发表机构
Center for Mathematics and Interdisciplinary Sciences, Fudan University; Shanghai Institute for Mathematics and Interdisciplinary Sciences（复旦大学数学与交叉科学研究院; 上海数学与交叉科学研究院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出一种交替数值格式，构造有界扰动下非线性PDE（如NLS和NLW方程）的拟周期解，通过共振Q方程建立振幅与频率的微分同胚，并处理空间指标带来的测度与奇异集困难，数值实验验证了行波与驻波解的收敛性。
AI 中文摘要
本文提出了一种交替数值格式，用于构造一类具有有界扰动的非线性偏微分方程（PDEs）的拟周期解，包括在周期和Dirichlet边界条件下的一维非线性薛定谔（NLS）方程和非线性波（NLW）方程。关于共振$Q$-方程的一个关键观察使我们能够直接建立振幅与漂移频率之间的局部微分同胚。这使得证明更加简洁，因为它避免了经典KAM理论中所需的Birkhoff正规形及相关的坐标变换。然而在有限维设置中，格点盒子还必须包含空间指标，因此多尺度归纳必须在一个额外的维度中进行。这引入了两个困难。首先，对“坏”频率排除集的测度估计必须考虑空间指标。对于NLS和NLW方程，线性频率及其差值都是整数或接近整数；这使得空间依赖性可以通过时间指标的模来控制，从而排除的测度保持可控。其次，一个奇异集可能包含具有相同时间指标但空间指标相距很远的点，这阻碍了“反演蕴含局部化”的过程。然而，对于一维NLS和NLW方程，这样的点仅以具有相反空间指标的成对形式出现。因此，它们可以通过小盒子分离，之后通过预解恒等式从局部逆获得全局逆。数值实验展示了该格式用于非线性行波和驻波，既展示了计算出的拟周期解，也展示了迭代的收敛性。
英文摘要
In this paper, we propose an alternating numerical scheme for constructing quasi-periodic solutions to a class of nonlinear partial differential equations (PDEs) with bounded perturbations, including the one-dimensional nonlinear Schrödinger (NLS) equations and nonlinear wave (NLW) equations under periodic and Dirichlet boundary conditions. A key observation concerning the resonant $Q$-equations allows us to establish directly a local diffeomorphism between the amplitudes and the drifted frequencies. This renders the proof considerably cleaner, since it avoids the Birkhoff normal form and the associated coordinate transformations required in classical KAM theory for PDEs.Unlike in the finite-dimensional setting, the lattice boxes must also include the spatial index, so that the multiscale induction has to be carried out in one additional dimension. This introduces two difficulties. First, the measure estimate for the excluded set of ``bad'' frequencies must account for the spatial index. For both the NLS and NLW equations, the linear frequencies and their differences are integers or close to integers; this allows the spatial dependence to be controlled through the modulus of the temporal index, so that the excluded measure remains under control. Second, a singular set may contain sites with the same temporal index but widely separated spatial indices, which obstructs the ``inversion implies localization'' process. For the one-dimensional NLS and NLW equations, however, such sites occur only in pairs with opposite spatial indices. They can therefore be separated by small boxes, after which the global inverse is obtained from the local inverses via the resolvent identity. Numerical experiments illustrate the scheme for nonlinear traveling and standing waves, exhibiting both the computed quasi-periodic solutions and the convergence of the iteration.
Comments58 pages, 20 figures
