Reductions of the Benney equations

John Gibbons, Imperial College

The Benney equations, an integrable Hamiltonian system of hydrodynamic type, admit many reductions, in which only finitely many of the dynamical variables {A_n (x,t)} are independent. These reductions satisfy integrability conditions, which are themselves of hydrodynamic type, but with an inhomogeneous term. Similar constructions are possible for other integrable moment hierarchies, and it is possible to find Miura maps between reductions of the modified and unmodified Benney hierarchies. Many interesting open questions remain, in particular, whether the integrability condition is itself an integrable system.
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