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Comp."],"abstract":"<p>\n                    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 (PDEs). 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\n                    <italic>preservation of geometric properties of the flow<\/italic>\n                    . This is particularly drastic in the case of the Korteweg\u2013de Vries (KdV) equation and the nonlinear Schr\u00f6dinger 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\u2013Kutta resonance-based methods, for a large class of dispersive nonlinear equations which incorporate a much larger amount of degrees of freedom than prior resonance-based schemes while featuring similarly favourable low-regularity convergence properties. In particular, for the KdV and NLSE case, we are able to bridge the gap between low regularity and structure preservation by characterising 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.\n                  <\/p>","DOI":"10.1090\/mcom\/4105","type":"journal-article","created":{"date-parts":[[2025,7,8]],"date-time":"2025-07-08T16:17:47Z","timestamp":1751991467000},"page":"2249-2314","source":"Crossref","is-referenced-by-count":3,"title":["Bridging the gap: Symplecticity and low regularity in Runge\u2013Kutta resonance-based schemes"],"prefix":"10.1090","volume":"95","author":[{"given":"Georg","family":"Maierhofer","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Katharina","family":"Schratz","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"14","published-online":{"date-parts":[[2025,7,8]]},"reference":[{"key":"1","series-title":"Cambridge Texts in Applied Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511998324","volume-title":"Nonlinear dispersive waves","author":"Ablowitz, Mark J.","year":"2011","ISBN":"https:\/\/id.crossref.org\/isbn\/9781107664104"},{"issue":"6","key":"2","doi-asserted-by":"publisher","first-page":"3648","DOI":"10.1093\/imanum\/drad093","article-title":"A symmetric low-regularity integrator for the nonlinear Schr\u00f6dinger equation","volume":"44","author":"Alama Bronsard, Yvonne","year":"2024","journal-title":"IMA J. 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