BIFURCATIONS OF TRAVELLING WAVE SOLUTIONS FOR A COUPLED NONLINEAR WAVE SYSTEM

Introduction FangShaomeiandGuoBoling[1]consideredthefollowingtimeperiodicproblemof dampedcouplednonlinearwaveequations:ut f(u)x-αuxx βuxxx 2vvx=G1(u,v) h1(x),vt-γvxx 2(uv)x g(v)x=G2(u,v) h2(x),(1)whereα,β,γareconstants,andγ>0,β≠0.Undertheperiodicboundaryconditions,the authorsobtainedtheuniqueexistenceofstrongsolutionsfortheabovesystem.InthispaperweshallconsiderbifurcationbehaviorofthetravellingwavesolutionsofEq.(1)inthecaseGi(u,v)≡0,hi(u,v)≡0(i=1,2).Letξ=x-ct,u=u(x-ct),where cis…     

DYNAMICAL CHARACTER FOR A PERTURBED COUPLED NONLINEAR SCHRDINGER SYSTEM

Introduction WeconsidertheperturbedcouplednonlinearSchr dingerequations(CNLS)iq1t=q1xx 2[|q1|2 |q2|2-ω2]q1 iε(^D1q1-r1),iq2t=q2xx 2[|q1|2 |q2|2-ω2]q2 iε(^D2q2-r2),(1)whereqjis2πperiodicandeveninx.rjisconstant,^Djisboundeddissipativeoperatorand assumedtotaketheform^Djqj=-αjqj βj^Bqj,(2)αjandβjarepositiveconstants,j=1,2.^BisaFouriertruncationofLaplaceoperator xx,i.e.,^Bcos(kx)=-k2cos(kx),k0isasmallperturbation parameter.Th…     

EXISTENCE OF BOUNDED SOLUTIONS ON THE REAL LINE FOR LINARD SYSTEM

IntroductionInthispaper,weconsidertheexistenceofsolutionsoffollowingboundaryvalueproblemonreallineforLi啨ｎａｒｄｓｙｓｔｅｍ :u″-f(u)u′ g(u) =0 ,u(-∞ ) =0 ,　u( ∞ ) =1 ,(1 )wheref:R →Randg :R →RarefunctionsofclassC1.Amotivationforourresearchisconne     

A NEW ALGORITHM FOR SOLVING DIFFERENTIAL/ALGEBRAIC EQUATIONS OF MULTIBODY SYSTEM DYNAMICS

I.Intr0ductionTheequationsofmoti0nofconstrainedmultibodysystemdynamicscanbewrittenasfollows11lwherelistimeparameter*q6R'isgeneralizedcoordinatesvectorofsystem,M(q,t):RnXR-R"xnisgeneralizedmassn1atrixofsystem,k6RmisLagrangemultipliervector,fo(q,t):RnXR-R"(…     

ON THE EXISTENCE OF PERIODIC SOLUTIONS FOR NONLINEAR SYSTEM WITH MULTIPLE DELAYS

IntroductionInthispaper,westudyT_periodicsolutionsofthefollowingnonlinearsystemwithmultipledelays x(t) =f(t,x(t) ,x(t-τ1(t) ) ,… ,x(t -τm(t) ) ) ,(1 )wherex(t) ∈C(R ,R) ,fiscontinuous,f(t+T ,·) =f(t,·) ,τi(t) (i=1 ,2 ,… ,m)arecontinuousperiodicfunctionsofperiodT .AlemmaisintroducedfordiscussingtheexistenceofT_periodicsolutionofsystem (1 ) .LetXbeaBanachSpace ,considerthefollowingoperatorequation :Lx =λNx　　 (λ∈ [0 ,1 ] ) ,whereL :DomL∩X→Xisalinearoperator,λ∈ [0 ,1 ]isapa…     

SINGULAR PERTURBATION OF GENERAL BOUNDARY VALUE PROBLEM FOR NONLINEAR DIFFERENTIAL EQUATION SYSTEM

I.IntroductionAllphysicalsystemsarenonlineartosomeextent.Actually,Lineal.systemisimaginarymodelwherenonlinearfactorisomittedinnonlinearsystem.Insolvingtheautocontl'ol,nonlinearoscillationtheory,theboundarystagnationproblenloffluidIncchanicsandsollleproblemsofsemi-conducttheoryandquantummechanicsetc'.weOnlyncedtosolvethefollowingproblem,whichisnonlineardifferentialequationsystemwithtilesll,allparanletel'inhighestorderderivativeandnonlinearboundaryconditions.whereE>0isasmallparameter,teR,x,fi…     

STABILITY OF MOTION FOR A CONSTRAINED BIRKHOFF'S SYSTEM IN TERMS OF INDEPENDENT VARIABLES

I.IntroductionInl927,AmericanmathematicianG.D.Birkh0ffgaveakindofequationsofdynamicswhichweremoregeneralthantheHamilton'sequationsinhiswork"DynamicalSystems-l'].TheequationsarecalledBirkhofCsequationssuggested.byAmericanphysicianR.M.SantiIIiinl978[2j.Inl9…     

HOPF BIFURCATION IN A THREE-DIMENSIONAL SYSTEM

In this paper,Liapunor-Schmidt reduction and singularity theory are employed todiscuss Hopf and degenerate Hopf bifurcations in global parametric region in a three-dimensional system x=-βx y,y=-x-βy(1-kz),z=β[α(1-z)-ky~2].Theconditions on existence and stability are given.     

