A New Approach to Solve Nonlinear Wave Equations

From the nonlinear sine-Gordon equation, new transformations are obtained in this paper, which are applied to propose a new approach to construct exact periodic solutions to nonlinear wave equations. It is shown that more new periodic solutions can be obtained by this new approach, and more shock wave solutions or solitary wave solutions can be got under their limit conditions.     

General Symmetry Approach to Solve Variable—Coefficient Nonlinear Equations

After considering the variable coefficient of a nonlinear equation as a new dependent variable,some special types of variable-coefficient equation can be solved from the corresponding constant-coefficient equations by using the general classical Lie approach.Taking the nonlinear Schrodinger equation as a concrete example,the method is recommended in detail.     

New Criteria on Oscillatory and Asymptotic Behavior of Third-Order Nonlinear Dynamic Equations with Nonlinear Neutral Terms

In the paper, we provide sufficient conditions for the oscillatory and asymptotic behavior of a new type of third-order nonlinear dynamic equations with mixed nonlinear neutral terms. Our theorems not only improve and extend existing theorems in the literature but also provide a new approach as far as the nonlinear neutral terms are concerned. The main results are illustrated by some particular examples.     

A New Generalization of Extended Tanh-Function Method for Solving Nonlinear Evolution Equations

Making use of a new generalized ansatze and a proper transformation, we generalized the extended tanh-function method. Applying the generalized method with the aid of Maple, we consider some nonlinear evolution equations.As a result, we can successfully recover the previously known solitary wave solutions that had been found by the extendedtanh-function method and other more sophisticated methods. More importantly, for some equations, we also obtain othernew and more general solutions at the same time. The results include kink-profile solitary-wave solutions, bell-profilesolitary-wave solutions, periodic wave solutions, rational solutions, singular solutions and new formal solutions.     

Recursion Formulas to Find Solutions of Some Nonlinear Evolution Equations

New recursion formulas are obtained for finding solutions of some nonlinear evolution equations.     

A Direct Algebraic Method in Finding Particular Solutions to Some Nonlinear Evolution Equations

Firstly, a direct algebraic method and a routine way in finding traveling wave solutions to nonlinear evolution equations are explained. And then some new exact solutions for some evolution equations are obtained by using the method.     

A Direct Algebraic Method in Finding Particular Solutions to Some Nonlinear Evolution Equations

Firstly, a direct algebraic method and a routine way in finding traveling wave solutions to nonlinear evolution equations are explained. And then some new exact solutions for some evolution equations are obtained by using the method.     

Variable Separation Approach to Solve Nonlinear Systems

The variable separation approach method is very useful to solving (2 1 )-dimensional integrable systems. But the (1 1)-dimensional and (3 1 )-dimensional nonlinear systems are considered very little. In this letter, we extend this method to (1 1) dimensions by taking the Redekopp system as a simple example and (3 1)-dimensional Burgers system. The exact solutions are much general because they include some arbitrary functions and the form of the (3 1 )-dimensional universal formula obtained from many (2 1 )-dimensional systems is extended.     

Variable Separation Approach to Solve Nonlinear Systems

The variable separation approach method is very useful to solving (2 1)-dimensional integrable systems.But the (1 1)-dimensional and (3 1)-dimensional nonlinear systems are considered very little. In this letter, we extend this method to (1 1) dimensions by taking the Redekopp system as a simp!e example and (3 1)-dimensional Burgers system. The exact solutions are much general because they include some arbitrary functions and the form of the (3 1)-dimensional universal formula obtained from many (2 1)-dimensional systems is extended.     

Elliptic Equation and New Solutions to Nonlinear Wave Equations

The new solutions to elliptic equation are shown, and then the elliptic equation is taken as a transformationand is applied to solve nonlinear wave equations. It is shown that more kinds of solutions are derived, such as periodicsolutions of rational form, solitary wave solutions of rational form, and so on.     

Elliptic Equation and New Solutions to Nonlinear Wave Equations

The new solutions to elliptic equation are shown, and then the elliptic: equation is taken as a transformation and is applied to solve nonlinear wave equations. It is shown that more kinds of solutions are derived, such as periodic solutions of rational form, solitary wave solutions of rational form, and so on.     

