无穷维KAM理论在偏微分方程中的应用
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摘要
本文主要研究无穷维KAM理论在偏微分方程中的应用.全文分为四章:第一章是绪论,从第二章到第四章为论文主体部分.
在第二章,我们主要研究在Dirichlet-Neumann边值条件、Neumann边值条件和一般Sturm-Liouville边值条件下的非线性Schr(?)dinger方程.通过深入而细致地分析各种边值条件下Schr(?)dinger算子的谱的形式和渐近特征,给出了谱的分离性.然后得到非线性Schr(?)dinger方程对应Hamilton系统和Birkhoff法形,最后利用Cantor流形定理,得到其不变环面的存在性.在第三章,我们主要研究变系数非线性波动方程.波动方程周期解和拟周期存在性问题是当今数学物理和应用数学领域中非常活跃的研究课题之一.因其与物理等学科的密切联系,在数学和物理等学科中的重要意义与其困难程度,长期以来一直受到国内外众多知名数学家的广泛关注,并得到了很多有意义的研究结果.但是目前已有的结果大多是处理较为理想的常系数波动方程.这种波动方程很好的描述了在均匀介质中传播的波动现象.然而在现实生活中,很多波动现象的传播却是发生在非均匀介质中,例如非均匀弦的振动,在各处具有不同密度和弹性系数的地质层中传播的地震波等.对这些波动现象的描述,具有依赖于空间变量系数的波动方程无疑是一种更加合理的模型.对于这类变系数非线性波动方程,在Dirichlet边值条件、Dirichlet-Neumann边值条件、Neumann边值条件和一般Sturm-Liouville边值条件下,我们通过分析谱的分离性,得到一个解析的辛变换,将波动方程对应的Hamilton系统化为四阶Birkhoff法形形式,最后利用一个无穷维KAM定理得到其拟周期解的存在性和稳定性.在第四章,我们主要研究在Dirichlet边界条件下的高阶波动方程.通过约化其对应的Hamilton系统和Birkhoff法形,利用无穷维的Cantor流形定理得到其拟周期解的存在性和稳定性.
The classical KAM theory which is constructed by three famous mathematicians Kolmogorov. Arnold and Moser in the last century is the landmark of the development of Hamiltonian systems. It made the stability of solar system got reasonable explanation and brought a new method in to the study of Hamiltonian systems. The classical KAM theory which constructed on 2n- dimensional smoothly manifold asserts that under the Kolmogorov non-degenerate condition, the majority of the non-reasont tori of integrable system are persistent under small perturbation. In the later 1980's, the celebrated KAM theory was successfully generalized to infinite dimensional setting by Kuksin [37] and Wayne [58], where the Hamiltonian systems are those with infinite many normal frequencies. Later, P(?)schel[54] restated the result and applied it to the nonlinear Schr(?)dinger equations[38] and nonlinear wave equations[53] under the Dirichlet boundary condition. At first, we restate the infinite dimensional KAM theorem and Cantor manifold theorem by Poschel. We consider a small perturbation of infinite dimensional Hamiltonian in the parameter dependent form in n- dimensional angle-action coordinates (x, y) and infinite-dimensionalCartesian coordinate (u, v), with symplectic structure (?) dv_j, on the phase spacewhere T~n is the usual n- torus with 1≤n <∞. The tangent frequenciesω= (ω_1,…,ω_n) and the normal frequenciesΩ= (Ω_1,Ω_2,…) depend on parameters (?), O is a closed bounded set of positive Lebesgue measure. For each (?), there is an n-torusτ_0~n= T~n×{0,0,0}, with frequenciesω(ξ) = (ω_1(ξ),…,ω_n(ξ)) of the linear integrable Hamiltonian N. In its norm space, described by u-v coordinates the origin is an elliptic fixed point with characteristic frequenciesΩ(ξ). The KAM theorem by Poschel [54] showes that the existence of this linear stable rotational tori under small perturbation P. In order to obtain the result we make some assumptions:
(A1): Non-degeneracy.The real mapξ→ω(ξ) is Lipeomorphism between O and its image. Moreover for all integer vectors (k,l)∈Z~n×Z~∞with1≤|l|≤2,means (?)and〈l,Ω(ξ)〉≠0 on O, where |l| =∑_j |l|_j for integer vectors, and〈·,·〉is the usual scalar product.
