周期三次NLS方程Strang分裂方法的时间误差增长
Temporal Error Growth of Strang Splitting Method for the Periodic Cubic NLS
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- School of Mathematics and Statistics, Xi’an Jiaotong University（西安交通大学数学与统计学院）
- School of Mathematical Sciences, Nanjing Normal University（南京师范大学数学科学学院）
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中文总结 AI 辅助
研究Strang分裂法求解周期三次NLS方程的时间误差增长，揭示误差增长依赖于维数和非线性符号，并给出散焦与聚焦情形下的上界、下界及数值验证。
中文摘要 AI 辅助
我们研究了周期三次非线性薛定谔（NLS）方程的Strang分裂方法的时间误差增长。主导误差由一个强迫线性化方程控制，其增长强烈依赖于维数和非线性项的符号。在一维散焦情形下，利用全局Birkhoff变换以及线性化流的退化结构，我们证明了均匀二次上界$C(1+T^2)\tau^2$。在高维散焦情形下，通过构造任意小的不稳定驻波，我们得到了指数增长。此外，为证明高维散焦情形的上界，我们使用Killip和Vişan的周期Strichartz估计来证明$\int_0^T\\|u(t)\\|_{L^\infty}^2\\,dt\le C(u_0)(1+T)$。尽管高次Sobolev范数可能增长，这仍给出了与最终时间无关的指数速率。聚焦情形下的指数下界通过平面波周围的不稳定线性化动力学获得，其中Akhmediev呼吸子提供了潜在机制。另外，对于具有小初值的一维聚焦情形，局部Birkhoff变换的应用确保了误差至多随时间二次增长。各种数值实验证实了尖锐的线性、二次和指数误差增长率。
英文摘要
We study the temporal error growth of the Strang splitting method for the periodic cubic nonlinear Schrödinger (NLS) equation. The leading error is governed by a forced linearized equation, whose growth depends sharply on dimension and the sign of the nonlinearity. In the 1D defocusing case, we prove a uniform quadratic upper bound $C(1+T^2)τ^2$ using the global Birkhoff transformation and the resulting degenerate structure of the linearized flow. In the higher-dimensional defocusing case, the exponential growth is constructed using arbitrarily small unstable standing waves. Moreover, to prove the higher-dimensional defocusing upper bound, we use the periodic Strichartz estimates of Killip and Vişan to show that $\int_0^T\|u(t)\|_{L^\infty}^2\,dt\le C(u_0)(1+T)$. This yields an exponential rate independent of the final time, despite possible growth of higher Sobolev norms. The exponential lower bound in the focusing case is obtained from the unstable linearized dynamics around a plane wave, with the Akhmediev breather providing the underlying mechanism. In addition, for 1D focusing case with small initial data, the appliction of the local Birkhoff transformation ensures that the error grows at most quadratically in time. Various numerical experiments confirm the sharp linear, quadratic, and exponential error growth rates.
