Variational existence theory for hydroelastic solitary waves

Mark D. Groves; Benedikt Hewer; Erik Wahlén

doi:10.1016/j.crma.2016.10.004

Partial differential equations/Calculus of variations

Variational existence theory for hydroelastic solitary waves
[Une théorie variationnelle d'existence d'ondes solitaires hydroélastiques]

Présenté par : the Editorial Board

Mark D. Groves ^1,² ; Benedikt Hewer ¹ ; Erik Wahlén ³

¹ Fachrichtung Mathematik, Universität des Saarlandes, Postfach 151150, 66041 Saarbrücken, Germany
² Department of Mathematical Sciences, Loughborough University, Loughborough, Leics, LE11 3TU, UK
³ Centre for Mathematical Sciences, Lund University, PO Box 118, 22100 Lund, Sweden

Comptes Rendus. Mathématique, Volume 354 (2016) no. 11, pp. 1078-1086.

Résumé
Abstract

Cette note présente une théorie d'existence d'ondes solitaires à l'interface entre une couche de glace mince (modélisée par la théorie des coques hyperélastiques de Cosserat) et un fluide parfait (de profondeur finie et irrotationnel), pour des valeurs suffisamment grandes d'un paramètre sans dimension γ. Nous montrons l'existence d'un minimiseur de l'énergie $E$ de l'onde sous la contrainte $I = 2 μ$ , où $I$ représente l'impulsion horizontale et $0 < μ ≪ 1$ . Nous démontrons que les ondes solitaires trouvées par notre méthode variationnelle convergent (après un changement d'échelle approprié) vers des solutions de l'équation de Schrödinger cubique focalisante, lorsque $μ ↓ 0$ .

This paper presents an existence theory for solitary waves at the interface between a thin ice sheet (modelled using the Cosserat theory of hyperelastic shells) and an ideal fluid (of finite depth and in irrotational motion) for sufficiently large values of a dimensionless parameter γ. We establish the existence of a minimiser of the wave energy $E$ subject to the constraint $I = 2 μ$ , where $I$ is the horizontal impulse and $0 < μ ≪ 1$ , and show that the solitary waves detected by our variational method converge (after an appropriate rescaling) to solutions to the nonlinear Schrödinger equation with cubic focussing nonlinearity as $μ ↓ 0$ .

Reçu le : 2016-04-14
Accepté le : 2016-10-05
Publié le : 2016-10-13

DOI : 10.1016/j.crma.2016.10.004

Affiliations des auteurs :

Mark D. Groves ^{1, 2} ; Benedikt Hewer ¹ ; Erik Wahlén ³

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     author = {Mark D. Groves and Benedikt Hewer and Erik Wahl\'en},
     title = {Variational existence theory for hydroelastic solitary waves},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1078--1086},
     publisher = {Elsevier},
     volume = {354},
     number = {11},
     year = {2016},
     doi = {10.1016/j.crma.2016.10.004},
     language = {en},
}

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Mark D. Groves; Benedikt Hewer; Erik Wahlén. Variational existence theory for hydroelastic solitary waves. Comptes Rendus. Mathématique, Volume 354 (2016) no. 11, pp. 1078-1086. doi : 10.1016/j.crma.2016.10.004. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2016.10.004/

Bibliographie
Cité par

[1] D.M. Ambrose, M. Siegel, Well-posedness of two-dimensional hydroelastic waves, preprint, 2014.

[2] F. Dias; C. Kharif Nonlinear gravity and capillary–gravity waves, Annu. Rev. Fluid Mech., Volume 31 (1999), pp. 301-346

[3] M.D. Groves; E. Wahlén Existence and conditional energetic stability of solitary gravity–capillary water waves with constant vorticity, Proc. R. Soc. Edinb., Sect. A, Volume 145 (2015), pp. 791-883

[4] P. Guyenne; E. Parau Computations of fully nonlinear hydroelastic solitary waves on deep water, J. Fluid Mech., Volume 713 (2012), pp. 307-329

[5] L. Hörmander Lectures on Nonlinear Hyperbolic Differential Equations, Springer-Verlag, Heidelberg, Germany, 1997

[6] P.A. Milewski; Z. Wang Three dimensional flexural-gravity waves, Stud. Appl. Math., Volume 131 (2013), pp. 135-148

[7] P. Plotnikov; J.F. Toland Modelling nonlinear hydroelastic waves, Philos. Trans. R. Soc. Lond. Ser. A, Volume 369 (2011), pp. 2942-2956

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