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

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 x 4 matrix spectral problems. As an illustrative example, the Satsuma-Hirota hierarchy related to the 4 x 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 psi 1 and the meromorphic function phi 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.(c) 2023 Elsevier B.V. All rights reserved.