Exact solutions for small-amplitude capillary-gravity water waves
  We present explicit solutions for the ordinary differential equations system describing the motion of the particles beneath small-amplitude capillary-gravity waves which propagate on the surface of an irrotational water flow with a flat bottom. The required computations involve elliptic integrals of first kind, the Legendre normal form and a solvable Abel differential equation of the second kind. 
Delia Ionescu-Kruse
math-ph math.MP physics.flu-dyn