(A2): Spectral Asymptotics. There exist d≥1 andδ< d - 1 such thatwhere the dots stands for fixed lower order terms in j, allowing also negative exponents. More precisely, there exists a fixed parameter independent sequence (?) with(?) such that (?) give rise to a Lipschitz map (?), where l_∞~p is the space of all real sequences with finite norm |ω|_p = sup_j |ω_j|j~p.
(A3):Regularity. The perturbation P(x,y,u1v) is real analytic for real argument (x, y, u, v)∈D(r, s) for given r,s > 0, and Lipschitz in the parameterξ∈O. For eachξ∈O, its gradients with respect to z = (?), (?) satisfyfor d＞1;for d= 1,where (?) denotes the class of maps from some neighborhood ofthe origin in l~(a,p) into (?), which is real analytic in the real and imaginary parts of the complex coordinates. To state Poschel 's theorem we assume thatMoreover we introduce the notations (?), A_k = 1+ |k|~T,whereτ> n + 1 is fixed later. Finally, let (?).We now state the basic KAM Theorem which is recited from P(?)schel [54].
Theorem 1 (Infinite Dimensional KAM Theorem ) Suppose Hamiltonian H = N + P satisfies (A1)-(A3)where 0 <α≤1 is another parameter, andγdepends on n,τand s. Then there exist a Cantor set (?), with meas (O_α/O)→0 asα→0, a Lipschitz continuous family of torus embeddingΦ: T~n×O_α→(?), and a Lipschitz continuous map (?), such that for eachξ∈O_α,the mapΦrestricted T~n×{ξ} is a real analytic embedding of rotational torus with frequencies (?)(ξ) for the Hamiltonian H atξ.
Each embedding is analytic on |Imx| < s/2, anduniformly on that domain and O_α, whereΦ_0 : T~ n×O→τ_0~n is the trivial embedding, and c≤γ~(-1) depends on the same parameters as 7.
Moreover there exist Lipschitz mapω_v andΩ_v on O for v≥1 satisfyingω_0 =ω,Ω_0=Ω, andsuch that meas (?) ,whereand the union is taken over all j≥0 and (k, l)∈Z such that |k| > K_02~(j-1) for j≥1 with a constant K_0≥1 depending only on n andτ.
Theorem 2 (The Cantor Manifold Theorem) Suppose the Hamiltonian H = A + Q + R satisfies assumptions (A1)-(A3), andwith Then there exists a Cantor manifoldεof real analytic, elliptic diophantine n- tori given by a Lipschitz continuous embeddingΨ: T[C]→ε, where C has full density at the origin, andΨis close to the inclusion mapΨ_0:Consequently,εis tangent to E at the origin.
In this thesis, by the Infinite KAM Theorem and the Cantor Manifold Theorem, under the Sturm-Liouville boundary conditions, the existence and linear stability of quasi-periodic solutions for nonlinear wave equation with x dependent coefficients, and the nonlinear Schr(?)dinger equation are obtained. Furthermore, we also obtain the existence and linear stability of quasi-periodic solutions for higher order nonlinear wave equation under the Dirichlet boundary condition. More specially,
Theorem 3 Let us consider the nonlinear Schrodinger equationunder the Sturm-Liouville boundary conditionsSuppose the nonlinearity f is real analytic and non-degenerate and V(x) is positive analytic function in the strip domain |Imx| < r with r > 0. Fix d∈N. Then we can find index set J = {j_1<…< j_d} with j_1 large enough to confirm that there exists a Cantor manifoldε_J of real analytic, linearly stable, diophantine d- tori for the equation above given by a Lipschitz continuous embeddingφ: T_J[C]→ε_J. which is a higher order perturbation of the inclusion mapφ_0 : T_J[C]→P 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 toε_J. Moreover,ε_J is contained in the space of analytic functions on [0,π], the Diophantine tori carry quasi-periodic motions of high mode.
