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

Dominic Breit, Erik Wahlen

Research output: Contribution to journalArticle

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 D μ of minimisers: solutions starting near D μ remain close to D μ 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.

LanguageEnglish
Pages1-55
Number of pages55
JournalJournal of Nonlinear Science
Early online date15 Jun 2019
DOIs
Publication statusE-pub ahead of print - 15 Jun 2019

Fingerprint

Variational Approach
Gravity
Gravitation
Solitary Waves
Solitons
Nonlinear equations
Nonlinear Schrödinger Equation
Existence Theory
Perfect Fluid
Conserved Quantity
Stability Theory
Rescaling
Global Minimum
Dispersion Relation
Energy
Variational Principle
Variational Methods
Momentum
Branch
Converge

Keywords

  • Interfacial waves
  • Solitary waves
  • Stability
  • Variational methods
  • Water waves

ASJC Scopus subject areas

  • Modelling and Simulation
  • Engineering(all)
  • Applied Mathematics

Cite this

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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 D μ of minimisers: solutions starting near D μ remain close to D μ 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 Schr{\"o}dinger 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 Schr{\"o}dinger 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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A Variational Approach to Solitary Gravity–Capillary Interfacial Waves with Infinite Depth. / Breit, Dominic; Wahlen, Erik.

In: Journal of Nonlinear Science, 15.06.2019, p. 1-55.

Research output: Contribution to journalArticle

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AU - Wahlen, Erik

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N2 - 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 D μ of minimisers: solutions starting near D μ remain close to D μ 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.

AB - 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 D μ of minimisers: solutions starting near D μ remain close to D μ 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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