非自治的非线性Schr(?)dinger方程Dirichlet边值问题
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摘要
本文主要利用Hamiltonian系统中经典的KAM理论，对非自治的非线性Schr(?)dinger方程的Dirichlet边值问题进行了较为深入的研究．文中通过适当的变换u=εv(0＜ε＜＜1)，有效地处理了方程中出现的小除数问题，建立了非自治的非线性Schr(?)dinger方程的Hamiltonian函数，给出了该方程的Birkhoff标准形式及相应的Cantor流形定理，较好的解决了非自治的非线性Schr(?)dinger方程的狄氏边值问题．
Schrodinger Equtions is a classics of partial differential equations.We achieve many scientific result for study of its quantity theoty.But when we looking for the quasi-periodic solutions of d-dimension partial differential equation(d> 2),we have to meet a small divison problems. The Hamiltonian systems and KAM theory is a very powerful tool to deal with the problems. The classical KAM theory is concerned with the existence of invariant tori (thus quasi-periodic solutions) for nearly integrable Hamiltonian systems. In order to obtain the quasi-periodic solutions of a partial differential equation, one may show the existence of the lower(finite) dimensional invariant tori for the infinite dimensional Hamiltonian systems defined by the partial differential equation.
The paper research the Dirichlet boundary value problem for nonlinear schrodinger equtions of nonautonomous the followingwhere nonlinear function f is required to be real analytic and nondegenerate in some neighbourhood of the origin in C.
Thus equation (1) is expressed the following form
We use the transformation u =εv(0 <ε<< 1) and desert the higher order term of v,the equation (2) is expressed the following form
After that, To research the dirichlet boundary value problem for (1) change to looking for the quasi-periodic solution of problem (4).
The frist, the equation (3) is replaced by the equation (5)
Step 2, We gived the Hamiltonian function (6) of the equation (5) where A
Step 3, We obtain the Birkhoff normal form for the Hamiltonian function (6) is expressed the following3.4.
Lemma 3.4 For the Hamiltonian H =εG with nonlinearity ,there exists a real analytic symplectic change of coordinates F in a neighbourhood of the origin in l~(a,p) that for all real values of m takes it into its Birkhoff normal form up to order four. That iswhere m determinid coefficients
Step 4, We established the Cantor manifold theorem .
For the existence ofεthe following assumptions are made.
(1) Nondegeneracy.The normal form A + Q' is nondegeneracy in the sense thatwhere
(2) Spectral Asymptotics.There exist d≥1,δ< d - 1,such thatwhere the dots stand for terms of order less than d in j.
(3) Regularity.where A(l~(a,p), l~(a,p)) denotes the class of all maps from some neighbourhood of the origin in l~(a,p) into l~(a,p). which are real analytic in the real and imaginary parts of the complex coordinate q.
Theorem3.2(Cantor manifold theorem) Suppose the HamiltonianH = A + Q' + R satisfies assumptions(1)-(3), andwith g > ,Δ= min(p - p, 1). Then there exist a Cantor manifoldεof real analytic,elliptic diophantine n-tori given by a Lipschitz continuous embedding ,where C has full density at the origin,and (?) is close to the inclusion map (?)_0
Consequently,εis tangent to E at the origin .
Finally,We obtain the important conclusion of the paper by using the Lemma 3.4 and Cantor manifold theorem
Theorem3.1 Suppose the nonlinearity f is real analytic and nonde-gence,Then for all m∈M, n∈N, and all J = {j_1 <…< j_n} (?) N, there exist a Cantor manifoldε_J of real analytic,linearly stable, diophantine-n tori for the equation (1) given by a Lipschitz continuous embedding .which is a higher order perturbation of the inclusion map restricted to T_J[C].The Cantor set C has full density at the origin, whenceε_J has a tangent space at the origin equal to E_J. Moreover,ε_J is contained in the space of analytic functions on [0,π].
Schrodinger Equtions is a classics of partial differential equations.We achieve many scientific result for study of its quantity theoty.But when we looking for the quasi-periodic solutions of d-dimension partial differential equation(d> 2),we have to meet a small divison problems. The Hamiltonian systems and KAM theory is a very powerful tool to deal with the problems. The classical KAM theory is concerned with the existence of invariant tori (thus quasi-periodic solutions) for nearly integrable Hamiltonian systems. In order to obtain the quasi-periodic solutions of a partial differential equation, one may show the existence of the lower(finite) dimensional invariant tori for the infinite dimensional Hamiltonian systems defined by the partial differential equation.
