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…     

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     

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

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

Local attractors for weakly damped forced KdV equation in thin 2D domains

ＩｎｔｒｏｄｕｃｔｉｏｎＩｎｒｅｃｅｎｔｙｅａｒｓｍｕｃｈｐｒｏｇｒｅｓｓｈａｓｂｅｅｎｍａｄｅｏｎｔｈｅｈｉｇｈｅｒａｎｄｉｎｆｉｎｉｔｅｄｉｍｅｎｓｉｏｎａｌｄｙｎａｍｉｃａｌｓｙｓｔｅｍｓ (ｓｅｅ ,ｅ .ｇ .[1 ]～ [1 0 ] ) ,ａｎｄｍａｉｎｌｙｂａｓｅｄｏｎｒｅａｃｔｉｏｎ＿ｄｉｆｆｕｓｉｏｎｅｑｕａｔｉｏｎｓ,Ｋｕｒａｍｏｔｏ＿ＳｉｖａｓｈｉｎｓｋｙｅｑｕａｔｉｏｎｓＮａｖｉｅｒ＿Ｓｔｏｋｅｓｅｑｕａｔｉｏｎｓ ,ｅｔｃ .Ｂｅｃａｕｓｅｏｆｔｈｅｆａｃｔｔｈａｔｔｈｅｐｒｉｎｃｉｐａｌｏｐｅｒ…     

The research of blow-up in 2D weakly damped forced KdV equation on thin domain

ＩｎｔｒｏｄｕｃｔｉｏｎＷｉｔｈｃｏｍｐｌｅｘｉｔｙｏｆｔｈｅｄｙｎａｍｉｃｓｏｆｔｈｅｈｉｇｈｅｒ＿ｄｉｍｅｎｓｉｏｎａｌｄｉｓｓｉｐａｔｉｖｅｓｙｓｔｅｍｓｏｌｉｔａｒｙｗａｖｅ ,ｔｈｅｒｅｉｓｍｕｃｈｄｉｆｆｉｃｕｌｔｙｉｎｓｔｕｄｙｉｎｇｔｈｅｓｅｐｒｏｂｌｅｍｓ.Ｔｈｅｐａｐｅｒｓ [1 ]～ [3]ｍｅｎｔｉｏｎｅｄｔｈｅｔｗｏ＿ｄｉｍｅｎｓｉｏｎａｌＫｄＶｅｑｕａｔｉｏｎｓｄｒｉｖｅｎｂｙＬａｘａｎｄ [3]ｏｂｔａｉｎｅｄｔｈｅｅｘｉｓｔｅｎｃｅｏｆｓｏｌｕｔｉｏｎｔｏｔｈｅｅｑｕａｔｉｏｎｓ…     

The fractal structure of attractor for the generalized Kuramoto-Sivashinsky equations

I.IntroductionSincethereexistspectralbarriersandspectralgapconditions,theexistenceofaninertialmanifoldformanynonlineardissipativeevolutionequationsisstillamystery.Recently,Edenetal[5]havediscoveredthatfornonlinearsemigroup,definedbynonlineardissipativeevolutionequationsincludingZDNavier-Stokesequations,thereexistsatinliefractaldimensionalinertialsetwhichmayberepresentedbyaunionoffractillsetsandattractor,ifitisLipschitzcontinuousandissqueezingonacompacti,ositiveinvariantset.Ontileotherhand,S…     

The theory of fractal interpolated surface and its applications

I.IntroductionFI.actillilltcrpolationwastlrstpEltforwardbyunA]ncrica1llathematician,M.F.Barllsley.in1986.ItgivesanewInethodologytardataf'ittillg,whichnotonlyopedsupanewresearch11eldfol'tilnctiollappl'oachingtheory,butalsoprovidespowerfultoolsforcolnptltcrgraphicsThistool'sapplicabilityisnowfilllyappreciated.Theuseoflinearfunctions,polynomialfunctionsandSurtllcespringfunctionstoestablishvariousmeterialobjectmodelsinreallifefi-omtraditionalEuclideangeometryisnowcommonpractice.Theavailabilit…     

Wavelet basis analysis in perturbed periodic KdV equation

I.IntroductionandSignsThephenomenaofnaturealwaysoccursatacertaintimeandacertainspace.Thecomplexityoftimeandspaceisduetodescribingtherelationbetweentimeandspaceandthelongtimebehaviorofsystembyusingthepartialdifferentialequation.Traditionally,thestatefunctionisexpandedinFotlrierseries,andthepartialdifferentialequationistransformedintotheordinarydifIYrentialequationwiththeevolutionofFouriercoefficientastimegoeson.ButitisverydifficulttoknowwelltheFouriercoefficientwhichplaysanimportantroleasthe…     

