Quasi‐Periodic Solutions of Nonlinear Evolution Equations Associated with a 3 × 3 Matrix Spectral Problem

Abstract

A hierarchy of new nonlinear evolution equations associated with a 3 × 3 matrix spectral problem with three potentials is proposed. With the aid of the characteristic polynomial of Lax matrix for the hierarchy, we introduce an algebraic curve of arithmetic genus m − 1, and discuss in detail the properties of the associated Baker–Akhiezer function and meromorphic function. On the basis of the theory of algebraic curves, we obtain the explicit theta function representations of the Baker–Akhiezer function, the meromorphic function, and, in particular, that of solutions for the entire hierarchy of nonlinear evolution equations.