Traveling water waves in three dimensions

Much work has been done on (1+1) traveling water waves. However, comparatively little is known about traveling water waves in (2+1) dimensions, in part because complex variables are no longer available as a tool. To overcome this difficulty, we use the nonlocal formulation of full water waves developed by Ablowitz, Fokas, and Musslimani. By employing the Poincare-Stokes method, we obtain equations for an asymptotic series for the velocity potential. The first equation is the KP equation, which has a well known rational solution. The other equations consist of a linear fourth-order non-constant coefficient PDE and a forcing term. These equations are then solved numerically.
 

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