The initial value problem for reductions of the Benney equations

We consider a family of N-parameter reductions of Benney's equations, introduced in Gibbons and Kodama (Gibbons J and Kodama Y 1994 Solving dispersionless Lax equations Proc. Singular Limits of Dispersive Waves (Nato ASI Adv. Sci. Inst. Ser. B: Phys. vol 320) (New York: Plenum) p 61) as a generalization of the dispersionless Lax equations. Using Geogdzhaev's method (Geogjaev V V [Geogdzhaev V V]1994 The quasiclassical limit of the inverse scattering problem method Proc. Singular Limits of Dispersive Wave (Nato ASI Adv. Sci. Inst. Ser. B: Phys. vol 20) (New York: Plenum) p 53). we solve the initial value problem for the reduced system. This construction is carried out explicitly for the reduction associated with an elliptic curve.