Spatial dynamics and solitary hydroelastic surface waves
type: Item
instance of: scholarly article
title: Spatial dynamics and solitary hydroelastic surface waves
zbMATH Open document ID: 1545.76021
author: R. Ahmad
author: Mark D. Groves
published in: Water Waves
publication date: 2024-06-17T00:00:00Z
review text: The paper presents a comprehensive existence theory for solitary waves occurring at the interface between a thin ice sheet, modeled using the Cosserat theory of hyperelastic shells, and an ideal fluid of finite depth in irrotational motion. This study is framed within the context of the Kirchgässner reduction to a finite-dimensional Hamiltonian system, showcasing the refinements and advancements in the theory over time. Notably, the paper introduces novel elements such as the application of a higher-order Legendre transformation to reformulate the problem as a spatial Hamiltonian system, alongside the utilization of a Riesz basis for the phase space, thus enhancing the analogy with dynamical systems.\N\NThe scientific problem addressed in the paper is the propagation of solitary waves on the surface of an ocean beneath a thin ice sheet, where the water is considered as a perfect fluid in irrotational flow, and the ice sheet is treated as an elastic shell that bends with the surface without stretching, friction, or cavitation with the fluid beneath. This study is grounded in the model developed by \textit{P. I. Plotnikov} and \textit{J. F. Toland} [Philos. Trans. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 369, No. 1947, 2942--2956 (2011; Zbl 1228.76027)], which employs the Euler equations for inviscid fluid flow in conjunction with the Cosserat theory of hyperelastic shells. The problem is described by a set of governing equations for the hydrodynamic problem in dimensionless coordinates and a moving coordinate system, with travelling waves represented by a velocity potential \(\varphi\).\N\NTo solve this problem, the authors reformulate the governing equations using a change of variable that maps the variable fluid domain to a fixed strip, transforming the equations into a more manageable form. This approach allows for a precise proof of the persistence of two homoclinic solutions as solutions to the unapproximated reduced system, corresponding to symmetric hydroe…
reviewed by: Denys Dutykh
Mathematics Subject Classification ID: 76B25
Mathematics Subject Classification ID: 35Q35
Mathematics Subject Classification ID: 37N10
Mathematics Subject Classification ID: 74F10
Mathematics Subject Classification ID: 86A05
Mathematics Subject Classification ID: 86A40
zbMATH DE Number: 7866604
zbMATH Keywords: nonlinear Schrödinger equation
zbMATH Keywords: finite-dimensional Hamiltonian system
zbMATH Keywords: Kirchgässner centre-manifold reduction
zbMATH Keywords: hyperelastic Cosserat shell
zbMATH Keywords: higher-order Legendre transform
zbMATH Keywords: Riesz basis
zbMATH Keywords: homoclinic solution
MaRDI profile type: Publication
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DOI: 10.1007/S42286-023-00077-9
description: scientific article; zbMATH DE number 7866604