Narrow Results

In this paper the moduli space of Higgs pairs over a fixed smooth projective curve with extra formal data is defined and is endowed with a scheme structure. We introduce a relative version of the Krichever map using a fibration of Sato Grassmannians and show that this map is injective. This, together with the characterization of the points of the image of the Krichever map, allows us to prove that this moduli space is a closed subscheme of the product of the moduli of vector bundles (with formal extra data) and a formal anologue of the Hitchin base. This characterization also provides us with a method for explicitly computing KP-type equations that describe the moduli space of Higgs pairs. Finally, for the case where the spectral cover is totally ramified at a fixed point of the curve, these equations are given in terms of the characteristic coefficients of the Higgs field.

In this article, we will present main results proved in,^26,27 where we rigorously justify Kadomtsev-Petviashvili (KP) Approximation to the three dimensional water wave equations with surface tension in the transverse long wave regime. In particular, when the Bond number is positive and away from ⅓, the solutions of the water wave system are accurately approximated by sums of solutions of two decoupled KP equations up to a time interval [0, T/ɛ]. In the case when the Bond number α = ⅓, the solutions are accurately approximated by sums of solutions of two decoupled fifth-order Kadomtsev-Petviashvili (KP^5th)^± equations on [0, T/ε²].