The application of trigonal curve to the Mikhailov–Shabat–Sokolov flows

Abstract

Resorting to the characteristic polynomial of Lax matrix for the Mikhailov–Shabat–Sokolov hierarchy associated with a \({3 \times 3}\) matrix spectral problem, we introduce a trigonal curve, from which we deduce the associated Baker–Akhiezer function, meromorphic functions and Dubrovin-type equations. The straightening out of the Mikhailov–Shabat–Sokolov flows is exactly given through the Abel map. On the basis of these results and the theory of trigonal curve, we obtain the explicit theta function representations of the Baker–Akhiezer function, the meromorphic functions, and in particular, that of solutions for the entire Mikhailov–Shabat–Sokolov hierarchy.