The paper research the Dirichlet boundary value problem for nonlinear schrodinger equtions of nonautonomous the followingwhere nonlinear function f is required to be real analytic and nondegenerate in some neighbourhood of the origin in C.
Thus equation (1) is expressed the following form
We use the transformation u =εv(0 <ε<< 1) and desert the higher order term of v,the equation (2) is expressed the following form
After that, To research the dirichlet boundary value problem for (1) change to looking for the quasi-periodic solution of problem (4).
The frist, the equation (3) is replaced by the equation (5)
Step 2, We gived the Hamiltonian function (6) of the equation (5) where A
Step 3, We obtain the Birkhoff normal form for the Hamiltonian function (6) is expressed the following3.4.
Lemma 3.4 For the Hamiltonian H =εG with nonlinearity ,there exists a real analytic symplectic change of coordinates F in a neighbourhood of the origin in l~(a,p) that for all real values of m takes it into its Birkhoff normal form up to order four. That iswhere m determinid coefficients
Step 4, We established the Cantor manifold theorem .
For the existence ofεthe following assumptions are made.
(1) Nondegeneracy.The normal form A + Q' is nondegeneracy in the sense thatwhere
(2) Spectral Asymptotics.There exist d≥1,δ< d - 1,such thatwhere the dots stand for terms of order less than d in j.
(3) Regularity.where A(l~(a,p), l~(a,p)) denotes the class of all maps from some neighbourhood of the origin in l~(a,p) into l~(a,p). which are real analytic in the real and imaginary parts of the complex coordinate q.
Theorem3.2(Cantor manifold theorem) Suppose the HamiltonianH = A + Q' + R satisfies assumptions(1)-(3), andwith g > ,Δ= min(p - p, 1). Then there exist a Cantor manifoldεof real analytic,elliptic diophantine n-tori given by a Lipschitz continuous embedding ,where C has full density at the origin,and (?) is close to the inclusion map (?)_0
Consequently,εis tangent to E at the origin .
Finally,We obtain the important conclusion of the paper by using the Lemma 3.4 and Cantor manifold theorem
Theorem3.1 Suppose the nonlinearity f is real analytic and nonde-gence,Then for all m∈M, n∈N, and all J = {j_1 <…< j_n} (?) N, there exist a Cantor manifoldε_J of real analytic,linearly stable, diophantine-n tori for the equation (1) given by a Lipschitz continuous embedding .which is a higher order perturbation of the inclusion map restricted to T_J[C].The Cantor set C has full density at the origin, whenceε_J has a tangent space at the origin equal to E_J. Moreover,ε_J is contained in the space of analytic functions on [0,π].
引文
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[4] H.Federer,Geometric measure theory,grundlehren 153,Springer,Heidelberg, 1969.
[5] Z.Liang,J.You,Quasiperiodic solutions for 1D Schrodinger equation with higher order nonlinearity,in press,SIAM,J.Math.Anal.(2005).
[6] A.I.Bobenko and S.B.Kuksin,The nonlinear Klein-Gordon equation on an interval as a perturbed sine-Gordon equation,Comm.Math.Helv.70(1995), 63-112.
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[8]W.Craig and C.E.Wayne,Newton's method and periodic solutions of nonlinear wave equations,Commun.Pure Appl.Math.46 (1993), 1409-1498.
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[12]T.Nishida.A note on a theorem of Nirenberg,J.Diff.Geom.12(1977),629-633.
[13]S.B.Kuksin,Nearly integrable infinite dimensional Hamiltonian systems,Lecture Notes in Math.1556,Berlin:Springer,1993.
[14]H.Ito,Convergence of Birkhoff normal forms for integrable systems,Comm.Math.Helvetici 64(1989),412-461.
[15]J.Poschel,On elliptic lower-dimensional tori in Hamiltonian systems, Math.Z.202(1989),559-608.
[16]S.B.Kuksin and J.Poschel,Invariant Cantor manifolds of quasiperiodic oscillations for a nonlinear Schrodinger equation, Ann.of Math.142(1995),149-179.
