The wild solutions of the induced form under the spline wavelet basis in weakly damped forced KdV equation

In the paper what is studied is the wild solution of the induced form under the spline wavelet basis in weakly damped forced KdV equation. Project supported by the National Natural Science Foundation of China (19601020) and the Science-Technology Foundation of Minitry of Machine-building Industry of P. R. China     

The research of longtime dynamic behavior in weakly damped forced KdV equation

I.IntroductionTheKorteweg-deVries(KdV)equationul uu, u.;:=o(l.1)wasinitiallyder1vedasamodelforonedirectionallongwaterwavesofsmal1amplitudepropagatinginachannel.SincetheworkofKorteweganddeVries,ithasbeenshownthatthisequationoccursinalargevarietyofphysicals…     

Numerical analysis of longtime dynamic behavior in weakly damped forced KdV equation

ＩｎｔｒｏｄｕｃｔｉｏｎＯｎｅｏｆｔｈｅｍｏｓｔｉｍｐｏｒｔａｎｔａｎｄｉｎｔｅｒｅｓｔｉｎｇｐｒｏｂｌｅｍｓｉｎｔｈｅｆｉｅｌｄｏｆｎｏｎｌｉｎｅａｒｓｃｉｅｎｃｅｉｓｔｈｅｄｙｎａｍｉｃｓａｎａｌｙｓｉｓｏｆｓｐａｃｅ＿ｔｉｍｅｃｏｍｐｌｅｘｉｔｙ .Ｔｈｅｓｔｕｄｙｏｆｔｈｉｓｓｕｂｊｅｃｔｉｓｄｅｖｅｌｏｐｉｎｇａｌｏｎｇｔｗｏｄｉｒｅｃｔｉｏｎｓ.Ｏｎｏｎｅｈａｎｄ ,ｔｈｅｔｈｅｏｒｙｅｓｔａｂｌｉｓｈｅｄｂｙＴｅｍａｍ[1,2 ]ａｎｄｈｉｓｃｏ＿ｗｏｒｋｅｒｓｏｆｔｈｅｕｎｉｑｕｅｇｌｏｂ…     

WAVELET APPROXIMATE INERTIAL MANIFOLD AND NUMERICAL SOLUTION OF BURGERS'''' EQUATION

ＩｎｔｒｏｄｕｃｔｉｏｎＢｕｒｇｅｒｓ’ｅｑｕａｔｉｏｎｉｓｏｎｅｏｆｖｅｒｙｉｍｐｏｒｔａｎｔｆｌｕｉｄｅｑｕａｔｉｏｎｓ.Ｉｔｓｄｙｎａｍｉｃａｌｂｅｈａｖｉｏｒｉｓｒａｔｈｅｒｔｈａｎｃｏｍｐｌｅｘｉｔｙｏｗｉｎｇｔｏｔｈｅｎｏｎｌｉｎｅａｒｔｅｒｍ .Ｔｅｍａｎｅｔａｌ.[1] ｓｔｕｄｉｅｄｔｈｅｅｘｉｓｔｅｎｃｅｏｆａｇｌｏｂａｌａｔｔｒａｃｔｏｒ,ｉｎｅｒｔｉａｌｍａｎｉｆｏｌｄａｎｄａｐｐｒｏｘｉｍａｔｅｉｎｅｒｔｉａｌｍａｎｉｆｏｌｄａｓｓｏｃｉａｔｅｄｗｉｔｈｄｙｎａｍｉｃａｌｓｙｓ…     

A numerical method for KdV equation using collocation and radial basis functions

Recently, there has been an increasing interest in the study of initial boundary value problems for Korteweg–de Vries (KdV) equations. In this paper, we propose a numerical scheme to solve the third-order nonlinear KdV equation using collocation points and approximating the solution using multiquadric (MQ) radial basis function (RBF). The scheme works in a similar fashion as finite-difference methods. Numerical examples are given to confirm the good accuracy of the presented scheme.     

Localization and approximation of attractors for the Ginzburg-Landau equation

This paper studies the long-term behavior of solutions to the Ginzburg-Landau partial differential equation. For each positive integerm we explicitly produce a sequence of approximate inertial manifolds _m,j ,j = 1, 2,..., of dimensionm and associate with each manifold a thin neighborhood into which the orbits enter with an exponential speed and in a finite time. Of course this neighborhood contains the universal attractor which embodies the large time dynamics of the equations. The thickness of these neighborhoods decreases with increasingm for a fixed orderj; however, for a fixedm no conclusion can be made about the thickness of the neighborhoods associated to two differentj's. The neighborhoods associated to the manifolds localize the universal attractor and provide computabie large time approximations to solutions of the Ginzburg-Landau equation.     

