The Novikov-Veselov hierarchy of equations and integrable deformations of minimal Lagrangian tori in CP^2

Abstract

We associate a periodic two-dimensional Schrodinger operator to every Lagrangian torus in CP^2 and define the spectral curve of a torus as the Floquet spectrum of this operator on the zero energy level. In this event minimal Lagrangian tori correspond to potential operators. We show that Novikov-Veselov hierarchy of equations induces integrable deformations of minimal Lagrangian torus in CP^2 preserving the spectral curve. We also show that the highest flows on the space of smooth periodic solutions of the Tzizeica equation are given by the Novikov-Veselov hierarchy.