Algebro-geometric solutions to the modified Sawada-Kotera hierarchy

Abstract

Based on solving the Lenard recursion equation and the zero-curvature equation, we derive the modified Sawada-Kotera (SK) hierarchy associated with a 3 × 3 matrix spectral problem. Resorting to the characteristic polynomial of Lax matrix for the modified SK hierarchy, we introduce a trigonal curve \documentclass[12pt]{minimal}\begin{document}$\mathcal {K}_{m-1}$\end{document}Km−1 and present the corresponding Baker-Akhiezer function and meromorphic function on it. The Abel map is introduced to straighten out the modified SK flows. With the aid of the property of the Baker-Akhiezer function and the meromorphic function and their asymptotic expansions, we arrive at their explicit Riemann theta function representations. Algebro-geometric solutions of the entire modified SK hierarchy are obtained by using asymptotic expansion of the meromorphic function and its Riemann theta function representation. As an application, we give two simple examples.