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We consider the initial-value problem for the bidirectional Whitham equation,a system which combines the full two-way dispersion relation from theincompressible Euler equations with a canonical shallow-water nonlinearity. Weprove local well-posedness in classical Sobolev spaces in the localised as wellas the periodic case, using a square-root type transformation to symmetrise thesystem. The existence theory requires a non-vanishing surface elevation,indicating that the problem is ill-posed for more general initial data.

A conditional well-posedness result for the bidirectional Whitham equation