Integrable nonlinear evolution equations in three spatial dimensions

Abstract

There are integrable nonlinear evolution equations in two spatial variables. The solution of the initial value problem of these equations necessitated the introduction of novel mathematical formalisms. Indeed, the classical Riemann–Hilbert problem used for the solution of integrable equations in one spatial variable was replaced by a non-local Riemann–Hilbert problem or, more importantly, by the so-calledd-bar formalism. The construction of integrable nonlinear evolution equations in three spatial dimensions has remained the key open problem in the area of integrability. For example, the two versions of the Kadomtsev–Petviashvili (KP) equation constitute two-dimensional generalizations of the celebrated Korteweg–de Vries equation. Are there three-dimensional generalizations of the KP equations? Here, we present such equations. Furthermore, we introduce a novel non-locald-bar formalism for solving the associated initial value problem.