Theorem 4 Consider the nonlinear wave equationwith the Sturm-Liouville boundary conditions. Let d be positive integers, N_d = {i_p∈N : p = 1,2.…,d}, K and N be positive constants large enough. Assumemin N_d > NK, max N_d≤C_0dNk, K_1≤|i_p - i_q|≤K_2, for p≠q,where C_0 > 1 is an absolute constant and K_1, K_2 large enough, positive constants depending on K. Then for given compact set C~* in P~d with positive Lebesgue measure, there is a set (?)with meas C > 0, a family of d- toriand a Lipschitz continuous embeddingΦ: T_(N_d)→H_0~1([0,π]), which is a higher order perturbation of the trivial embedding mapΦ_0: E_(N_d)→H_0~1([O,π]) restricted to T_(N_d)(C), such that the restriction ofΦto each T_(N_d) in the family is an embedding of rotational invariant d- torus for the nonlinear wave equation with x dependent coefficients.
Theorem 5 We are going to study the nonlinear higher order wave equationon the finite x- interval [0,π] with Dirichlet boundary conditions Suppose the nonlinearity f = au~3 +∑_(n≥5) f_nu~n, a≠0 is real analytic. Then for all m > 0, we can find index set J = {j_1 <…< j_n} with (?) to confirm that there exists a Cantor manifoldε_Jgiven by a Whitney smooth embeddingΦ: T_J[O]→ε_J, which is a higher order perturbation of the inclusion mapΦ_0: E_J→P restricted to T_J[O]. Moreover, the Cantor manifoldε_J is foliated by real analytic, linearly stable,n- dimensional invariant tori carrying quasi-periodic solutions.
在第二章,我们主要研究在Dirichlet-Neumann边值条件、Neumann边值条件和一般Sturm-Liouville边值条件下的非线性Schr(?)dinger方程.通过深入而细致地分析各种边值条件下Schr(?)dinger算子的谱的形式和渐近特征,给出了谱的分离性.然后得到非线性Schr(?)dinger方程对应Hamilton系统和Birkhoff法形,最后利用Cantor流形定理,得到其不变环面的存在性.在第三章,我们主要研究变系数非线性波动方程.波动方程周期解和拟周期存在性问题是当今数学物理和应用数学领域中非常活跃的研究课题之一.因其与物理等学科的密切联系,在数学和物理等学科中的重要意义与其困难程度,长期以来一直受到国内外众多知名数学家的广泛关注,并得到了很多有意义的研究结果.但是目前已有的结果大多是处理较为理想的常系数波动方程.这种波动方程很好的描述了在均匀介质中传播的波动现象.然而在现实生活中,很多波动现象的传播却是发生在非均匀介质中,例如非均匀弦的振动,在各处具有不同密度和弹性系数的地质层中传播的地震波等.对这些波动现象的描述,具有依赖于空间变量系数的波动方程无疑是一种更加合理的模型.对于这类变系数非线性波动方程,在Dirichlet边值条件、Dirichlet-Neumann边值条件、Neumann边值条件和一般Sturm-Liouville边值条件下,我们通过分析谱的分离性,得到一个解析的辛变换,将波动方程对应的Hamilton系统化为四阶Birkhoff法形形式,最后利用一个无穷维KAM定理得到其拟周期解的存在性和稳定性.在第四章,我们主要研究在Dirichlet边界条件下的高阶波动方程.通过约化其对应的Hamilton系统和Birkhoff法形,利用无穷维的Cantor流形定理得到其拟周期解的存在性和稳定性.