Direct Approach to Construct the Periodic Wave Solutions for Two Nonlinear Evolution Equations

With symbolic computation, the Hirota method and Riemann theta function are employed to directly construct the periodic wave solutions for the Hirota-Satsuma equation for shallow water waves and Boiti-Leon-Manna- Pempinelli equation. Then, the corresponding figures of the periodic wave solutions are given. Fhrthermore, it is shown that the known soliton solutions can be reduced from the periodic wave solutions.     

Conformally Invariant Generalization of Einstein Equations and the Causality Principle

The paper presents an improved derivation of the dynamic equations for the conformally invariant generalization of Einstein's equations. The consistency of the variational procedure with the causality principle is studied. The well-posedness of the Cauchy problem in the synchronous coordinate system is proved as applied to the generalized equations. The possibility of generalized equations at finding quantitative relations between observed values is noted.     

Extension of Variable Separable Solutions for Nonlinear Evolution Equations

We give the generalized definitions of variable separable solutions to nonlinear evolution equations, and characterize the relation between the functional separable solution and the derivative-dependent functional separable solution. The new definitions can unify various kinds of variable separable solutions appearing in references. As application, we classify the generalized nonlinear diffusion equations that admit special functional separable solutions and obtain some exact solutions to the resulting equations.     

Applications of Computer Algebra in Solving Nonlinear Evolution Equations

With the use of computer algebra, the method that straightforwardly leads to travelling wave solutions is presented. The compound KdV--Burgers equation and KP--B equation are chosen to illustrate this approach. As a result, their abundant new soliton-like solutions and period form solutions are found.     

Direct Approach to Construct the Periodic Wave Solutions for Two Nonlinear Evolution Equations

With symbolic computation, the Hirota method and Riemann theta function are employed to directly construct the periodic wave solutions for the Hirota-Satsuma equation for shallow water waves and Boiti-Leon-Manna-Pempinelli equation. Then, the corresponding figures of the periodic wave solutions are given. Furthermore, it is shown that the known soliton solutions can be reduced from the periodic wave solutions.     

Trial Equation Method to Nonlinear Evolution Equations with Rank Inhomogeneous:Mathematical Discussions and Its Applications

A trial equation method to nonlinear evolution equation with rank inhomogeneous is given. As applications, the exact traveling wave solutions to some higher-order nonlinear equations such as generalized Boussinesq equation, generalized Pochhammer-Chree equation, KdV-Burgers equation, and KS equation and so on, are obtained. Among these, some results are new. The proposed method is based on the idea of reduction of the order of ODE. Some mathematical details of the proposed method are discussed.     

Extended Riccati Equation Rational Expansion Method and Its Application to Nonlinear Stochastic Evolution Equations

In this work, by means of a new more general ansatz and the symbolic computation system Maple, we extend the Riccati equation rational expansion method [Chaos, Solitons & Fractals 25 (2005) 1019] to uniformly construct a series of stochastic nontravelling wave solutions for nonlinear stochastic evolution equation. To illustrate the effectiveness of our method, we take the stochastic mKdV equation as an example, and successfully construct some new and more general solutions including a series of rational formal nontraveling wave and coefficient functions' soliton-like solutions and trigonometric-like function solutions. The method can also be applied to solve other nonlinear stochastic evolution equation or equations.     

The Peridic Wave Solutions for Two Nonlinear Evolution Equations

By using the F-expansion method proposed recently, the periodic wave solutions expressed by Jacobielliptic functions for two nonlinear evolution equations are derived. In the limit cases, the solitary wave solutions andthe other type of traveling wave solutions for the system are obtained.     

New Rational Form Solutions to Coupled Nonlinear Wave Equations

The new rational form solutions to the elliptic equation are shown, and then these solutions to the elliptic equation are taken as a transformation and applied to solve nonlinear coupled wave equations. It is shown that more novel kinds of solutions are derived, such as periodic solutions of rational form, solitary wave solutions of rational form,and so on.