On the Existence and Instability of Solitary Water Waves with a Finite Dipole
Abstract
This paper considers the existence and stability properties of two-dimensional solitary waves traversing an infinitely deep body of water. We assume that above the water is air and that the waves are acted upon by gravity with surface tension effects on the air-water interface. In particular, we study the case where there is a finite dipole in the bulk of the fluid, that is, the vorticity is a sum of two weighted $\delta$-functions. Using an implicit function theorem argument, we construct a family of solitary waves solutions for this system that is exhaustive in a neighborhood of $0$. Our main result is that this family is conditionally orbitally unstable. This is proved using a modification of the Grillakis--Shatah--Strauss method recently introduced by Varholm, Wahlén, and Walsh.
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Information & Authors
Information
Published In
SIAM Journal on Mathematical Analysis
Pages: 4074 - 4104
DOI: 10.1137/18M1231638
ISSN (online): 1095-7154
Copyright
© 2019, Society for Industrial and Applied Mathematics.
History
Submitted: 10 December 2018
Accepted: 9 September 2019
Published online: 17 October 2019
Keywords
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Authors
Funding Information
National Science Foundation https://doi.org/10.13039/100000001 : DMS-1549934, DMS-1710989
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