[17]S.B.Kuksin,A KAM-theorem for equations of the Korteweg-de Veries type,Rev.Math.Phys. 10(1998),1-64.
[18]M.Herman, Sur les courbes invariants par les diffeomorphisms de I'anneau,Asterisque 103(1983),1-221.
[19]T.Kappeler, J. Poschel, Kdv and KAM, Springer, to appear,2003.
[20]G.L.Siegel and J.K.Moser,Lectures an celestial mechanics,Grundlehren 187,Springer,Berlin, 1971.
[21]I.M.Sigal,Nonlinear wave and Schrodinger equations I.Instability of periodic and quasiperiodic solutions.Comm.Math.Phys.153(1993),297-320.
[22]J.Poschel,Quasi-periodic solutions for nonlinear wave equations,Comm. Math.Helv.71 (1996),269-296.
[23]J.Xu,J.You,Q.Qiu,A KAM theorem of degenerate infinite dimensional Hamiltonian systems(Ⅰ)(Ⅱ),Science in China(Series A)Vol.39 No.4(1996),372-394.
[24]C.E.Wayne,Periodic and quasi-periodicsolutions of nonlinear wave equations via KAM theory, Comm.Math.Phys. 127(1990),479-528.
[25]J.Moser,On inveriant curves of area preserving mappings of an annulus, Nachr.Akad.Wiss.Gott.,Math.Phys.Kl. (1962), 1-20.
[26] J.Bourgain Construction of quasi-periodic solutions for Hamiltonian perturbations of linear equation and applications to nonlinear PDE;Inter.Math.Res. Not.11(1994),475-497.
[27] D.Bernstein,A.Katok,Birkhoff periodic orbits for samll perturbations of completely integrable Hamiltonian systems with conver Hamiltonian, Invent.Math.88(1987),225-241.
[28] D.Bambusi,On Lyapunov center theorem for some nonlinear PDEs:Asimple proof,Ann.Sc.Norm.Sup.Pisa CL.Sci.29(2000),823-837.
[29] D.Bambusi,S.Paleari,Families of periodic solutions of resonant PDEs, J. Nonlinear Sci.,11(2001),69-87.
[30] J.You,Perturbations of lower dimensional tori for Hamiltonian systems, J.Diff.Eq.152(1999),1-29.
[31] J.Geng,J.You,KAM tori of Hamiltonian perturbations of 1D linear beam equations,J.Math.Anal.Appl.277(2003),104-121.
[32] M.B.Sevryuk,The classical KAM theory at the dawn of the twenty-first century,Moscow Math.J,2003.
[33] H.Riissmann,Invariant tori in non-gegenerate nearly integrable Hamiltonian systems,Reg.Chaot.Dynamics 6(2001),119-204.
[34] F. H.Clarke,Periodic Solutions to Hamiltonian Inclusions, J. Diff. Equa., 40(1981),1-6.
[35] J.Xu,J.You,Q.Qiu,Invariant tori for nearly integrable Hamiltonian systems with degeneracy,Math.Z.,226(1997),375-387.
[36] M.Berti,P.Bolle,Multiplicity of periodic solutions of nonlinear wave equa-tions,Nonlinear Analysis 57/7(2004),1011-1046.
[37] J.Bourgain, On Melnikov's persistncy problem,Math.Res.Lett.4(1997),445-458.
[38] A.I.Bobenko and S.B.Kuksin,The nonlinear Klein-Gordon equation on an interval as a perturbed sine-Gordon equation,Comm.Math.Helv.70 (1995),63-112.
[39] M.Berti,P.Bolle,Cantor families of periodic solutions for completely resonant nonlinear wave equations,preprint,2004.
[40] J.Bourgain,Quasi-periodic solutions of Hamiltonian perturbations of 2D linear Schrodinger equations,Annals of Math.l48(1998),363-439.
[41] J.Bourgain,Green's function estimates for lattice Schrodinger operators and applications,Annals of Math.Studies 158,Princeton Universi ty Press,Princeton,NJ,2005.
[42] J.Bourgain,Nonlinear Schrodinger equations,Park city series 5,American Math.Soc.(Providence,Rhode Island 1999).
[43] J.Mawhin, Periodic Solutions of Nonlinear Functional Differential Equations, J. Diff.Equa., 10(1971),240-261.
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