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Quasi-periodic solutions of completely resonant forced wave equations

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arXiv:math/0504406v1 [math.FA] 20 Apr 2005
Keywords: Nonlinear Wave Equation, Quasi-Periodic Solutions, Variational Methods, Lyapunov-Schmidt reduction, Infinite Dimensional Hamiltonian Systems.1 2000AMS subject classification: 35L05, 37K50, 58E05.
QUASI-PERIODIC SOLUTIONS OF COMPLETELY RESONANT FORCED WAVE EQUATIONS
MASSIMILIANO BERTI, MICHELA PROCESI Abstract. We prove existence of quasi-periodic solutions with two frequencies of completely resonant, periodically forced nonlinear wave equations with periodic spatial boundary conditions. We consider both the cases the forcing frequency is: (Case A) a rational number and (Case B) an irrational number.
and therefore (1.4)
2 2 2 2 + 2ω1ω2 ∂ϕ1 ∂ϕ2 u(ϕ) + f (ϕ1 , u) = 0 . − 1)∂ϕ ω1 ∂ϕ1 + (ω2 2
We assume that the forcing term f : T × R → R f (ϕ1 , u) = a2d−1 (ϕ1 )u2d−1 + O (u2d ), d ∈ N+ , d > 1
Hale Waihona Puke is analytic in u but has only finite regularity in ϕ1 . More precisely
k + • (H) f (ϕ1 , u) := ∞ k =2d−1 ak (ϕ1 )u , d ∈ N , d > 1 and the coefficients ak (ϕ1 ) ∈ ∞ H 1 (T) verify, for some r > 0, k=2d−1 |ak |H 1 r k < ∞. The function f (ϕ1 , u) is not identically constant in ϕ1 .
1. Introduction We prove existence of small amplitude quasi-periodic solutions for completely resonant forced nonlinear wave equations like (1.1) vtt − vxx + f (ω1 t, v ) = 0 v (t, x) = v (t, x + 2π ) f (ω1 t, v ) = a(ω1 t)v 2d−1 + O (v 2d ), d > 1 , d ∈ N+
l∈Z2
|u ˆl |e|l2 |σ [l1 ]s < +∞
where [l1 ] := max{|l1 |, 1} and σ > 0, s ≥ 0. The space Hσ,s is a Banach algebra with respect to multiplications of functions (see Lemma 4.1 in the Appendix), namely u1 , u2 ∈ Hσ,s =⇒ u1 u2 ∈ Hσ,s and |u1 u2 |σ,s ≤ C |u1 |σ,s |u2|σ,s .
where the nonlinear forcing term is 2π/ω1-periodic in time. We shall consider both the cases • A) the forcing frequency ω1 ∈ Q • B ) the forcing frequency ω1 ∈ R \ Q. Existence of periodic solutions for completely resonant forced wave equations was first proved in the pioneering papers [R1], [R2] (with Dirichlet boundary conditions) if the forcing frequency is a rational number (ω1 = 1 in [R1]-[R2]). This requires to solve an infinite dimensional bifurcation equation which lacks compactness property; see [BN], [C], [BBi]-[BBi1] and references therein for other results. If the forcing frequency is an irrational number existence of periodic solutions has been proved in [PY]-[Mc]: here the bifurcation equation is trivial but a “small divisors problem” appears. To prove existence of small amplitude quasi-periodic solutions for completely resonant PDE’s like (1.1) one generally has to deal with a small divisor problem as well; however the main difficulty is to understand from which solutions of the linearized equation at v = 0, vtt − vxx = 0 ,
QUASI-PERIODIC SOLUTIONS OF FORCED WAVE EQUATIONS
3
We look for solutions u of (1.4) in the Banach space2 Hσ,s := u(ϕ) =
l∈Z2
u ˆl eil·ϕ : u ˆ∗ ˆ−l and |u|σ,s := l = u
1
2
MASSIMILIANO BERTI, MICHELA PROCESI
quasi-periodic solutions branch-off: such linearized equation possesses only 2π -periodic solutions q+ (t + x)+ q− (t − x) where q+ (·), q− (·) are 2π -periodic (completely resonant PDE). Here is the main difference w.r.t non-resonant PDE’s for which a developed existence theory of periodic and quasi-periodic solutions has been established, see e.g. [K], [Wa], [CW], [P], [B1] and references therein. For completely resonant autonomous PDE’s, existence of periodic solutions has been proved in [LS], [BP], [BB1], [BB2], [BB3], [GMP], [GP], and quasi-periodic solutions with 2-frequencies have been recently obtained in [P1]-[P2] for the specific nonlinearities f = u3 + O (u5). Here the bifurcation equation is solved by ODE methods. In this paper we prove existence of quasi-periodic solutions with two frequencies ω1 , ω2 for the completely resonant forced equation (1.1) in both the two cases: Case A): ω1 ∈ Q; Case B ): ω1 ∈ R \ Q. The more interesting case is ω1 ∈ Q (case A) when the forcing frequency ω1 enters in resonance with the linear frequency 1. To find out from which solutions of the linearized equation quasi-periodic solutions of (1.1) branch-off, requires to solve an infinite dimensional bifurcation equation which can not be solved in general by ODE techniques (it is a system of integro-differential equations). However, exploiting the variational nature of equation (1.1) like in [BB1]-[BB2], the bifurcation problem can be reduced to finding critical points of a suitable action functional which, in this case, possesses the infinite dimensional linking geometry [BR]. 1.1. Main results. We look for quasi-periodic solutions v (t, x) of equation (1.1) of the form v (t, x) = u(ω1 t, ω2 t + x) (1.2) u(ϕ1 + 2k1 π, ϕ2 + 2k2 π ) = u(ϕ1 , ϕ2 ), ∀k1 , k2 ∈ Z with frequencies ω = (ω1 , ω2 ) = (ω1 , 1 + ε) , imposing the frequency ω2 = 1 + ε to be close to the linear frequency 1. Writing ∂tt − ∂xx = (∂t − ∂x ) ◦ (∂t + ∂x ) we get (1.3) ω1 ∂ϕ1 + (ω2 − 1)∂ϕ2 ◦ ω1 ∂ϕ1 + (ω2 + 1)∂ϕ2 u + f (ϕ1 , u) = 0

