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

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.