A couple of BO equations as a normal form for the interface problem

Dario Bambusi; Simone Paleari; Dario Bambusi; Simone Paleari

doi:10.3934/math.20241118

AIMS Mathematics

2024, Volume 9, Issue 8: 23012-23026. doi: 10.3934/math.20241118

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A couple of BO equations as a normal form for the interface problem

Dario Bambusi ^,,
Simone Paleari

Dipartimento di Matematica, Università degli Studi di Milano, Via Saldini 50, I-20133 Milano, Italy

Received: 21 May 2024 Revised: 12 July 2024 Accepted: 15 July 2024 Published: 26 July 2024
MSC : 35Q35, 37K55, 76B55

We consider two fluids in a 2-dimensional region: The lower fluid occupies an infinitely depth region, while the upper fluid occupies a region with a fixed upper boundary. We study the dynamics of the interface between the two fluids (interface problem) in the limit in which the interface has a space periodic profile, is close to horizontal, and has a "long wave profile". We use a Hamiltonian normal form approach to show that up to corrections of second order, the equations are approximated by two decoupled Benjamin-Ono equations.
- water waves,
- Benjamin-Ono,
- Hamiltonian partial differential equations,
- normal form
Citation: Dario Bambusi, Simone Paleari. A couple of BO equations as a normal form for the interface problem[J]. AIMS Mathematics, 2024, 9(8): 23012-23026. doi: 10.3934/math.20241118

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Abstract

We consider two fluids in a 2-dimensional region: The lower fluid occupies an infinitely depth region, while the upper fluid occupies a region with a fixed upper boundary. We study the dynamics of the interface between the two fluids (interface problem) in the limit in which the interface has a space periodic profile, is close to horizontal, and has a "long wave profile". We use a Hamiltonian normal form approach to show that up to corrections of second order, the equations are approximated by two decoupled Benjamin-Ono equations.

References

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