Hamiltonian Structure, Soliton Solution and Conservation Laws for a New Fifth-Order Nonlinear Evolution Equation Which Describes Pseudo-Spherical Surfaces

In this paper, we shall show that the Hamiltonian structure can be defined for any nonlinear evolution equations which describe surfaces of a constant negative curvature, so that the densities of conservation laws can be considered as corresponding Hamiltonians. This paper obtains the soliton solution and conserved quantities of a new fifth-order nonlinear evolution equation by the aid of inverse scattering method.

The homotopic mapping solution for the solitary wave for a generalized nonlinear evolution equation

This paper studies a generalized nonlinear evolution equation. Using the homotopic mapping method, it constructs a corresponding homotopic mapping transform. Selecting a suitable initial approximation and using homotopic mapping, it obtains an approximate solution with an arbitrary degree of accuracy for the solitary wave. From the approximate solution obtained by using the homotopic mapping method, it possesses a good accuracy.

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

Making use of a new generalized ans?tze 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 extended tanh-function method and other more sophisticated methods. More importantly, for some equations, we also obtain other new and more general solutions at the same time. The results include kink-profile solitary-wave solutions, bell-profile solitary-wave solutions, periodic wave solutions, rational solutions, singular solutions and new formal 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 appncations, 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.

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.

TRAVELLING WAVE SOLUTIONS OF NONLINEAR EVOLUTION EQUATIONS BY USING SYMBOLIC COMPUTATION

A Riccati equation involving a parameter and symbolic computation are used to uniformly construct the different forms of travelling wave solutions for nonlinear evolution equations.It is shown that the sign of the parameter can be applied in judging the existence of various forms of travelling wave solutions.An efficiency of this method is demonstrated on some equations,which include Burgers Huxley equation,Caudrey Dodd Gibbon Kawada equation,generalized Benjamin Bona Mahony equation and generalized Fisher equation.

Constructing infinite sequence exact solutions of nonlinear evolution equations

To construct the infinite sequence new exact solutions of nonlinear evolution equations and study the first kind of elliptic function, new solutions and the corresponding B^cklund transformation of the equation are presented. Based on this, the generalized pentavalent KdV equation and the breaking soliton equation are chosen as applicable examples and infinite sequence smooth soliton solutions, infinite sequence peak solitary wave solutions and infinite sequence compact soliton solutions are obtained with the help of symbolic computation system Mathematica. The method is of significance to search for infinite sequence new exact solutions to other nonlinear evolution equations.

Travelling Wave Solutions to a Special Type of Nonlinear Evolution Equation

A unified approach is presented for finding the travelling wave solutions to one kind of nonlinear evolution equation by introducing a concept of 'rank'. The key idea of this method is to make use of the arbitrariness of the manifold in Painlevé analysis. We selected a new expansion variable and thus obtained a rich variety of travelling wave solutions to nonlinear evolution equation, which covered solitary wave solutions, periodic wave solutions, Weierstrass elliptic function solutions, and rational solutions. Three illustrative equations are investigated by this means, and abundant travelling wave solutions are obtained in a systematic way. In addition, some new solutions are firstly reported here.

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.

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 solution.s and trigonometric-like function solutions. The method can also be applied to solve other nonlinear stochastic evolution equation or equations.

A more general form of lump solution,lumpoff,and instanton/rogue wave solutions of a reduced (3+1)-dimensional nonlinear evolution equation

In this manuscript,a reduced(3+1)-dimensional nonlinear evolution equation is studied.We first construct the bilinear formalism of the equation by using the binary Bell polynomials theory,then explore a lump solution to the special case for z=x.Furthermore,a more general form of lump solution of the equation is found which possesses seven arbitrary parameters and four constraint conditions.By cutting the lump by the induced soliton(s),lumpoff and instanton/rogue wave solutions are also constructed by the more general form of lump solution.

TAYLOR EXPANSION METHOD FOR NONLINEAR EVOLUTION EQUATIONS

A new numerical method of integrating the nonlinear evolution equations, namely the Taylor expansion method, was presented. The standard Galerkin method can be viewed as the 0_th order Taylor expansion method; while the nonlinear Galerkin method can be viewed as the 1_st order modified Taylor expansion method. Moreover, the existence of the numerical solution and its convergence rate were proven. Finally, a concrete example, namely, the two_dimensional Navier_Stokes equations with a non slip boundary condition,was provided. The result is that the higher order Taylor expansion method is of the higher convergence rate under some assumptions about the regularity of the solution.

