Algebro-geometric quasi-periodic solutions to the Satsuma–Hirota hierarchy

Abstract

In this paper, we study the theory of the resulting tetragonal curve and derive three kinds of Abel differentials, Baker–Akhiezer function and meromorphic function and so on, from which a systematic method is developed to construct algebro-geometric quasi-periodic solutions of soliton equations associated with the 4 × 4 matrix spectral problems. As an illustrative example, the Satsuma–Hirota hierarchy related to the 4 × 4 matrix spectral problem is obtained by utilizing the Lenard recursion equation and zero-curvature equation. Based on the characteristic polynomial of Lax matrix for the Satsuma–Hirota hierarchy, we introduce a tetragonal curve Kg of genus g and study the asymptotic properties of the Baker–Akhiezer function ψ1 and the meromorphic function ϕ near the infinite points on the tetragonal curve. The straightening out of various flows is exactly given through the Abel map and Abel–Jacobi coordinates. Using the theory of the tetragonal curve and the properties of the three kinds of Abel differentials, we obtain the explicit Riemann theta function representations of the Baker–Akhiezer function and the meromorphic function, and in particular, that of solutions for the entire Satsuma–Hirota hierarchy.