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.

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.

Using a New Auxiliary Equation to Construct Abundant Solutions for Nonlinear Evolution Equations

In this paper, a new auxiliary equation method is proposed. Combined with the mapping method, abundant periodic wave solutions for generalized Klein-Gordon equation and Benjamin equation are obtained. They are new types of periodic wave solutions which are rarely found in previous studies. As m → 0 and m → 1, some new types of trigonometric solutions and solitary solutions are also obtained correspondingly. This method is promising for constructing abundant periodic wave solutions and solitary solutions of nonlinear evolution equations (NLEEs) in mathematical physics.

BOUNDARY LAYER AND VANISHING DIFFUSION LIMIT FOR NONLINEAR EVOLUTION EQUATIONS

In this paper, we consider an initial-boundary value problem for some nonlinear evolution equations with damping and diffusion. The main purpose is to investigate the boundary layer effect and the convergence rates as the diffusion parameter α goes to zero.

STUDY ON EXACT ANALYTICAL SOLUTIONS FOR TWO SYSTEMS OF NONLINEAR EVOLUTION EQUATIONS

The homogeneous balance method was improved and applied to two systems Of nonlinear evolution equations. As a result, several families of exact analytic solutions are derived by some new ansatzs. These solutions contain Wang's and Zhang's results and other new types of analytical solutions, such as rational fraction solutions and periodic solutions. The way can also be applied to solve more nonlinear partial differential equations.

Exact traveling wave solutions for an integrable nonlinear evolution equation given by M.Wadati

By using the method of dynamical systems, the travelling wave solutions of for an integrable nonlinear evolution equation is studied. Exact explicit parametric representations of kink and anti-kink wave, periodic wave solutions and uncountably infinite many smooth solitary wave solutions are given.

The Wronskian technique for nonlinear evolution equations

The investigation of the exact traveling wave solutions to the nonlinear evolution equations plays an important role in the study of nonlinear physical phenomena. To understand the mechanisms of those physical phenomena, it is necessary to explore their solutions and properties. The Wronskian technique is a powerful tool to construct multi-soliton solutions for many nonlinear evolution equations possessing Hirota bilinear forms. In the process of utilizing the Wronskian technique, the main difficulty lies in the construction of a system of linear differential conditions, which is not unique. In this paper, we give a universal method to construct a system of linear differential conditions.

NOTE ON AN OPEN PROBLEM OF HIGHER ORDER NONLINEAR EVOLUTION EQUATIONS

In view of a new idea on initial conditions, an open problem of nonlinear evolution equations with higher order, which was given by J. L. Lions, is solved. Effect of our results is shown on an example.

GLOBAL SOLUTION OF THE INVERSE PROBLEM FOR ACLASS OF NONLINEAR EVOLUTION EQUATIONSOF DISPERSIVE TYPE

The inverse problem for a class of nonlinear evolution equations of dispersive type type was reduced to Cauchy problem of nonlinear evolution equation in an abstract space. By means of the semigroup method and equipping equivalent norm technique, the existence and uniqueness theorem of global solution was obtained for this class of abstract evolution equations, and was applied to the inverse problem discussed here. The existence and uniqueness theorem of global solution it,as given for this class of nonlinear evolution equations of dispersive type. The results extend and generalize essentially the related results of the existence and uniqueness of local solution presented by YUAN Zhong-xin.

Multiple exp-function method for soliton solutions of nonlinear evolution equations

We applied the multiple exp-function scheme to the（2＋1）-dimensional Sawada-Kotera（SK） equation and（3＋1）-dimensional nonlinear evolution equation and analytic particular solutions have been deduced. The analytic particular solutions contain one-soliton, two-soliton, and three-soliton type solutions. With the assistance of Maple, we demonstrated the efficiency and advantages of the procedure that generalizes Hirota＇s perturbation scheme. The obtained solutions can be used as a benchmark for numerical solutions and describe the physical phenomena behind the model.

A NEW APPROACH TO SOLVE PERTURBED NONLINEAR EVOLUTION EQUATIONS THROUGH LIE-B■CKLUND SYMMETRY METHOD

A new method based on Lie-Backlund symmetry method to solve the perturbed nonlinear evolution equations is presented. New approximate solutions of perturbed nonlinear evolution equations stemming from the exact solutions of unperturbed equations are obtained. This method is a generalization of Burde＇s Lie point symmetry technique.

Fermionic Covariant Prolongation Structure for a Super Nonlinear Evolution Equation in 2+1 Dimensions

The integrability of a （2＋1）-dimensional super nonlinear evolution equation is analyzed in the framework of the fermionie covariant prolongation structure theory. We construct the prolongation structure of the multidimen- sional super integrable equation and investigate its Lax representation. Furthermore, the Backlund transformation is presented and we derive a solution to the super integrable equation.

NEW PERIODIC SOLUTIONS OF NONLINEAR EVOLUTION EQUATIONS

In this paper,some new periodic solutions of nonlinear evolution equations and corresponding travelling wave solutions are obtained by using the double function method and Jacobi elliptic functions.

Nonlinear Evolution Equations and Its Application to a Tumour Invasion Model

We consider nonlinear evolution equations with logistic term satisfying initial Neumann-boundary condition and show global existence in time of solutions to the problem in arbitrary space dimension by using the method of energy. Applying the result to a mathematical model of tumour invasion, we discuss the property of the rigorous solution to the model. Finally we will show the time depending relationship and interaction between tumour cells, the surrounding tissue and matrix degradation enzymes in the model by computer simulations. It is seen that our mathematical result of the existence and asymptotic behaviour of solutions verifies our simulations, which also confirm the mathematical result visibly.

A NEW APPROACH TO BACKLUND TRANSFORMATIONSOF NONLINEAR EVOLUTION EQUATIONS

In this paper, a new approach to Backlund transformations of nonlinear evolution equations is presented. The results obtained by this procedure are completely the same as that by Painleve truncating expansion.

DYNAMIC BIFURCATION OF NONLINEAR EVOLUTION EQUATIONS

The authors introduce a notion of dynamic bifurcation for nonlinear evolution equa- tions, which can be called attractor bifurcation. It is proved that as the control pa- rameter crosses certain critical value, the system bifurcates from a trivial steady state solution to an attractor with dimension between m and m + 1, where m + 1 is the number of eigenvalues crossing the imaginary axis. The attractor bifurcation theory presented in this article generalizes the existing steady state bifurcations and the Hopf bifurcations. It provides a uni?ed point of view on dynamic bifurcation and can be applied to many problems in physics and mechanics.

A deep learning method for solving third-order nonlinear evolution equations

It has still been difficult to solve nonlinear evolution equations analytically.In this paper,we present a deep learning method for recovering the intrinsic nonlinear dynamics from spatiotemporal data directly.Specifically,the model uses a deep neural network constrained with given governing equations to try to learn all optimal parameters.In particular,numerical experiments on several third-order nonlinear evolution equations,including the Korteweg-de Vries(KdV)equation,modified KdV equation,KdV-Burgers equation and Sharma-Tasso-Olver equation,demonstrate that the presented method is able to uncover the solitons and their interaction behaviors fairly well.

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.