KAM理论及其在微分方程中的应用
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摘要
本文主要应用有限维KAM理论证明了一类拟周期系数的Lotka-Volterra模型存在正拟周期解,以及应用无穷维KAM理论证明了高维Ginzburg-Landau方程和一维带有非线性项|u|~(2p)u的Ginzburg-Landau方程存在2维不变环面,即2维拟周期解。全文共分四章。
第一章,主要介绍了KAM理论的背景,意义,国内外研究现状以及本文的主要工作。
第二章,证明了一类拟周期系数的Lotka-Volterra模型存在正拟周期解。Lotka-Volterra模型是生态系统中一个很重要的模型,它成功地描述了两个生物种群在各种条件下种群数量随时间变化的情况,如分别是捕食者和食物,两种群竞争,两种群合作,寄生和寄主等。我们假设环境因素是拟周期变化的,这样更能符合现实的生态系统,并用KAM迭代证明该系统具有正拟周期解。
第三章,考虑高维Ginzburg-Landau方程。Ginzburg-Landau方程具有十分丰富的物理背景和内涵。它可以描述Banerd对流,Tayor-Couette流,平面Poiseuille流以及化学湍流等问题,在非平衡态的相变,超导和超流理论中均有应用。我们用一个修正的KAM定理证明了高维Ginzburg-Landau方程2维不变环面的存在性。
第四章,证明了一维带有非线性项|u|~(2p)u的Ginzburg-Landau方程存在2维不变环面。
We are mainly interested in KAM theory,and apply KAM theory to prove that there exists a positive quasi-periodic solution for Lotka-Volterra system with quasiperiodic coefficients and there exist 2-dimensional invariant tori both for the cubic complex Ginzburg-Landau equation with higher spatial dimension and 1D Ginzburg-Landau equation with nonlinearity |u|~(2p)u.This paper is divided into four parts.
In Chapter 1,we introduce the historical background,some recent results of KAM theory obtained in the literature and our main work in this paper.
In Chapter 2,it is proven that the existence of quasi-periodic solutions for Lotka-Volterra system with quasi-periodic coefficients.Lotka-Volterra system plays an important role in the research field of ecosystem.It includes predator-prey system,competitive system,cooperative systems and host-parasite systems.Many work mostly discuss periodic Lotka-Volterra system.In fact,quasi-periodic system describes our world more realistic and more accurate than periodic one.We use KAM iteration to prove the existence of quasi-periodic solutions for the system with quasi-periodic coefficients.
In Chapter 3,we consider the cubic complex Ginzburg-Landau equation with higher spatial dimension.Ginzburg-Landau equation is an important equation with very rich physical and mathematical background.It can be described Banerd convection, Tayor-Couette flow,Plane Poiseuille flow,as well as issues such as chemical turbulence, and also applied to non-equilibrium phase transition,theory of superconductivity and superfluidity.It is proven that the existence of 2-dimensional invariant tori for this equation by a modified KAM theorem.
In Chapter 4,it is proven that the existence of 2-dimensional invariant tori for 1D Ginzburg-Landau equation with nonlinearity |u|~(2p)u by a modified KAM theorem.
第一章,主要介绍了KAM理论的背景,意义,国内外研究现状以及本文的主要工作。
第二章,证明了一类拟周期系数的Lotka-Volterra模型存在正拟周期解。Lotka-Volterra模型是生态系统中一个很重要的模型,它成功地描述了两个生物种群在各种条件下种群数量随时间变化的情况,如分别是捕食者和食物,两种群竞争,两种群合作,寄生和寄主等。我们假设环境因素是拟周期变化的,这样更能符合现实的生态系统,并用KAM迭代证明该系统具有正拟周期解。
第三章,考虑高维Ginzburg-Landau方程。Ginzburg-Landau方程具有十分丰富的物理背景和内涵。它可以描述Banerd对流,Tayor-Couette流,平面Poiseuille流以及化学湍流等问题,在非平衡态的相变,超导和超流理论中均有应用。我们用一个修正的KAM定理证明了高维Ginzburg-Landau方程2维不变环面的存在性。
第四章,证明了一维带有非线性项|u|~(2p)u的Ginzburg-Landau方程存在2维不变环面。
We are mainly interested in KAM theory,and apply KAM theory to prove that there exists a positive quasi-periodic solution for Lotka-Volterra system with quasiperiodic coefficients and there exist 2-dimensional invariant tori both for the cubic complex Ginzburg-Landau equation with higher spatial dimension and 1D Ginzburg-Landau equation with nonlinearity |u|~(2p)u.This paper is divided into four parts.
