On invariant tori of full dimension for 1D periodic NLS
On invariant tori of full dimension for 1D periodic NLS
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DOI：
10.1016/j.jfa.2004.10.019
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发表时间：
2005-12
影响因子：
1.7
通讯作者：
J. Bourgain
中科院分区：
文献类型：
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作者：
J. Bourgain
考虑一维周期边界条件下的非线性最小二乘问题，其中M是定义为的随机Fourier乘子，（Vn）n∈ Z在[-1，1]中独立选取.（0.1）中的五次非线性不重要，可以用u| u| p-2，p∈2Z，p <$4.我们证明以下事实。定理.对于适当的M，（0.1）有一个（全维）不变量torIT满足（r> 0是任意的）。备注。这个陈述事实上对大多数（Vn）n∈Z∈[-1，1]Z成立，尽管这里没有明确证明。定理1的证明将沿着“通常的”KAM方案进行，其中扰动最终通过相空间的连续正则变换被去除。在无限维相空间的本上下文中最相关的文献是Fröhlich等人的论文[Fröhlich，Spencer，韦恩，无序非线性动力系统中的局部化，J. Statist. 42（1986）247-274]，尤其是Pöschel [Pöschel，无限维Hamilton系统中具有空间结构的小因子，CMP 127（1990）351-393]关于无序系统。[Fröhlich，Spencer，韦恩，无序非线性动力系统中的局部化，J.统计学。42（1986）247-274，Pöschel，Small divisor with spatial structure in infinite dimensional Hamiltonian systems，CMP 127（1990）351-393]考虑了具有短程相互作用的Hamiltonian，因此这些结果不适用于我们的问题。事实证明，然而，该计划，详细阐述了[Pöschel，小因子与空间结构在无限维哈密顿系统，CMP 127（1990）351-393]，仍然适用于（0.3），由于特殊的算术功能，将在下一节中解释。粗略地说，关键点是下面的观察。设（ni）是有限模集，|N1| ⩾| N2|在“近”共振的情况下，也有一个关系，除非n1=n2，然后可以控制|N1| +| N2|从（0.4），（0.5）乘∑j <$3| NJ|.这个特征特别是一维的，我们现在还不知道如何证明定理1的二维模拟，例如考虑三次NLS iut+Δu±u| u| T2上的2=0。还应该指出的是，在早期的工作中，构造了一维NLS和NLW的全组频率上的概周期解（参见[Bourgain，Construction of approximative and almost periodic solutions of perturbed linear Schr dinger and wave equations，GAFA 6（2）（1996）201-230]和[Pöschel，On the construction of almost periodic solutions for nonlinear Schr dinger equations，遍历理论动力系统22（5）（2002）1537-1559]）。这些不变环面（全维的）是通过对有限维环面的连续小扰动而获得的，导致了非常强的紧性，事实上，对于n→∞，作用变量Inf有一个非显式的衰减率。另一方面，本文中的构造（类似于[Pöschel，Small divisor with space structure in infinite dimensional Hamiltonian systems，CMP 127（1990）351-393]）一次处理所有傅立叶模式，并需要明确和现实的衰减条件。式（0.3）中的乘数M=（Vn）被认为是一个参数，式（0.1）中的乘数M =（Vn）被认为是一个依赖于参数的方程。此参数的作用是必不可少的，以确保适当的非共振特性的（调制）频率沿着迭代。在没有外部参数的情况下，这些条件需要通过幅频调制和对动作变量的适当限制来实现。这个问题比较难。事实上，动作变量的快速衰减（增强过程的收敛）允许更少的......
Consider the NLS with periodic boundary conditions in 1Dwhere M is a random Fourier multiplier defined byand (Vn)n∈Zare independently chosen in [-1,1]. The quintic nonlinearity in (0.1) is unimportant and may be replaced by u|u|p-2,p∈2Z,p⩾4. We give a proof of the following fact. Theorem.For appropriateM, (0.1) has an invariant toriT (of full dimension) satisfying(r>0is arbitrary). Remark. The statement holds in fact for most (Vn)n∈Z∈[-1,1]Z, although not explicitly proven here. Written in Fourier modes (qn)n∈Z, the Hamiltonian corresponding to (0.1) is given byThe proof of Theorem 1 will proceed along the ‘usual’ KAM scheme where the perturbation is eventually removed by consecutive canonical transformations of phase space. The most relevant literature in the present context of an infinite dimensional phase space are the papers of Fröhlich et al. [Fröhlich, Spencer, Wayne, Localization in disordered, nonlinear dynamical systems, J. Statist. Phys. 42 (1986) 247–274] and especially Pöschel [Pöschel, Small divisors with spatial structure in infinite dimensional Hamiltonian systems, CMP 127 (1990) 351–393] on disordered systems. Both [Fröhlich, Spencer, Wayne, Localization in disordered, nonlinear dynamical systems, J. Statist. Phys. 42 (1986) 247–274, Pöschel, Small divisors with spatial structure in infinite dimensional Hamiltonian systems, CMP 127 (1990) 351–393] consider Hamiltonians with short-range interactions and hence these results do not apply to our problem. It turns out, however that the scheme, as elaborated on in great detail in [Pöschel, Small divisors with spatial structure in infinite dimensional Hamiltonian systems, CMP 127 (1990) 351–393], is still applicable to (0.3), due to special arithmetical features as will be explained in the next section. Roughly speaking, the key point is the following observation. Let (ni) be a finite set of modes, |n1|⩾|n2|⩾⋯ andIn the case of a ‘near’ resonance, there is also a relationUnless n1=n2, one may then control |n1|+|n2| from (0.4), (0.5) by ∑j⩾3|nj|. This feature is specifically 1-dimensional and we do not know at this time how to prove a 2D-analogue of Theorem 1, considering for instance the cubic NLS iut+Δu±u|u|2=0 on T2. It should also be pointed out that almost periodic solutions on a full set of frequencies for NLS and NLW in 1D were constructed in earlier works (see [Bourgain, Construction of approximative and almost periodic solutions of perturbed linear Schrödinger and wave equations, GAFA 6 (2) (1996) 201–230] and [Pöschel, On the construction of almost periodic solutions for nonlinear Schrödinger equations, Ergodic Theory Dynamical Systems 22 (5) (2002) 1537–1559]). These invariant tori (of full dimension) were obtained by successive small perturbations of finite-dimensional tori, resulting in very strong compactness properties and in fact a nonexplicit decay rate of the action variables Infor n→∞. On the other hand, the construction in this paper (similarly to [Pöschel, Small divisors with spatial structure in infinite dimensional Hamiltonian systems, CMP 127 (1990) 351–393]) treats all Fourier modes at once and requires explicit and realistic decay conditions. The multiplier M=(Vn) in (0.3) is to be considered as a parameter and (0.1) a parameter-dependent equation. The role of this parameter is essential to ensure appropriate nonresonance properties of the (modulated) frequencies along the iteration. In the absence of exterior parameters, these conditions need to be realized from amplitude–frequency modulation and suitable restriction of the action-variables. This problem is harder. Indeed, a fast decay of the action-variables (enhancing convergence of the process) allows less …