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.


  1. finite dipole
  2. point vortices
  3. solitary water waves
  4. existence
  5. instability
  6. spectrum

MSC codes

  1. 35Q35
  2. 37K45
  3. 76B25
  4. 35B35

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Information & Authors


Published In

cover image SIAM Journal on Mathematical Analysis
SIAM Journal on Mathematical Analysis
Pages: 4074 - 4104
ISSN (online): 1095-7154


Submitted: 10 December 2018
Accepted: 9 September 2019
Published online: 17 October 2019


  1. finite dipole
  2. point vortices
  3. solitary water waves
  4. existence
  5. instability
  6. spectrum

MSC codes

  1. 35Q35
  2. 37K45
  3. 76B25
  4. 35B35



Funding Information

National Science Foundation https://doi.org/10.13039/100000001 : DMS-1549934, DMS-1710989

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