An Asymptotic Expansion for Bloch Functions on Riemann Surfaces of Infinite Genus and Almost Periodicity of the Kadomcev–Petviashvilli Flow

Abstract

This article describes the solution of the Kadomcev–Petviashvilli equation with C10 real periodic initial data in terms of an asymptotic expansion of Bloch functions. The Bloch functions are parametrized by the spectral variety of a heat equation (heat curves) with an external potential. The mentioned spectral variety is a Riemann surface of in general infinite genus; the Kadomcev–Petviashvilli flow is represented by a one-parameter-subgroup in the real part of the Jacobi variety of this Riemann surface. It is shown that the KP-I flow with these initial data propagates almost periodically.