Three‐Phase Solutions of the Kadomtsev–Petviashvili Equation

Abstract

The Kadomtsev–Petviashvili (KP) equation is known to admit explicit periodic and quasiperiodic solutions with N independent phases, for any integer N , based on a Riemann theta‐function of N variables. For N =1 and 2, these solutions have been used successfully in physical applications. This article addresses mathematical problems that arise in the computation of theta‐functions of three variables and with the corresponding solutions of the KP equation. We identify a set of parameters and their corresponding ranges, such that every real‐valued, smooth KP solution associated with a Riemann theta‐function of three variables corresponds to exactly one choice of these parameters in the proper range. Our results are embodied in a program that computes these solutions efficiently and that is available to the reader. We also discuss some properties of three‐phase solutions.