On the existence and instability of solitary water waves with a finite dipole

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 vacuum, 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, Wahl\'en, and Walsh.