The Long Wave Limit for a Boussinesq Equation

We consider the long wave limit for a Boussinesq equation. It is shown that for spatially localized initial conditions the dynamics are under a Korteweg--de Vries (KdV) regime on a large time interval; i.e., in this limit the solutions split up into two counterpropagating wave packets, where each of the wave packets evolve independently and approximately as a solution of a KdV equation. For the proof, exact estimates between the long wave solutions of the Boussinesq equation and the approximations obtained via the decoupled set of KdV equations are demonstrated. We expect that this result holds for the water wave problem, too.

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