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宇宙之最 中英文稿

宇宙之最 中英文稿

宇宙之最1.封面2宙中最冷的行星OGLE-BLG-390L是迄今发现最寒冷的行星，其质量是地球的5倍，被认为是一颗岩石行星，它也是距离地球最遥远的行星之一，距离地球大约28000光年。

它表面温度仅为零下220℃，低于液氮的沸点，接近于绝对零度（-273.15℃）Zhou in the coldest planet to OGLE - BLG - 390 - l is the most cold planets discovered so far, the quality is 5 times that of earth, is considered to be a rocky planet, it is also one of the most distant planets from earth, about 28000 light-years from earth. It surface temperature is below 220 ℃, below the boiling point of liquid nitrogen, close to absolute zero (273.15 ℃)3.宇宙中迄今观测到的最大行星TrES-4是观测到的最大行星，它的体积是太阳系“巨人”木星的1.7倍，距离地球1400光年。

它的质量非常轻，密度仅相当于家用的软木塞，可能拖曳着一个由自身大气层构成的彗星状尾部。

Has observed the planets in the universe, 4 are observed the planets, it is the volume of the solar system "giant" 1.7 times the mass of Jupiter, 1400light-years from earth. Its quality is very light, density isequivalent to the home of cork, may drag one of the atmosphere of themselves at the comet tail.4.宇宙中最重的黑洞OJ287是迄今观测到最重的无形黑洞，它的质量是太阳质量的180亿倍。