1/3 PURE SUB-HARMONIC SOLUTION AND FRACTAL CHARACTERISTIC OF TRANSIENT PROCESS FOR DUFFING＇S EQUATION

The 1/3 sub-harmonic solution for the Duffing＇s with damping equation was investigated by using the methods of harmonic balance and numerical integration. The assumed solution is introduced, and the domain of sub-harmonic frequencies was found. The asymptotical stability of the subharmonic resonances and the sensitivity of the amplitude responses to the variation of damping coefficient were examined. Then, the subharmonic resonances were analyzed by using the techniques from the general fractal theory. The analysis indicates that the sensitive dimensions of the system time-field responses show sensitivity to the conditions of changed initial perturbation, changed damping coefficient or the amplitude of excitation, thus the sensitive dimension can clearly describe the characteristic of the transient process of the subharmonic resonances.     

Attactors of dissipative soliton equation

ＡＴＴＡＣＴＯＲＳＯＦＤＩＳＳＩＰＡＴＩＶＥＳＯＬＩＴＯＮＥＱＵＡＴＩＯＮＴｉａｎＬｉ－ｘｉｎ（田立新）（ＤｅｐａｒｔｍｅｎｔｏｆＭａｔｈｅｍａｔｉｃｓａｎｄＰｈｙｓｉｃｓ,ＪｉａｎｇｓｕＵｎｉｖｅｒｓｉｔｙｏｆＳｃｉｅｎｃｅａｎｄＴｅｃｈｎｏｌｏｇ...     

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.     

Convergent families of approximate inertial manifolds for nonautonomous evolution equations

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

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.     

The fractal research and predicating on the times series of sunspot relative number

In this paper, with the theory of nonlinear dynamic systems, It is analyzed that the dynamic behavior and the predictabilityfor the monthly mean variationsof the sunspot relative number recorded from January 1891 to December 1996. Inthe progress, the fractal dimension (D=3. 3±0.2) for the variation process wascomputed. This helped us to determine the embedded dimension [2×D+1]=7.By computing the Lyapunov index (λ₁=0.863), it was indicated that the variationprocess is a chaotic system. The Kolmogorov entropy (K=0.0260) was also computed, which provides, theoretically, the predicable time scale. And at the end, according to the result of the analysis above, an experimental predication is made,whose data was a part cut from the sample data.     

A numerical scheme for solving a periodically forced Reynolds equation

A modified Reynolds equation is used to model the air‐film in a high‐speed squeeze‐film bearing. The axial position of the bearing stator is prescribed as a finite amplitude periodic oscillation. A numerical approach is considered for solving the uncoupled and coupled periodic problems associated with this model. The uncoupled problem requires the computation of the squeeze‐film dynamics when the rotor is held at a fixed axial position and the coupled problem incorporates the additional air–rotor interaction since the rotor position is unknown and modelled as a spring‐mass‐damper system. The details of a Fourier spectral collocation scheme are provided for the reduction of the modified Reynolds equation to a system of non‐linear, first‐order ordinary differential equations in space. Using the Matlab boundary value problem solver bvp4c this system of equations is solved to give the periodic pressure distributions and rotor heights. The high degree of accuracy in the spectral collocation scheme is demonstrated through comparison with an appropriate analytical solution. Further analysis indicates that the direct periodic solver is at least 10 times faster than the equivalent Crank–Nicholson finite‐difference scheme. For changing values of a selected physical parameter the method of arc‐length continuation is employed to track branches of solutions computed using the spectral collocation scheme. A selection of results is presented to demonstrate the range of accessible solutions and the robust nature of the numerical scheme. Copyright © 2010 John Wiley & Sons, Ltd.     

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.     

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.     

GLOBAL ATTRACTOR FOR HASEGAWA-MIMA EQUATION

The long time behavior of solution of the Hasegawa-Mima equation with dissipation term was considered. The global attractor problem of the Hasegawa-Mima equation with initial periodic boundary condition was studied. Applying the uniform a priori estimates method, the existence of global attractor of this problem was proved, and also the dimensions of the global attractor was estimated.     

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.     

On Finite Fractal Dimension of the Global Attractor for the Wave Equation with Nonlinear Damping

We prove that the global attractor to a semilinear damped wave equation has finite fractal dimension provided that the damping function and the lower order nonlinearity are smooth with certain polynomial growth.