The classical KAM theory which is constructed by three famous mathematicians Kolmogorov. Arnold and Moser in the last century is the landmark of the development of Hamiltonian systems. It made the stability of solar system got reasonable explanation and brought a new method in to the study of Hamiltonian systems. The classical KAM theory which constructed on 2n- dimensional smoothly manifold asserts that under the Kolmogorov non-degenerate condition, the majority of the non-reasont tori of integrable system are persistent under small perturbation. In the later 1980's, the celebrated KAM theory was successfully generalized to infinite dimensional setting by Kuksin [37] and Wayne [58], where the Hamiltonian systems are those with infinite many normal frequencies. Later, P(?)schel[54] restated the result and applied it to the nonlinear Schr(?)dinger equations[38] and nonlinear wave equations[53] under the Dirichlet boundary condition. At first, we restate the infinite dimensional KAM theorem and Cantor manifold theorem by Poschel. We consider a small perturbation of infinite dimensional Hamiltonian in the parameter dependent form in n- dimensional angle-action coordinates (x, y) and infinite-dimensionalCartesian coordinate (u, v), with symplectic structure (?) dv_j, on the phase spacewhere T~n is the usual n- torus with 1≤n <∞. The tangent frequenciesω= (ω_1,…,ω_n) and the normal frequenciesΩ= (Ω_1,Ω_2,…) depend on parameters (?), O is a closed bounded set of positive Lebesgue measure. For each (?), there is an n-torusτ_0~n= T~n×{0,0,0}, with frequenciesω(ξ) = (ω_1(ξ),…,ω_n(ξ)) of the linear integrable Hamiltonian N. In its norm space, described by u-v coordinates the origin is an elliptic fixed point with characteristic frequenciesΩ(ξ). The KAM theorem by Poschel [54] showes that the existence of this linear stable rotational tori under small perturbation P. In order to obtain the result we make some assumptions:
(A1): Non-degeneracy.The real mapξ→ω(ξ) is Lipeomorphism between O and its image. Moreover for all integer vectors (k,l)∈Z~n×Z~∞with1≤|l|≤2,means (?)and〈l,Ω(ξ)〉≠0 on O, where |l| =∑_j |l|_j for integer vectors, and〈·,·〉is the usual scalar product.
(A2): Spectral Asymptotics. There exist d≥1 andδ< d - 1 such thatwhere the dots stands for fixed lower order terms in j, allowing also negative exponents. More precisely, there exists a fixed parameter independent sequence (?) with(?) such that (?) give rise to a Lipschitz map (?), where l_∞~p is the space of all real sequences with finite norm |ω|_p = sup_j |ω_j|j~p.
(A3):Regularity. The perturbation P(x,y,u1v) is real analytic for real argument (x, y, u, v)∈D(r, s) for given r,s > 0, and Lipschitz in the parameterξ∈O. For eachξ∈O, its gradients with respect to z = (?), (?) satisfyfor d＞1;for d= 1,where (?) denotes the class of maps from some neighborhood ofthe origin in l~(a,p) into (?), which is real analytic in the real and imaginary parts of the complex coordinates. To state Poschel 's theorem we assume thatMoreover we introduce the notations (?), A_k = 1+ |k|~T,whereτ> n + 1 is fixed later. Finally, let (?).We now state the basic KAM Theorem which is recited from P(?)schel [54].
Theorem 1 (Infinite Dimensional KAM Theorem ) Suppose Hamiltonian H = N + P satisfies (A1)-(A3)where 0 <α≤1 is another parameter, andγdepends on n,τand s. Then there exist a Cantor set (?), with meas (O_α/O)→0 asα→0, a Lipschitz continuous family of torus embeddingΦ: T~n×O_α→(?), and a Lipschitz continuous map (?), such that for eachξ∈O_α,the mapΦrestricted T~n×{ξ} is a real analytic embedding of rotational torus with frequencies (?)(ξ) for the Hamiltonian H atξ.