完整且严格的多层光盘矢量衍射模型

完整且严格的多层光盘矢量衍射模型


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As shown in Fig. 1, lenses L1 L2 and L2 L3 compose separately the read-in and the readout systems of a multilayered optical disc. For L1 L2 , the incident light is focused on the detected bit A in the multilayered optical disc that together with the air layers is divided into ( N+1) layers by N planes perpendicular to the z axis. Dielectric interfaces are placed at z z0 , , z N 1 and the 0th and Nth layers are semiinfinite homogenous. Bits may locate any layer of the multilayered optical disc. The laser is viewed as a point source Oo with the electric field (r )e o (0) xeox yeoy zeoz and the time dependence exp( jt ) . Assume that the coordinate of the detected bit A is ( xo , 0, z A ) . As a stratified media results in spherical aberrations, the focus of the lens L 2 should be at the point B ( xo , 0, z B ) where the size of the dispersed spot formed by virtually extended rays of the incident light is the smallest. At this time, only the virtual plane z z B and the source plane z Oo are corrected for L1 L2 so as to obey the sine condition and for the virtual plane z z B the image space of L1 L2 is homogenous. Therefore, one may first find the vector plane-wave spectrum (VPWS) of the incident fields at the virtual plane z z B that have been given by Eqs. (19) and (21) in Ref. [9], and then from which derive the VPWS of the incident electric field immediate to the z z N 1 interface at the Nth layer, namely
电子与通信专业英语第二版李霞 课后翻译