PLANE INFINITE ANALYTICAL ELEMENT AND HAMILTONIAN SYSTEM

IntroductionManyinfiniteproblemscanbefoundincivilengineering ,suchastunnelconstruction ,structurefoundation ,etc ..Forotherengineeringproblems,whenthephysicaldimensionsofanobjectaresmallandthesurroundingmediaorstructuresaremuchbiggerthantheobject,thenumericalcalculatingmodelcanbetreatedasoneinaninfinitefield .Sofar,onlyafewanalyticalsolutionsforinfinitefieldproblemscanbefound[1- 3].ManyprojectsrelatedtoinfinitefieldproblemsaresolvedbytheFEM ,whereinfiniteelementmethodsareused[4 ,5 ].Sometime…     

SAFETY MARGIN CRITERION OF NONLINEAR UNBALANCE ELASTIC AXLE SYSTEM

IntroductionAtpresentanewanddevelopingsubject—chaoticdynamicsstartsabroadprospectforanalysisofnonlinearsystem[1～ 5 ].Largerotatingmachineryisatypicalnonlinearnon_autonomoussystem .Thesaferunofrotorsystemisofgreatsignificancetosociallifeandeconomicdevelopment.Thestabilityisthekeytosafeoperation .Thesafestabilityanalysisandcontrolforlargesystemisnotonlyamajorbasicresearchbutalsoisveryimportanttosolvethesafeproblemsinlifeandproduction[6 ,7].Soar,althoughmanymathematicians,mechanistsandengineer…     

THE TRANSFORMATION FUNCTION Φ AND THE CONDITION NEEDED FOR KUR SPACE HAVING THE FIXED POINT

In the last several years some progress has been made in the study of the properties of the extent of Banach space: In 1979 for example, when Suillivan discussed a related characterization of real L^p (X) space, he used uniform behavior of all two-dimensional subspace and defined this concept of a KUR space. In 1980 Huff used the concept of a NUC space when he discussed the property of generalizing uniform convexity which was defined in terms of sequence. And in 1980 Yu Xin-tai stated certainly and proved that the KUR space is equal to the NUC space^[1]. However, the following quite interesting questions raised respectively by Suillivan and Huff merit attention: Does every super-reflexive space have the fixed point property?^[2] The purpose of this paper is to study the characterization of transformation function^[4] and relationships between transformation function and the two questions above.     

Fixed points of a pair of asymptotically regular mappings

I.IntroductionTheconceptofasymptoticregularitywasintroducedbyBrowderandPetryshynl'].LetEbearealBanachspacewithnorn11l'l'AmappingT:E-Eissaidtobeasymptotical1yregulariflimijT" 'x~T'x:l=ofora1lxinE,whereTffxdenotesthenthiteratio11of--ooT.ltiswell-knownthatif…     

THE PROBLEMS OF THE NONLINEAR UNSYMMETRICAL BENDING FOR CYLINDRICALLY ORTHOTROPIC CIRCULAR PLATE（II）

ＴＨＥＰＲＯＢＬＥＭＳＯＦＴＨＥＮＯＮＬＩＮＥＡＲＵＮＳＹＭＭＥＴＲＩＣＡＬ．ＢＥＮＤＩＮＧＦＯＲＣＹＬＩＮＤＲＩＣＡＬＬＹＯＲＴＨＯＴＲＯＰＩＣＣＩＲＣＵＬＡＲＰＬＡＴＥ（ＩＩ）ＨｕａｎｇＪｉａｙｉｎ（黄家寅）；ＱｉｎＳｈｅｎｇｌｉ（秦圣立）；Ｘｉ...     

Singular periodic impulse problems

We establish an existence principle for the impulsive periodic boundary-value problem {fx029-01}, where g ∈ C(0, ∞) can have a strong singularity at the origin. Furthermore, we assume that 0 < t ₁ < … < t _m < T, e ∈ L ₁[0, T], c ∈ ℝ, J _i and M _i , i = 1, 2, …, m, are continuous mappings of G[0, T] × G[0, T] into ℝ, and G[0, T] denotes the space of functions regulated on [0, T]. The presented principle is based on an averaging procedure similar to that introduced by Manásevich and Mawhin for singular periodic problems with p-Laplacian. Published in Neliniini Kolyvannya, Vol. 11, No. 1, pp. 32–44, January–March, 2007.     