The Relationship between Nonconservative Schemes and Initial Values of Nonlinear Evolution Equations

For the nonconservative schemes of the nonlinear evolution equations, taking the one-dimensional shallow water wave equation as an example, the necessary conditions of computational stability are given. Based on numerical tests, the relationship between the nonlinear computational stability and the construction of difference schemes, as well as the form of initial values, is further discussed. It is proved through both theoretical analysis and numerical tests that if the construction of difference schemes is definite, the computational stability of nonconservative schemes is decided by the form of initial values.

Computational Stability of the Explicit Difference Schemes of the Forced Dissipative Nonlinear Evolution Equations

The computational stability of the explicit difference schemes of the forced dissipative nonlinear evolution equations is analyzed and the computational quasi-stability criterion of explicit difference schemes of the forced dissipative nonlinear atmospheric equations is obtained on account of the concept of computational quasi-stability, Therefore, it provides the new train of thought and theoretical basis for designing computational stable difference scheme of the forced dissipative nonlinear atmospheric equations. Key words Computational quasi-stability - Computational stability - Forced dissipative nonlinear evolution equation - Explicit difference scheme This work was supported by the National Outstanding Youth Scientist Foundation of China (Grant No. 49825109), the Key Innovation Project of Chinese Academy of Sciences (KZCX1-10-07), the National Natural Science Foundation of China (Grant Nos, 49905007 and 49975020) and the Outstanding State Key Laboratory Project (Grant No. 40023001).

Construction of Explicit Quasi-complete Square Conservative Difference Schemes of Forced Dissipative Nonlinear Evolution Equations

Based on the forced dissipetive nonlinear evolution equations for describing the motion of atmosphere and ocean, the computational stability of the explicit difference schemes of the forced dissipotive nonlinear atmospheric and oceanic equations is analyzed and the computationally stable explicit complete square conservative difference schemes are constructed. The theoretical analysis and numerical experiment prove that the explicit complete square conservative difference schemes are computationally stable and deserve to be disseminated.

Generalized Dromion Structures of New（2＋1）—Dimensional Nonlinear Evolution Equation

We derive the generalized dromions of the new (2 + 1)-dimensional nonlinear evolution equation by the arbitrary function presented in the bilinearized linear equations. The rich soliton and dromion structures for this system are released.

The Impacts of Initial Perturbations on the Computational Stability of Nonlinear Evolution Equations

The impacts of initial perturbations on the computational stability of nonlinear evolution equations for non-conservative difference schemes and non-periodic boundary conditions are studied through theoretical analysis and numerical experiments for the case of onedimensional equations.The sensitivity of the difference scheme to initial values is further analyzed.The results show that the computational stability primarily depends on the form of the initial values if the difference scheme and boundary conditions are determined.Thus,the computational stability is sensitive to the initial perturbations.

A Generalized Hirota Ansatz to Obtain Soliton-Like Solutions for a (3+l)-Dimensional Nonlinear Evolution Equation

Instead of the usual Hirota ansatz,i.e.,the functions in bilinear equations being chosen as exponentialtypes,a generalized Hirota ansatz is proposed for a (3+1)-dimensional nonlinear evolution equation.Based on theresulting generalized Hirota ansatz,a family of new explicit solutions for the equation are derived.

Variable Separation for(1+1)-Dimensional Nonlinear Evolution Equations with Mixed Partial Derivatives

We present basic theory of variable separation for (1 + 1)-dimensional nonlinear evolution equations withmixed partial derivatives.As an application,we classify equations u_(xt)=A(u,u_x)u_(xxx)+B(u,u_x) that admits derivative-dependent functional separable solutions (DDFSSs) and illustrate how to construct those DDFSSs with some examples.

Controllability of Nonlinear Neutral Evolution Equations with Nonlocal Conditions

In this paper, we establish sufficient conditions for the controllability of nonlinear neutral evolution equations with nonlocal conditions. The result is obtained by using Krasnoselski-Schaefer type fixed point theorem.