A new auxiliary equation method for finding travelling wave solutions to KdV equation

In this paper, a new auxiliary equation method is used to find exact travelling wave solutions to the （1＋1）-dimensional KdV equation. Some exact travelling wave solu- tions with parameters have been obtained, which cover the existing solutions. Compared to other methods, the presented method is more direct, more concise, more effective, and easier for calculations. In addition, it can be used to solve other nonlinear evolution equations in mathematical physics.     

Alternating segment explicit-implicit scheme for nonlinear third-order KdV equation

A group of asymmetric difference schemes to approach the Korteweg-de Vries(KdV)equation is given here.According to such schemes,the full explicit difference scheme and the fun implicit one,an alternating segment explicit-implicit difference scheme for solving the KdV equation is constructed.The scheme is linear unconditionally stable by the analysis of linearization procedure,and is used directly on the parallel computer. The numerical experiments show that the method has high accuracy.     

Ordinary differential equations (ODEs) on inertial manifolds for reaction-diffusion systems in a singularly perturbed domain with several thin channels

Dynamics of solutions to a reaction-diffusion system in a domain of specific shape is investigated under the homogeneous Neumann boundary conditions. It is assumed that the domain hasN large regionsD _i ,i=1,...,N, and thin channelsQ _i,j () connectingD _i andD _j , which approach a line segment as 0 in some sense. In such a domain the firstN eigenvalues of – with the Neumann boundary conditions tend to zero as 0, while the (N + 1)-th eigenvalue is bounded away from zero. By virtue of this gap of the eigenvalues, an inertial manifold which is invariant and attracts every solution exponentially can be constructed under a certain condition. Moreover, the ODE describing the dynamics on the inertial manifold can be given in quite an explicit form through the analysis of the limit of the manifold as 0.     

Convergent families of approximate inertial manifolds for nonautonomous evolution equations

I.IntroduCtionAtpresent.autollomousinfillitedimensionaldynamicalsystemshavebeenthoroughlystLldicdilltllcory.andwidelyappliedinpracticel'-'1.Forthe11onautonomouscase,[5--91havesttldicd1ilocxistenceanddimensionestimateofattractorsofnonautonomouscase;[12].hasconsideredtileexislellceofinertialmanifolds.Theoretically,inertialmanifoldisaveryusefulInethodtodiscussthelongtimebehaviorofthesolutionstononautonomousinfinitedimcnsiollaldynamicalsystems.Butitcannotbeexpressedexplicitly.Soitisnotconvenient…     

The numerical solution of the unsteady natural convection flow in a square cavity at high Rayleigh number using SADI method

The unsteady natural convection flow in a square cavity at high Rayleigh number Ra=10 ⁷ and 2×10 ⁷ has been computed using cubic spline integration. The required solutions to the two dimensional Navier-Stokes and energy equations have been obtained using two alternate numerical formulations on non-uniform grids. The main features of the transient flow have been briefly discussed. The results obtained by using the present method are in good agreement with the theoretical predictions ^[1,2].The steady state results have been compared with accurate solutions presented recently for Ra=10 ⁷.     

A correction of the 2D KdV equation of Djordjevic ＆ Redekopp in exponentially stratified fluid＊

To further study the fission laws of initial internal solitons on the continental shelf/slope, we rederive and correct the 2D KdV equation of Djordjevic & Redekopp for exponentially stratified fluid (or ocean) and with two-dimensional topography. Through a combination of theoretical study and numerical experiments, we show that solitons in the odd vertical modes can fission. However, because of the corrections, the fission conditions are different from those of Djordjevic & Redekopp. The even modes cannot fission unless the initial internal solitons propagate from shallow sea to deep sea. This conclusion is entirely opposite to that of Djordjevic & Redekopp.The project supported by the National Natural Science Foundation of China (40276008) and the Grant of Key Laboratory of Marine Science and Numerical Modeling, SOA (0201(2003))The English text was polished by Keren Wang.     

High-accuracy algorithms for solving the discrete-time periodic Riccati equation

The paper proposes an algorithm for solving the discrete-time periodic Riccati equation, which arises, for instance, in designing a stabilization system for a hopping robot. This algorithm does not have a number of constraints on the matrix coefficients peculiar to other algorithms. It is significant that variable-precision arithmetic can be used in numerical implementation of the algorithm. This makes it possible to solve discrete-time periodic Riccati equations with high accuracy. Some examples are given __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 9, pp. 98–105, September 2007.     

