2018/12/08 by Hung Le, Le, Hung · 1 citation
Earth and Planetary Sciences · Mathematics · #35B35 #35Q35 #37K45 #76B25 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #Argument (complex analysis) #Classical mechanics #Computer science #Construct (python library) #Dipole #FOS: Mathematics #Free surface #Function (biology) #Instability #Mathematical analysis #Mathematical physics #Mathematics #Mechanics #Navier-Stokes equation solutions #Ocean Waves and Remote Sensing #Physics #Potential vorticity #Quantum mechanics #Stability (learning theory) #Surface tension #Vortex #Vorticity #math.AP #msc:35B35 #msc:35Q35 #msc:37K45 #msc:76B25
paper · pdf · doi:10.48550/arxiv.1812.03341
published in arXiv (Cornell University) (Cornell University) · 30 pages
arxiv created 2018/12/08 · openalex publication_date 2018/12/08 · arxiv updated 2018/12/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
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 δ-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.