AI 中文总结
本文针对中间长波方程等一类偏微分方程，提出结合Birkhoff坐标的新型分裂方法，改善了正则性要求，可高效精确进行长时间模拟，并数值探究了两类方程的孤子分辨率猜想。
AI 中文摘要
近年来，新颖的可积性技术，尤其是Birkhoff坐标结合显式公式，已被引入用于求解Benjamin--Ono（BO）方程，取得了重要的理论和计算进展。我们提出通过一种新型分裂方法对该显式BO公式进行摄动，以数值求解一类偏微分方程，这类方程包括中间长波（ILW）方程以及其他不一定可积的拟线性方程。利用BO的Birkhoff坐标，我们在H^s范数（s≥0）下建立了一阶收敛性，同时初始数据u₀∈H^{s+1}_0仅需额外一个导数，显著改善了非线性色散方程典型分裂方法的正则性要求。此外，我们证明在深水极限下，ILW方程的格式收敛到BO解。数值模拟显示了这些格式的计算优势：与经典分裂方法不同，我们的方法无需严格的二次时间步长条件即可在长时间尺度上近似保持能量，从而实现高效且精确的长时间模拟。作为应用，我们对ILW方程和KdV--BO方程的孤子分辨率猜想进行了数值探究。
英文摘要
Recently, novel integrability techniques, and most notably Birkhoff coordinates followed by an explicit formula, have been introduced to solve the Benjamin--Ono (BO) equation, leading to major theoretical and computational advances. We propose to perturb this explicit BO formula via a novel splitting method for numerically solving a class of PDEs, which includes the Intermediate Long Wave (ILW) equation, as well as other quasilinear equations that are not necessarily integrable. Using the Birkhoff coordinates of BO we establish first-order convergence in the $H^s$-norm ($s \ge 0$) while requiring only one additional derivative on the initial data $u_0 \in H^{s+1}_{0}$, significantly improving upon the regularity requirements of typical splitting methods for nonlinear dispersive equations. Furthermore, we prove that in the deep-water limit, the scheme for the ILW equation converges to the BO solution. Computational advantages of these schemes are shown in simulations: unlike classical splitting methods, our approach does not require a restrictive quadratic time step condition to nearly preserve energy over long time scales, thus rendering efficient and accurate long-time simulations feasible. As an application, we numerically explore the soliton resolution conjecture for both the ILW and a KdV--BO equation.
