Recent years have seen an increasing amount of research devoted to the development of so-called resonance-based methods for dispersive nonlinear partial differential equations. In many situations, this new class of methods allows for approximations in a much more general setting (e.g. for rough data) than, for instance, classical splitting or exponential integrator methods. However, they lack one important property: the preservation of geometric properties of the flow. This is particularly drastic in the case of the Korteweg--de Vries (KdV) equation and the nonlinear Schr\"odinger equation (NLSE) which are fundamental models in the broad field of dispersive infinite-dimensional Hamiltonian systems, possessing infinitely many conserved quantities, an important property which we wish to capture - at least up to some degree - also on the discrete level. Nowadays, a wide range of structure preserving integrators for Hamiltonian systems are available, however, typically these existing algorithms can only approximate highly regular solutions efficiently. State-of-the-art low-regularity integrators, on the other hand, poorly preserve the geometric structure of the underlying PDE. In this work we introduce a novel framework, so-called Runge-Kutta resonance-based methods, which are able to bridge the gap between low regularity and structure preservation in the KdV and NLSE case. In particular, we are able to characterise a large class of symplectic (in the Hamiltonian picture) resonance-based methods for both equations that allow for low-regularity approximations to the solution while preserving the underlying geometric structure of the continuous problem on the discrete level.

翻译：近年来，针对色散非线性偏微分方程的所谓共振类方法研究日益增多。在许多情形下，这类新方法允许在比经典分裂法或指数积分器方法更广泛的框架（如处理粗糙数据）中进行近似求解。然而，它们缺乏一个关键特性：对流动几何性质的保持。这一点在Korteweg-de Vries（KdV）方程和非线性薛定谔方程（NLSE）中尤为突出——这两个色散无穷维哈密顿系统领域的核心模型拥有无穷多个守恒量，这一重要性质我们希望在离散层面（至少在一定程度上）得以保留。当前学界已发展出丰富的哈密顿系统保结构积分器，但现有算法通常仅能高效近似高度正则解；而前沿的低正则性积分器却难以有效保持原偏微分方程的几何结构。本研究提出名为"Runge-Kutta共振法"的新框架，成功弥合了KdV与NLSE情形中低正则性与结构保持之间的鸿沟。特别地，我们刻画了两类方程的辛（哈密顿框架下）共振法大家族，这些方法既能实现解的低正则性近似，又能在离散层面保持原连续问题的几何结构。