Approximate Generalized Conditional Symmetries for Perturbed Evolution Equations

The concept of approximate generalized conditional symmetry （AGCS） for the perturbed evolution equations is introduced, and how to derive approximate conditional invariant solutions to the perturbed equations via their A GCSs is illustrated with examples.

OSCILLATION THEOREMS FOR SECOND ORDER QUASILINEAR PERTURBED DIFFERENTIAL EQUATIONS

New oscillation criteria for the second order perturbed differential equation are presented. The special case of the results includes the corresponding results in previous papers, extends and unifies a number of known results.

Classification and Approximate Solutions to Perturbed Nonlinear Diffusion-Convection Equations

This paper studies the perturbed nonlinear diffusion-convection equation with source term via the approximate generalized conditional symmetry （A GCS）. Complete classification of those perturbed equations which admit certain types of AGCSs is derived. Some approximate invariant solutions to the resulting equations can also be obtained.

Oscillation theorems for second-order nonlinear perturbed differential equations with damping

A class of second-order nonlinear damped perturbed differential equations is considered and its oscillation theorems are studied.These theorems are more general and deal with the cases which are not covered by the known criteria.Particularly,these criteria extend and unify some existing results.An example is given to verify the results.

Approximate Solutions of Perturbed Nonlinear Schroedinger Equations

By applying Lou＇s direct perturbation method to perturbed nonlinear Schroedinger equation and the critical nonlinear SchrSdinger equation with a small dispersion, their approximate analytical solutions including the zero-order and the first-order solutions are obtained. Based on these approximate solutions, the analytical forms of parameters of solitons are expressed and the effects of perturbations on solitons are briefly analyzed at the same time. In addition, the perturbed nonlinear Schroedinger equations is directly simulated by split-step Fourier method to check the validity of the direct perturbation method. It turns out that the analytical results given by the direct perturbation method are well supported by numerical calculations.

SECOND-ORDER ACCURATE DIFFERENCE METHOD FOR THE SINGULARLY PERTURBED PROBLEM OF FOURTH-ORDER ORDINARY DIFFERENTIAL EQUATIONS

In this paper, we construct a uniform second-order difference scheme for a class of boundary value problems of fourth-order ordinary differential equations. Finally, a numerical example is given.

UNIQUENESS OF SOLUTIONS OF SINGULARLY PERTURBED BOUNDARY VALUE PROBLEMS FOR THIRD QRDER NONLINEAR ORDINARY DIFFERENTIAL EQUATIONS

By making use of the differential inequalities, in this paper we study the uniqueness of solutions of the two kinds of the singularly perturbed boundary value problems for the nonlinear third order ordinary differential equation with a small parameter ε>0: where i=1, 2; a(?)(ε), β(ε) and γ(ε) are functions defined on (0, ε_o], while ε_o>0 is a constant.This paper is the continuation of our works [4, 6].

Approximate Symmetries and Infinite Series Symmetry Reduction Solutions to Perturbed Kuramoto-Sivashinsky Equation

Starting from Lie symmetry theory and combining with the approximate symmetry method, and using the package LieSYMGRP proposed by us, we restudy the perturbed Kuramoto-Sivashinsky （KS） equation. The approximate symmetry reduction and the infinite series symmetry reduction solutions of the perturbed KS equation are constructed. Specially, if selecting the tanh-type travelling wave solution as initial approximate, we not only obtain the general formula of the physical approximate similarity solutions, but also obtain several new explicit solutions of the given equation, which are first reported here.

A STABILITY PROBLEM FOR THE 3D MAGNETOHYDRODYNAMIC EQUATIONS NEAR EQUILIBRIUM

This paper is concerned with a stability problem on perturbations near a physically important steady state solution of the 3D MHD system.We obtain three major results.The first assesses the existence of global solutions with small initial data.Second,we derive the temporal decay estimate of the solution in the L^(2)-norm,where to prove the result,we need to overcome the difficulty caused by the presence of linear terms from perturbation.Finally,the decay rate in L^(2) space for higher order derivatives of the solution is established.

Asymptotical solutions of coupled nonlinear Schrodinger equations with perturbations

In this paper Lou＇s direct perturbation method is applied to the perturbed coupled nonlinear Schrodinger equations to obtain their asymptotical solutions, which include not only the zero-order solutions but also the first-order modifications. Based on the asymptotical solutions, the effects of perturbations on soliton parameters and the collision between two solitons are then discussed in brief. Furthermore, we directly simulate the perturbed coupled nonlinear SchrSdinger equations by split-step Fourier method to check the validity of the direct perturbation method. It turns out that our analytical results are well supported by the numerical calculations.

