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Comptes Rendus Mathématique
Volume 354, n° 11
pages 1078-1086 (novembre 2016)
Doi : 10.1016/j.crma.2016.10.004
Received : 14 April 2016 ;  accepted : 5 October 2016
Variational existence theory for hydroelastic solitary waves
Une théorie variationnelle d'existence d'ondes solitaires hydroélastiques
 

Mark D. Groves a, b , Benedikt Hewer a, Erik Wahlén c
a Fachrichtung Mathematik, Universität des Saarlandes, Postfach 151150, 66041 Saarbrücken, Germany 
b Department of Mathematical Sciences, Loughborough University, Loughborough, Leics, LE11 3TU, UK 
c Centre for Mathematical Sciences, Lund University, PO Box 118, 22100 Lund, Sweden 

Abstract

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   subject to the constraint  , where   is the horizontal impulse and  , 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  .

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Résumé

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   de l'onde sous la contrainte  , où   représente l'impulsion horizontale et  . 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  .

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