A Parameterization of Small Amplitude KP Finite Gap Solutions via Classical Schottky Uniformization and Persistence of One and Two Gap Solutions

We develop a parameterization of small amplitude finite gap solutions to the KP equation with prescribed spatial wavenumber vectors using classical Schottky uniformization. This parameterization is suitable for a Lyapunov-Schmidt reduction. Using this parameterization, we prove the persistence, under Hamiltonian perturbations, of periodic one-gap solutions for both KP-I and KP-II and of bi-periodic two-gap solutions for KP-I.