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A Variational Approach to Solitary Gravity–Capillary Interfacial Waves with Infinite Depth

Breit, D. and Wahlén, E. LU (2019) In Journal of Nonlinear Science 29(6). p.2601-2655
Abstract

We present an existence and stability theory for gravity–capillary solitary waves on the top surface of and interface between two perfect fluids of different densities, the lower one being of infinite depth. Exploiting a classical variational principle, we prove the existence of a minimiser of the wave energy E subject to the constraint I= 2 μ, where I is the wave momentum and 0 < μ< μ, where μ is chosen small enough for the validity of our calculations. Since E and I are both conserved quantities a standard argument asserts the stability of the set Dμ of minimisers: solutions starting near Dμ remain close to Dμ in a suitably defined energy space over their interval of existence. The solitary waves... (More)

We present an existence and stability theory for gravity–capillary solitary waves on the top surface of and interface between two perfect fluids of different densities, the lower one being of infinite depth. Exploiting a classical variational principle, we prove the existence of a minimiser of the wave energy E subject to the constraint I= 2 μ, where I is the wave momentum and 0 < μ< μ, where μ is chosen small enough for the validity of our calculations. Since E and I are both conserved quantities a standard argument asserts the stability of the set Dμ of minimisers: solutions starting near Dμ remain close to Dμ in a suitably defined energy space over their interval of existence. The solitary waves which we construct are of small amplitude and are to leading order described by the cubic nonlinear Schrödinger equation. They exist in a parameter region in which the ‘slow’ branch of the dispersion relation has a strict non-degenerate global minimum and the corresponding nonlinear Schrödinger equation is of focussing type. The waves detected by our variational method converge (after an appropriate rescaling) to solutions of the model equation as μ↓ 0.

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author
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organization
publishing date
type
Contribution to journal
publication status
published
subject
keywords
Interfacial waves, Solitary waves, Stability, Variational methods, Water waves
in
Journal of Nonlinear Science
volume
29
issue
6
pages
2601 - 2655
publisher
Springer
external identifiers
  • scopus:85067803815
ISSN
0938-8974
DOI
10.1007/s00332-019-09553-4
project
Nonlinear water waves and nonlocal model equations
Nonlinear Water Waves
language
English
LU publication?
yes
id
c74ecd6c-6c48-4319-aa7b-58621d52c5a7
date added to LUP
2019-07-08 15:59:32
date last changed
2022-05-04 00:08:54
@article{c74ecd6c-6c48-4319-aa7b-58621d52c5a7,
  abstract     = {{<p>We present an existence and stability theory for gravity–capillary solitary waves on the top surface of and interface between two perfect fluids of different densities, the lower one being of infinite depth. Exploiting a classical variational principle, we prove the existence of a minimiser of the wave energy E subject to the constraint I= 2 μ, where I is the wave momentum and 0 &lt; μ&lt; μ, where μ is chosen small enough for the validity of our calculations. Since E and I are both conserved quantities a standard argument asserts the stability of the set D<sub>μ</sub> of minimisers: solutions starting near D<sub>μ</sub> remain close to D<sub>μ</sub> in a suitably defined energy space over their interval of existence. The solitary waves which we construct are of small amplitude and are to leading order described by the cubic nonlinear Schrödinger equation. They exist in a parameter region in which the ‘slow’ branch of the dispersion relation has a strict non-degenerate global minimum and the corresponding nonlinear Schrödinger equation is of focussing type. The waves detected by our variational method converge (after an appropriate rescaling) to solutions of the model equation as μ↓ 0.</p>}},
  author       = {{Breit, D. and Wahlén, E.}},
  issn         = {{0938-8974}},
  keywords     = {{Interfacial waves; Solitary waves; Stability; Variational methods; Water waves}},
  language     = {{eng}},
  number       = {{6}},
  pages        = {{2601--2655}},
  publisher    = {{Springer}},
  series       = {{Journal of Nonlinear Science}},
  title        = {{A Variational Approach to Solitary Gravity–Capillary Interfacial Waves with Infinite Depth}},
  url          = {{http://dx.doi.org/10.1007/s00332-019-09553-4}},
  doi          = {{10.1007/s00332-019-09553-4}},
  volume       = {{29}},
  year         = {{2019}},
}