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(Stud Appl Math 207:1\u201320, 2018), Favrie and Gavrilyuk (Nonlinearity 30:2718\u20132736, 2017). We consider the hyperbolic reformulations of two different nonlinear dispersive systems, namely the Serre\u2013Green\u2013Naghdi model of dispersive water waves and the defocusing nonlinear Schr\u00f6dinger equation. The first order hyperbolic reformulation of the Schr\u00f6dinger equation is endowed with a curl involution constraint that needs to be properly accounted for in multiple space dimensions. We show that the original model proposed in Dhaouadi et al. (2018) is only<jats:italic>weakly hyperbolic<\/jats:italic>in the multi-dimensional case and that strong hyperbolicity can be restored at the aid of a novel thermodynamically compatible GLM curl cleaning approach that accounts for the curl involution constraint in the PDE system. We show one and two-dimensional numerical results applied to both systems and compare them with available exact, numerical and experimental reference solutions whenever possible.<\/jats:p>","DOI":"10.1007\/s10915-021-01429-8","type":"journal-article","created":{"date-parts":[[2021,3,25]],"date-time":"2021-03-25T23:40:09Z","timestamp":1616715609000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":54,"title":["On High Order ADER Discontinuous Galerkin Schemes for First Order Hyperbolic Reformulations of Nonlinear Dispersive Systems"],"prefix":"10.1007","volume":"87","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-6509-4269","authenticated-orcid":false,"given":"Saray","family":"Busto","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8201-8372","authenticated-orcid":false,"given":"Michael","family":"Dumbser","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7602-5920","authenticated-orcid":false,"given":"Cipriano","family":"Escalante","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3788-2927","authenticated-orcid":false,"given":"Nicolas","family":"Favrie","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4605-8104","authenticated-orcid":false,"given":"Sergey","family":"Gavrilyuk","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2021,3,25]]},"reference":[{"key":"1429_CR1","unstructured":"Balsara, D., K\u00e4ppeli, R., Boscheri, W., Dumbser, M.: Curl constraint-preserving reconstruction and the guidance it gives for mimetic scheme design. 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