电子与通信专业英语第二版李霞 课后翻译

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渐近方法

渐近方法

Asymptotic MethodsLecturer:Xuesong Wu,ProfessorPre-requisitesLinear algebra,calculus(including Fourier/Laplace transforms),analysis of complex variable, ordinary and partial differential equations.(Knowledge of classical mechanics andfluid dynamics would be helpful but not essential.) Basic competency in English language:familiarity with mathematic terminology in English. Useful referencesO.M.Bender&S.A.Orszag Advanced Mathematical Methods for Scientists and Engineers. M.H.Holmes Introduction to Perturbation Methods.E.J.Hinch Perturbation Methods.M.Van Dyke Perturbation Methods in Fluid Mechanics.J.Kevorkian&J.D.Cole Perturbation Methods in Applied Mathematics.A.H.Nayfeh Perturbation Methods.Note:This course may be called asymptotic analysis or perturbation methods. Teaching arrangement30lectures and plus5tutorials.Extensive printed notes;5Question Sheets plus solutions(no assessed coursework).General descriptionMany mathematical problems arising from sciences and engineering often consist of small or large parameters.In exceptional cases(e.g.certain types of linear ordinary/partial differential equations),the problem can be solved exactly,often by means of an integral transformation(place or Fourier transform),and the solution may then be expressed in the form of an integral,referred to as integral representation.Numerical evaluation by using a computer,though straightforward nowadays,may not be most efficient.Moreover,the interest is often in the analytical property of the solution when the parameter or variable is large,and it is then desirable to derive an ana-lytical formula which approximates the integral for asymptotically large values of the parameter of variable.Such a formula is referred to as asymptotic representation of the integral.In general,the mathematical problems formulated are so complex(e.g.nonlinear)thatfinding an exact analytical solution in closed form is impossible.However,because of the presence of a large or small parameter,approximate solutions may be constructed in the form of asymptotic expansions or asymptotic series,and the original system systematically reduced to a simpler one,which can be solved analytically or numerically.The resulting solution represents afirst-order approximation,which may be further improved by seeking higher-order corrections.A range of powerful techniques and methods have been developed,each pertaining to a specific mathematical feature of the problem.They represent some of the most effective and important tools,which are being widely used by researches in both natural and engineering sciences. Aims&objectivesThis advanced course presents a systematical introduction to asymptotic methods,which form one of cornerstones of modern applied mathematics.The main ideas and various techniquesfor deriving asymptotic representations of integrals,and for constructing appropriate solutions will be explained.This is primarily a method course,but some important applications to classical mechanics andfluid mechanics will be demonstrated.Historically these subjects directly motivated the development of several key methods.The intended learning outcomes for students are•To familiarise with major asymptotic methods available,and to acquire a thorough un-derstanding of key ideas involved and mathematical contexts to which these methods are applicable.•To acquire basic skills for solving problems of‘textbook level’.•To acquire the ability of applying/modifying/extending relevant methods to unfamiliar problems,such as those emerging from research topics or practical applications.Main topics1.Asymptotic approximationConventional converging series,divergent series and asymptotic series;asymptotic versus Taylor expansions.2.Asymptotic representation of integralsIntegrals of Laplace type:Laplace’s method,Watson’s Lemma.Integrals of Fourier type:Method of stationary phase.Integral in complex plane:Method of steepest descent.Applications.3.Matched asymptotic expansionOrdinary and partial differential equations with small parameters,regular expansions.Singular asymptotic perturbations:notion of‘boundary layer’,inner and outer expansions.Overlap region.Matching of the asymptotic expansions.Location of the boundary layer.Composite approximation.Singular perturbation problem with an inner layer.4.Methods of multiple scalesMethod of strained coordinates;Poincare-Lindstedt method,periodic solutions.Quasi-periodic solutions,secular terms&solvability condition.Multiple-scale methods,averaging method.Nonlinear oscillation:frequency response,sub/super-harmonic resonance,hysteresis.5.WKB MethodTurning point problems,and analysis of transition layers.Ray tracing,caustics.Importance of asymptotic methodsThe importance of asymptotic methods may be appreciated by looking at how they were originated.1.Classical analysis and special functions(Laplace,Legendre,Poisson,Cauchy)2.Specific applications in science and engineering,e.g.celestial mechanics(Poincare),wave mechanics(classical or quantum),fluid mechanics(Prandtl).Methods developed have become•A fundamental analytical tool that has been,and is still being,used to prove rigorous theorems in many branches of pure or applied mathematics(e.g.existence&regularity of solutions to differential equations,approximation theory).•Powerful methods for tackling many increasingly challenging problems in applied sciences and engineering,which often require a sophisticated combination/extension of existing methods.Most methods were developed before1970s,i.e.before computers became powerful and easily accessible.The availability of powerful computers and software raises the equation: what is the role of asymptotic methods in the era of computer?1.Analytical properties cannot be readily extracted from numerics.2.Guide numerical computations:mesh size,algorithms,validation,framework for interpret-ing numerical results.3.Asymptotic analysis and reduction allow us to identify the key physical factors,and pro-vides essential insights into complex problems.4.Many classical and emerging problems remain unsolvable even by the most powerful com-puters because they involve multi-scales and multi-physics(e.g.turbulentflows around aircrafts;weather forecast;combustion,aeroacoustics).Computers broaden the role of asymptotic methods:Before the era of computer:asymptotically reduced systems had to be analytically tractable;Now in the era of computer:reduced systems only need to be numerically tractable sys-tems.5.A general methodology in scienceWell-known theories viewed from the perspective of asymptotic reduction,e.g.•Einstein’s Special Relativity theory−−→Newton’s mechanics in the limit of V/c 1.•Boltzmann’s equation−−→Continuum model,pressible Navier-Stokes equa-tions when Knutsen number Kn=L/λ 1−−>Compressible Euler equation if the Reynolds number Re 1−−>Boundary layer theory−−>Incompressible Navier-Stokes equations the Mach number M=V/a 1.Often works in the opposite direction,i.e.extend a less comprehensive/complete theory in such a fashion that the original theory is an asymptotic limit.Examples(To corroborate the points,and to illustrate the‘flavour’of the problems tackled and the general‘philosophy’involved.)Gamma functionΓ(x)≡ ∞0t x−1e−t dt(x>0is real).If x=n(an integer),Γ(n)=(n−1)!,a factorial,suggesting thatΓmust be very large for x 1.But how large precisely?Can we express it in terms of elementary functions?Stirling’s formula for x 1:Γ(x)∼x x e−x(2π2.Remark:Computing the valueΓfor any given x is trivial,but results such as Stirling’s formula cannot be established purely by numerical means.Airy equation&functionw (x)−xw=0(x is real)Two linear independent solutions:Ai(x)and Bi(x).Ai(x)has the integral representationAi(x)=13+xt)dt.Evaluation is straightforward for any moderate values of x,but tricky for large x.(Why?) Asymptotic representation for x>0and x 1,Ai(x)∼e−ζ2x 1ζs,Ai(−x)∼12x 14) s=0(−1)s a2s4) s=0(−1)s a2s+1 3x3(216)s s!.Remarks:1.Evaluating Ai for large x;complementary to numerical means.2.Asymptote cannot readily be extracted from numerics;need analytic methods.3.The series: s=0(−1)s a s。