ASYMPTOTIC SOLUTION OF SINGULAR PERTURBATION PROBLEMS FOR THE FOURTH-ORDER ELLIPTIC DIFFERENTIAL EQUATIONS

In this paper we consider the singularly perturbed boundary value problem for the fourth-order elliptic differential equation, establish the energy estimates of the solutionand its derivatives and construct the formal asymptotic solution by Lyuternik- Vishik 's method. Finally, by means of the energy estimates we obtain the bound of the remainder of the asymptotic solution.

LARGE TIME BEHAVIOR OF SOLUTIONS TO THE PERTURBED HASEGAWA-MIMA EQUATION

The large time behavior of solutions to the two-dimensional perturbed Hasegawa- Mima equation with large initial data is studied in this paper. Based on the time-frequency decomposition and the method of Green function, we not only obtain the optimal decay rate but also establish the pointwise estimate of global classical solutions.

Asymptotic Bifurcation Solutions for Perturbed Kuramoto-Sivashinsky Equation

Stability and dynamic bifurcation in the perturbed Kuramoto-Sivashinsky （KS） equation with Dirichlet boundary condition are investigated by using central manifold reduction procedure. The result shows, as the bifurcation parameter crosses a critical value, the system undergoes a pitchfork bifurcation to produce two asymptotically stable solutions. Furthermore, when the distance from bifurcation is of comparable order ε2 （｜ε｜≤1）, the first two terms in e-expansions for the new asymptotic bifurcation solutions are derived by multiscale expansion method. Such information is useful to the bifurcation control.

Approximate symmetry reduction for perturbed nonlinear Schrdinger equation

We investigate the one-dimensional nonlinear SchrSdinger equation with a perturbation of polynomial type. The approximate symmetries and approximate symmetry reduction equations are obtained with the approximate symmetry perturbation theory.

Approximate Similarity Reduction for Perturbed Kaup-Kupershmidt Equation via Lie Symmetry Method and Direct Method

The perturbed Kaup-Kupershmidt equation is investigated in terms of the approximate symmetry perturbationmethod and the approximate direct method.The similarity reduction solutions of different orders are obtainedfor both methods, series reduction solutions are consequently derived.Higher order similarity reduction equations arelinear variable coefficients ordinary differential equations.By comparison, it is find that the results generated from theapproximate direct method are more general than the results generated from the approximate symmetry perturbationmethod.

Approximate direct reduction method:infinite series reductions to the perturbed mKdV equation

The approximate direct reduction method is applied to the perturbed mKdV equation with weak fourth order dispersion and weak dissipation. The similarity reduction solutions of different orders conform to formal coherence, accounting for infinite series reduction solutions to the original equation and general formulas of similarity reduction equations. Painleve Ⅱ type equations, hyperbolic secant and Jacobi elliptic function solutions are obtained for zeroorder similarity reduction equations. Higher order similarity reduction equations are linear variable coefficient ordinary differential equations.

Truncated series solutions to the(2+1)-dimensional perturbed Boussinesq equation by using the approximate symmetry method

In this paper, the（2＋1）-dimensional perturbed Boussinesq equation is transformed into a series of two-dimensional（2 D） similarity reduction equations by using the approximate symmetry method. A step-by-step procedure is used to acquire Jacobi elliptic function solutions to these similarity equations, which generate the truncated series solutions to the original perturbed Boussinesq equation. Aside from some singular area, the series solutions are convergent when the perturbation parameter is diminished.

Direct perturbation method for perturbed complex Burgers equation

So far, Lou＇s direct perturbation method has been applied successfully to solve the nonlinear SchrSdinger equation（NLSE） hierarchy, such as the NLSE, the coupled NLSE, the critical NLSE, and the derivative NLSE. But to our knowledge, this method for other types of perturbed nonlinear evolution equations has still been lacking. In this paper, Lou＇s direct perturbation method is applied to the study of perturbed complex Burgers equation. By this method, we calculate not only the zero-order adiabatic solution, but also the first order modification.

On Direct Transformation Approach to Asymptotical Analytical Solutions of Perturbed Partial Differential Equation

In this article, we will derive an equality, where the Taylor series expansion around ε= 0 for any asymptotical analytical solution of the perturbed partial differential equation （PDE） with perturbing parameter e must be admitted. By making use of the equality, we may obtain a transformation, which directly map the analytical solutions of a given unperturbed PDE to the asymptotical analytical solutions of the corresponding perturbed one. The notion of Lie-Bgcklund symmetries is introduced in order to obtain more transformations. Hence, we can directly create more transformations in virtue of known Lie-Bgcklund symmetries and recursion operators of corresponding unperturbed equation. The perturbed Burgers equation and the perturbed Korteweg-de Vries （KdV） equation are used as examples.

WAVELET BASIS ANALYSIS IN PERTURBEDPERIODIC KdV EQUATION

In the paper by using the spline wavelet basis to constructr the approximate inertial manifold, we study the longtime behavior of perturbed perodic KdV equation.