In Chapter 1,we introduce the historical background,some recent results of KAM theory obtained in the literature and our main work in this paper.
In Chapter 2,it is proven that the existence of quasi-periodic solutions for Lotka-Volterra system with quasi-periodic coefficients.Lotka-Volterra system plays an important role in the research field of ecosystem.It includes predator-prey system,competitive system,cooperative systems and host-parasite systems.Many work mostly discuss periodic Lotka-Volterra system.In fact,quasi-periodic system describes our world more realistic and more accurate than periodic one.We use KAM iteration to prove the existence of quasi-periodic solutions for the system with quasi-periodic coefficients.
In Chapter 3,we consider the cubic complex Ginzburg-Landau equation with higher spatial dimension.Ginzburg-Landau equation is an important equation with very rich physical and mathematical background.It can be described Banerd convection, Tayor-Couette flow,Plane Poiseuille flow,as well as issues such as chemical turbulence, and also applied to non-equilibrium phase transition,theory of superconductivity and superfluidity.It is proven that the existence of 2-dimensional invariant tori for this equation by a modified KAM theorem.
In Chapter 4,it is proven that the existence of 2-dimensional invariant tori for 1D Ginzburg-Landau equation with nonlinearity |u|~(2p)u by a modified KAM theorem.
引文
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[7]D.Bambusi and S.Paleari,Families of periodic orbits for resonant PDE's,J.Nonlinear Science,11,69-87(2001)
[8]D.Bambusi and S.Paleari,Families of periodic orbits for some PDE's in higher dimensions,Commun.Pure Appl.Anal.,1,269-279(2002)
[9]D.Bambusi,Birkhoff normal form for some nonlinear PDEs.Commun.Math.Phys.,234,253-285(2003)
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[79] X. Yuan, Quasi-periodic solutions of completely resonant nonlinear wave equation, J. Diff. Eq., 230,213-274(2006)
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[2]Alvarez,C.Lazer and C.Alan,An application of topological degree to the periodic competing species problem,J.Austral.Math.Soc.Ser.B,28,202-219(1986)
[3]V.I.Arnold,Proof of A.N.Kolmogorov's theorem on the preservation of quasi-periodic motions under small perturbations of the Hamiltonian,Uspekhi Mat.Nauk U.S.S.R.,18,Ser.5(113),13-40(1963).
[4]V.I.Arnold,Mathematical Methods of Classical Mechanics,Springer,Berlin,1978.