Each embedding is analytic on |Imx| < s/2, anduniformly on that domain and O_α, whereΦ_0 : T~ n×O→τ_0~n is the trivial embedding, and c≤γ~(-1) depends on the same parameters as 7.
Moreover there exist Lipschitz mapω_v andΩ_v on O for v≥1 satisfyingω_0 =ω,Ω_0=Ω, andsuch that meas (?) ,whereand the union is taken over all j≥0 and (k, l)∈Z such that |k| > K_02~(j-1) for j≥1 with a constant K_0≥1 depending only on n andτ.
Theorem 2 (The Cantor Manifold Theorem) Suppose the Hamiltonian H = A + Q + R satisfies assumptions (A1)-(A3), andwith Then there exists a Cantor manifoldεof real analytic, elliptic diophantine n- tori given by a Lipschitz continuous embeddingΨ: T[C]→ε, where C has full density at the origin, andΨis close to the inclusion mapΨ_0:Consequently,εis tangent to E at the origin.
In this thesis, by the Infinite KAM Theorem and the Cantor Manifold Theorem, under the Sturm-Liouville boundary conditions, the existence and linear stability of quasi-periodic solutions for nonlinear wave equation with x dependent coefficients, and the nonlinear Schr(?)dinger equation are obtained. Furthermore, we also obtain the existence and linear stability of quasi-periodic solutions for higher order nonlinear wave equation under the Dirichlet boundary condition. More specially,
Theorem 3 Let us consider the nonlinear Schrodinger equationunder the Sturm-Liouville boundary conditionsSuppose the nonlinearity f is real analytic and non-degenerate and V(x) is positive analytic function in the strip domain |Imx| < r with r > 0. Fix d∈N. Then we can find index set J = {j_1<…< j_d} with j_1 large enough to confirm that there exists a Cantor manifoldε_J of real analytic, linearly stable, diophantine d- tori for the equation above given by a Lipschitz continuous embeddingφ: T_J[C]→ε_J. which is a higher order perturbation of the inclusion mapφ_0 : T_J[C]→P 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 toε_J. Moreover,ε_J is contained in the space of analytic functions on [0,π], the Diophantine tori carry quasi-periodic motions of high mode.
Theorem 4 Consider the nonlinear wave equationwith the Sturm-Liouville boundary conditions. Let d be positive integers, N_d = {i_p∈N : p = 1,2.…,d}, K and N be positive constants large enough. Assumemin N_d > NK, max N_d≤C_0dNk, K_1≤|i_p - i_q|≤K_2, for p≠q,where C_0 > 1 is an absolute constant and K_1, K_2 large enough, positive constants depending on K. Then for given compact set C~* in P~d with positive Lebesgue measure, there is a set (?)with meas C > 0, a family of d- toriand a Lipschitz continuous embeddingΦ: T_(N_d)→H_0~1([0,π]), which is a higher order perturbation of the trivial embedding mapΦ_0: E_(N_d)→H_0~1([O,π]) restricted to T_(N_d)(C), such that the restriction ofΦto each T_(N_d) in the family is an embedding of rotational invariant d- torus for the nonlinear wave equation with x dependent coefficients.
Theorem 5 We are going to study the nonlinear higher order wave equationon the finite x- interval [0,π] with Dirichlet boundary conditions Suppose the nonlinearity f = au~3 +∑_(n≥5) f_nu~n, a≠0 is real analytic. Then for all m > 0, we can find index set J = {j_1 <…< j_n} with (?) to confirm that there exists a Cantor manifoldε_Jgiven by a Whitney smooth embeddingΦ: T_J[O]→ε_J, which is a higher order perturbation of the inclusion mapΦ_0: E_J→P restricted to T_J[O]. Moreover, the Cantor manifoldε_J is foliated by real analytic, linearly stable,n- dimensional invariant tori carrying quasi-periodic solutions.
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