A Resonant Lyapunov Centre Theorem with an Application to Doubly Periodic Travelling Hydroelastic Waves
(2024) In Journal of Nonlinear Science 34(6).- Abstract
We present a Lyapunov centre theorem for an antisymplectically reversible Hamiltonian system exhibiting a nondegenerate 1 : 1 or 1:-1 semisimple resonance as a detuning parameter is varied. The system can be finite- or infinite-dimensional (and quasilinear) and have a non-constant symplectic structure. We allow the origin to be a ‘trivial’ eigenvalue arising from a translational symmetry or, in an infinite-dimensional setting, to lie in the continuous spectrum of the linearised Hamiltonian vector field provided a compatibility condition on its range is satisfied. As an application, we show how Kirchgässner’s spatial dynamics approach can be used to construct doubly periodic travelling waves on the surface of a three-dimensional body of... (More)
We present a Lyapunov centre theorem for an antisymplectically reversible Hamiltonian system exhibiting a nondegenerate 1 : 1 or 1:-1 semisimple resonance as a detuning parameter is varied. The system can be finite- or infinite-dimensional (and quasilinear) and have a non-constant symplectic structure. We allow the origin to be a ‘trivial’ eigenvalue arising from a translational symmetry or, in an infinite-dimensional setting, to lie in the continuous spectrum of the linearised Hamiltonian vector field provided a compatibility condition on its range is satisfied. As an application, we show how Kirchgässner’s spatial dynamics approach can be used to construct doubly periodic travelling waves on the surface of a three-dimensional body of water (of finite or infinite depth) beneath a thin ice sheet (‘hydroelastic waves’). The hydrodynamic problem is formulated as a reversible Hamiltonian system in which an arbitrary horizontal spatial direction is the time-like variable, and the infinite-dimensional phase space consists of wave profiles which are periodic (with fixed period) in a second, different horizontal direction. Applying our Lyapunov centre theorem at a point in parameter space associated with a 1 : 1 or 1:-1 semisimple resonance yields a periodic solution of the spatial Hamiltonian system corresponding to a doubly periodic hydroelastic wave.
(Less)
- author
- Ahmad, R. ; Groves, M. D. and Nilsson, D. LU
- organization
- publishing date
- 2024-12
- type
- Contribution to journal
- publication status
- published
- subject
- keywords
- 35Q35, 37K50, 76B15, Doubly periodic waves, Hydroelastic waves, Lyapunov centre theorem, Spatial dynamics
- in
- Journal of Nonlinear Science
- volume
- 34
- issue
- 6
- article number
- 104
- publisher
- Springer
- external identifiers
-
- scopus:85203860697
- ISSN
- 0938-8974
- DOI
- 10.1007/s00332-024-10073-z
- language
- English
- LU publication?
- yes
- id
- 76c8283c-4f9f-4946-b1fc-0f5544df59f0
- date added to LUP
- 2024-11-12 15:48:12
- date last changed
- 2025-04-04 15:11:47
@article{76c8283c-4f9f-4946-b1fc-0f5544df59f0,
abstract = {{<p>We present a Lyapunov centre theorem for an antisymplectically reversible Hamiltonian system exhibiting a nondegenerate 1 : 1 or 1:-1 semisimple resonance as a detuning parameter is varied. The system can be finite- or infinite-dimensional (and quasilinear) and have a non-constant symplectic structure. We allow the origin to be a ‘trivial’ eigenvalue arising from a translational symmetry or, in an infinite-dimensional setting, to lie in the continuous spectrum of the linearised Hamiltonian vector field provided a compatibility condition on its range is satisfied. As an application, we show how Kirchgässner’s spatial dynamics approach can be used to construct doubly periodic travelling waves on the surface of a three-dimensional body of water (of finite or infinite depth) beneath a thin ice sheet (‘hydroelastic waves’). The hydrodynamic problem is formulated as a reversible Hamiltonian system in which an arbitrary horizontal spatial direction is the time-like variable, and the infinite-dimensional phase space consists of wave profiles which are periodic (with fixed period) in a second, different horizontal direction. Applying our Lyapunov centre theorem at a point in parameter space associated with a 1 : 1 or 1:-1 semisimple resonance yields a periodic solution of the spatial Hamiltonian system corresponding to a doubly periodic hydroelastic wave.</p>}},
author = {{Ahmad, R. and Groves, M. D. and Nilsson, D.}},
issn = {{0938-8974}},
keywords = {{35Q35; 37K50; 76B15; Doubly periodic waves; Hydroelastic waves; Lyapunov centre theorem; Spatial dynamics}},
language = {{eng}},
number = {{6}},
publisher = {{Springer}},
series = {{Journal of Nonlinear Science}},
title = {{A Resonant Lyapunov Centre Theorem with an Application to Doubly Periodic Travelling Hydroelastic Waves}},
url = {{http://dx.doi.org/10.1007/s00332-024-10073-z}},
doi = {{10.1007/s00332-024-10073-z}},
volume = {{34}},
year = {{2024}},
}