[5]D.Bambusi,On long time stability in Hamiltonian perturbations of non-resonant linear PDEs,Nonlinearity,12,823-850(1999)
[6]D.Bambusi and S.Graffi,Time quasi-periodic unbounded perturbations of Schrtinger operators and KAM methods,Commun.Math.Phys.,219,465-480(2001)
[7]D.Bambusi and S.Paleari,Families of periodic orbits for resonant PDE's,J.Nonlinear Science,11,69-87(2001)
[8]D.Bambusi and S.Paleari,Families of periodic orbits for some PDE's in higher dimensions,Commun.Pure Appl.Anal.,1,269-279(2002)
[9]D.Bambusi,Birkhoff normal form for some nonlinear PDEs.Commun.Math.Phys.,234,253-285(2003)
[10]D.Bambusi,An averaging theorem for quasilinear Hamiltonian PDEs,Ann.Henri Poincare,4,685-712(2003)
[11]D.Bambusi and B.Grebert,Birkhoff normal form for PDEs with Tame modulus,Duke Math.J.,135(3),507-567(2006)
[12]D.Bambusi,J.M.Delort and B.Grebert,Almost global existence for Hamiltonian semilinear Klein-Gordon equations with small Cauchy data on zoll manifolds,Commun.Pure Appl.Math.,60,1665-1690(2007)
[13] D. Bambusi and C. Bardelle, Invariant tori for commuting Hamiltonian PDEs, J. Diff. Eq., 246, 2484-2505 (2009)
[14] M. Berti and D. Bambusi, A Birkhoff-Lewis type theorem for some Hamiltonian PDEs, SIAM. J. Math. Anal., 37(1), 83-102 (2005)
[15] M. Berti and P. Bolle, Cantor families of periodic solutions for completely resonant nonlinear wave equations, Duke Math. J., 134(2), 359-419 (2006)
[16] A. I. Bobenko and S. B. Kuksin, The nonlinear Klein-Gordon equation on an interval as pertured sine-Gordon equation, Comment. Math. Helv., 70,63-112 (1995)
[17] N. N. Bogoljubov, Ju. A. Mitropoliskii, and A. M. Samoilenko, Methods of accelerated convergence in nonlinear mechanics, Springer-Verlag, New York, 1976; [Russian Original: Naukova Dumka, Kiev, 1969].
[18] J. Bourgain, Construction of quasi-periodic solutions for Hamiltonian perturbations of linear equations and application to nonlinear pde, Int. Math. Research Notice, 11,475-497 (1994)
[19] J. Bourgain, On Melnikov's persistence problem. Math. Res. Lett., 4,445-458 (1997)
[20] J. Bourgain, Quasi-periodic solutions of Hamiltonian perturbations for 2D linear Schrodinger equation, Ann. Math., 148,363-439 (1998)
[21] J. Bourgain, Periodic solutions of nonlinear wave equations, Harmonic analysis and partial equations, Chicago IL: Chicago Univ. Press, 69-97 (1999)
[22] J. Bourgain, Green's function estimates for lattice Schrodinger operators and applications, Annals of mathematics studies, 158 Princeton, NJ: Princeton University Press, 2005
[23] S. Chen and X. Yuan, A KAM theorem for reversible systems of infinite dimension, Acta Mathe-matica Sinica-English Series, 23,1777-1796 (2007)
[24] L. Chierchia and J. You, KAM tori for 1D nonlinear wave equations with periodic boundary conditions, Commun. Math. Phys., 211,497-525 (2000)
[25] K. W. Chung and X. Yuan, Periodic and quasi-periodic solutions for the complex Ginzburg-Landau equation, Nonlinearity, 21,435-451 (2008)
[26] W. Craig and C. Wayne, Newton's method and periodic solutions of nonlinear wave equation, Commun. Pure Appl. Math., 46,1409-1498 (1993)
[27]T.Ding and F.Zanolin,Periodic solutions and subharmonic solutions for a class of planar systems of Lotka-Volterra Type,Proceedings for the first world congress of nonlinear analysis,(Lakshmikantham,V.,ed.),19-26,Tampa,Florid,August,1992
[28]R.C.Diprima,W.Eckhans and L.A.Segel,Non-linear wave-number interaction in near-critical two-dimensional flows,J.Fluid.Mech.,49,705-744(1971)