The uniqueness and existence of solution of the characteristic problem on the generalized KdV equation

The generalized KdV equationu ₁+auu_a+μu_a3+eu_a5=0^[1] is a typical integrable equation. It is derived studying the dissemination of magnet sound wave in cold plasma^[2], the isolated wave in transmission line^[3], and the isolated wave in the boundary surface of the divided layer fluid^[4]. For the characteristic problem of the generalized KdV equation, this paper, based on the Riemann function, designs a suitable structure, then changes the characteristic problem to an equivalent integral and differential equation whose corresponding fixed point, the above integral differential equation has a unique regular solution, so the characteristic problem of the generalized KdV equation has a unique solution. The iteration solution derived from the integral differential equation sequence is uniformly convegent in .     

Vorticities in a LES Model for 3D Periodic Turbulent Flows

This paper is devoted to the study of a LES model to simulate turbulent 3D periodic flow. We focus our attention on the vorticity equation derived from this LES model for small values of the numerical grid size δ. We obtain entropy inequalities for the sequence of corresponding vorticities and corresponding pressures independent of δ, provided the initial velocity u₀ is in L_x² while the initial vorticity ω₀ = ∇ × u₀ is in L_x¹. When δ tends to zero, we show convergence, in a distributional sense, of the corresponding equations for the vorticities to the classical 3D equation for the vorticity.     

Finite-dimensional asymptotic behavior of some semilinear damped hyperbolic problems

We prove that the solution semigroup $$S_t \left[ {u_0 ,v_0 } \right] = \left[ {u(t),u_t (t)} \right]$$ generated by the evolutionary problem $$\left\{ P \right\}\left\{ \begin{gathered} u_{tt} + g(u_t ) + Lu + f(u) = 0, t \geqslant 0 \hfill \\ u(0) = u_0 , u_t (0) = \upsilon _0 \hfill \\ \end{gathered} \right.$$ possesses a global attractorA in the energy spaceE _o=V×L ²(Ω). Moreover,A is contained in a finite-dimensional inertial setA attracting bounded subsets ofE ₁=D(L)×V exponentially with growing time.     

Vorticity and Regularity for Viscous Incompressible Flows under the Dirichlet Boundary Condition. Results and Related Open Problems

In reference [7] it is proved that the solution of the evolution Navier–Stokes equations in the whole of R ³ must be smooth if the direction of the vorticity is Lipschitz continuous with respect to the space variables. In reference [5] the authors improve the above result by showing that Lipschitz continuity may be replaced by 1/2-H?lder continuity. A central point in the proofs is to estimate the integral of the term (ω · ∇)u · ω, where u is the velocity and ω = ∇ × u is the vorticity. In reference [4] we extend the main estimates on the above integral term to solutions under the slip boundary condition in the half-space R ₊³. This allows an immediate extension to this problem of the 1/2-H?lder sufficient condition. The aim of these notes is to show that under the non-slip boundary condition the above integral term may be estimated as well in a similar, even simpler, way. Nevertheless, without further hypotheses, we are not able now to extend to the non slip (or adherence) boundary condition the 1/2-H?lder sufficient condition. This is not due to the “nonlinear" term (ω · ∇)u · ω but to a boundary integral which is due to the combination of viscosity and adherence to the boundary. On the other hand, by appealing to the properties of Green functions, we are able to consider here a regular, arbitrary open set Ω.      

Nonlinear MHD Kelvin-Helmholtz instability in a pipe

Nonlinear MHD Kelvin-Helmholtz (K-H) instability in a pipe is treated with the derivative expansion method in the present paper. The linear stability problem was discussed in the past by Chandrasekhar (1961)^[1] and Xu et al. (1981).^[6]Nagano (1979)^[3] discussed the nonlinear MHD K-H instability with infinite depth. He used the singular perturbation method and extrapolated the obtained second order modifier of amplitude vs. frequency to seek the nonlinear effect on the instability growth rate γ. However, in our view, such an extrapolation is inappropriate. Because when the instability sets in, the growth rates of higher order terms on the right hand side of equations will exceed the corresponding secular producing terms, so the expansion will still become meaningless even if the secular producing terms are eliminated. Mathematically speaking, it's impossible to derive formula (39) when γ ₀ ² is negative in Nagano's paper.^[3]Moreover, even as early as γ ₀ ² → O⁺, the expansion becomes invalid because the 2nd order modifier γ₂ (in his formula (56)) tends to infinity. This weakness is removed in this paper, and the result is extended to the case of a pipe with finite depth. Theproject is supported by the National Natural Science Foundation of China.     

Isentropic Approximation¶of the Compressible Euler System¶in One Space Dimension

Using the stability results of Bressan & Colombo [BC] for strictly hyperbolic 2 × 2 systems in one space dimension, we prove that the solutions of isentropic and non-isentropic Euler equations in one space dimension with the respective initial data (ρ₀, u ₀) and (ρ₀, u ₀, &\theta;₀=ρ₀ ^{γ− 1}) remain close as soon as the total variation of (ρ₀, u ₀) is sufficiently small. Accepted April 25, 2000?Published online November 24, 2000