### 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.

Original language | English |
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Pages (from-to) | 1-55 |

Number of pages | 55 |

Journal | Journal of Nonlinear Science |

Early online date | 15 Jun 2019 |

DOIs | |

Publication status | E-pub ahead of print - 15 Jun 2019 |

### Fingerprint

### 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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**A Variational Approach to Solitary Gravity–Capillary Interfacial Waves with Infinite Depth.** / Breit, Dominic; Wahlen, Erik.

Research output: Contribution to journal › Article

TY - JOUR

T1 - A Variational Approach to Solitary Gravity–Capillary Interfacial Waves with Infinite Depth

AU - Breit, Dominic

AU - Wahlen, Erik

PY - 2019/6/15

Y1 - 2019/6/15

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.

KW - Interfacial waves

KW - Solitary waves

KW - Stability

KW - Variational methods

KW - Water waves

UR - http://www.scopus.com/inward/record.url?scp=85067803815&partnerID=8YFLogxK

U2 - 10.1007/s00332-019-09553-4

DO - 10.1007/s00332-019-09553-4

M3 - Article

SP - 1

EP - 55

JO - Journal of Nonlinear Science

JF - Journal of Nonlinear Science

SN - 0938-8974

ER -