Exact solutions of a KdV equation with variable coefficients via Exp-function method

In this paper, the Exp-function method is used to obtain generalized solitonary solutions and periodic solutions of a KdV equation with five arbitrary functions. The results show that the Exp-function method with the help of symbolic computation provides a very effective and powerful mathematical tool for solving nonlinear evolution equations in mathematical physics.     

THE NUMERICAL SOLUTION OF A SINGULARLY PERTURBED PROBLEM FOR SEMILINEAR PARABOLIC DIFFERENTIAL EQUATION

The numerical solution of a singularly perturbed problem for the semilinear parabolicdifferential equation with parabolic boundary layers is discussed.A nonlinear two-leveldifference scheme is constructed on the special non-uniform grids.The uniform convergenceof this scheme is proved and some numerical examples are given.     

The attractor of the scalar reaction diffusion equation is a smooth graph

For the scalar reaction diffusion equation with Dirichlet boundary conditions, it is proved that its maximal compact attractor is the graph of a C¹ function from a subset with nonempty interior of a subspace of the state space the dimension of which is equal to the maximal Morse index of the equilibria of the equation.     

SPECTRAL METHOD IN TIME FOR KdV EQUATIONS

ＳＰＥＣＴＲＡＬＭＥＴＨＯＤＩＮＴＩＭＥＦＯＲＫｄＶＥＱＵＡＴＩＯＮＳＳＰＥＣＴＲＡＬＭＥＴＨＯＤＩＮＴＩＭＥＦＯＲＫｄＶＥＱＵＡＴＩＯＮＳ￥ＷｕＳｈｅｎｇｃｈａｎｇ（吴声昌）；ＬｉｕＸｉａｏｑｉｎｇ（刘小清）（ＲｅｃｅｉｖｅｄＦｅｂ．２２，１９９５...     

Problem of periodic solutions for neutral functional differential equation

ＩｎｔｒｏｄｕｃｔｉｏｎＬｅｔＣ(ｋ- 1)2π =ｈ(ｔ) ｜ｈ :Ｒ →Ｒｉｓ (ｋ -1 )＿ｔｈｏｒｄｅｒｃｏｎｔｉｎｕｏｕｓｄｉｆｆｅｒｅｎｔｉａｂｌｅａｎｄｈ(ｔ+ 2π) ≡ｈ(ｔ) ,　　Ｃ2π =ｈ(ｔ) ｜ｈ :Ｒ →Ｒｉｓｃｏｎｔｉｎｕｏｕｓａｎｄｈ(ｔ+ 2π) ≡ｈ(ｔ) ,　　‖ｈ(ｔ)‖ =ｓｕｐｔ∈ [0 ,2π] ｜ｈ(ｔ) ｜ ,　　‖ｈ(ｔ)‖Ｐｋ- 1 =ｍａｘ‖ｈ(ｔ)‖ ,‖ｈ′(ｔ)‖ ,… ,‖ｈ(ｋ- 1) (ｔ)‖ ,　　ｘ(ｍ) (ｔ+ ·) (θ) =ｘ(ｍ) (ｔ+θ)　　θ∈Ｒ (ｍ =0 ,1 ,2 ,… ,ｋ-1 ) .Ｃｌｅａｒｌｙ ,ｘ(ｍ) (ｔ + ·) ∈Ｃ2π, …     

Orbital stability of periodic waves for the nonlinear Schrödinger equation

The nonlinear Schr?dinger equation has several families of quasi-periodic traveling waves, each of which can be parametrized up to symmetries by two real numbers: the period of the modulus of the wave profile, and the variation of its phase over a period (Floquet exponent). In the defocusing case, we show that these travelling waves are orbitally stable within the class of solutions having the same period and the same Floquet exponent. This generalizes a previous work (Gallay and Haragus, J. Diff. Equations, 2007) where only small amplitude solutions were considered. A similar result is obtained in the focusing case, under a non-degeneracy condition which can be checked numerically. The proof relies on the general approach to orbital stability as developed by Grillakis, Shatah, and Strauss, and requires a detailed analysis of the Hamiltonian system satisfied by the wave profile.     

A singular perturbation problem for periodic boundary partial differential equation

In this paper,we consider a singular perturbation elliptic-parabolic partial differentialequation for periodic boundary value problem,and construct a difference scheme.Using themethod of decomposing the singular term from its solution and combining an asymptoticexpansion of the equation,we prove that the scheme constructed by this paper convergesuniformly to the solution of its original problem with O(τ h~2).