[291 L.Du and X.Yuan,Invariant tori of nonlinear Schrodinger equations with a given potential,Dynamics of Partial Differential Equations,3,331-346(2006)
[30]L.H.Eliasson,Perturbations of stable invailant toil.Ann.So.Norm.Super.Pisa,15,115-147(1988)
[31]L.H.Eliasson,Normal forms for Hamiltonian systems with Poisson commuting integrals—Elliptic case,Comment.Math.Helv.65,4-35(1990)
[32]L.H.Eliasson and S.B.Kuksin,Infinite Toplitz-Lipschitz matrices and operators,Zeit.fur Ange.Math.and Phy.,59,24-50(2008)
[33]J.Frohlich,and T.Spencer,Absence of diffusion in the Anderson tight binding model for large disorder or lower energy,Commun.Math.Phys.,88,151-184(1983)
[34]J.Geng and J.You,A KAM theorem for Hamiltonian partial differential equations in higher dimensional spaces,Commun.Math.Phys.,262,343-372(2006)
[35]G.Gentile,V.Mastropietro and M.Procesi,Periodic solutions for completely resonant nonlinear wave equations,Commun.Math.Phys.,256,437-490(2005)
[36]V.L.Ginzburg and L.D.Landau,On the theory of superconductivity.Zh.Eksp.Teor.Fiz.(USSR),20,1064(1950)
[37]Hausrath,R.Alan and Raul F.Manasevich,Periodic solutions of a periodically perturbed Lotka-Volterra equation using the Poincare-Birkhoff theorem,J.Math.Anal.Appl.,157,1-9(1991)
[38]T.Kappeler and J.P(o|¨)schel,KdV & KAM,Berlin:Springer,2003
[39]K.Kato,Perturbation Theory for Linear Operators.(Corrected printing of the second edition) Berlin Heidelberg New York:Springer-Verlag,1980
[40]A.N.Kolmogorov,On the conservation of conditionally periodic motions for a small change in Hamilton's function Dokl.Akad.Nauk.SSSR.98,527-530(1954)
[41] S. B. Kuksin, Hamiltonian perturbations of infinite dimensional linear system with an imaginary spectrum. Funct. Anal. Appl., 21,192-205 (1987)
[42] S. B. Kuksin, Nearly integreble infinite dimensional Hamiltonian system, Lecture Notes in Math., 1556, New york: Springer-Veriag, 1993
[43] S. B. Kuksin and J. Poschel, Invariant cantor manifolds of quasi-periodic oscillations for a nonlinear Schrodinger equation, Ann. Math., 143,149-179 (1996)
[44] S. B. Kuksin, Elements of a qualitative theory of Hamiltonian PDEs, Doc. Math. J. DMV (Extra Volume ICM) 2,819-829 (1998)
[45] S. B. Kuksin, Analysis of Hamiltonian PDEs, Oxford Univ. Press, Oxford, 2000.
[46] S. B. Kuksin and A. L. Piatnitski, Khasminskii-Whitham averaging for randomly perturbed KdV equation, J. de Mathematiques Pures et Appliquees, 89,400-428 (2008)
[47] Z. Liang, Quasi-periodic solutions for 1D Schrodinger equation with the nonlinearity |u|~(2p)u, J. Diff. Eq., 244,2185-2225 (2008)
[48] B. P. Luce, Homoclinic explosions in the complex Ginzburg-Landau equation, Physica D, 84,553- 581 (1995)
[49] L. Liu, Z. Lu and D. Wang, The Structure of LaSalle's Invariant Set of Lotka-Volterra Systems, Science in China: Ser. A, 34(7): 783-790 (1991)
[50] C. D. Levermore and M. Olive, The complex Ginzburg-Landau Equation as a model Problem (Lectures in Applied Mathematics vol 31)ed P Deift et al (Providence, RI: American Mathematical Society 1996) 141-189
[51] B. V. Lidskij and E. Shulman, Periodic solutions of the equation u_(tt) - u_(xx) + u~3 = 0. Funct. Anal. Appl., 22,332-333 (1988)
[52] P. Lochak, Hamiltonian perturbation theory: periodic orbits, resonances and intermittency. Nonlinearity, 6, 885-904 (1993)
[53] P. Lochak and A. Neishtadt, Estimates of stability time for nearly integrable systems with a quasi-convex Hamiltonian. Chaos, 2,495-499 (1992)
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