorems improve some previous results.

Growth of Meromorphic Function Sharing Functions and Some Uniqueness Problems

Estimating the growth of meromorphic solutions has been an important topic of research in complex differential equations. In this paper, we devoted to considering uniqueness problems by estimating the growth of meromorphic functions. Further, some examples are given to show that the conclusions are meaningful.