Analytic Riemann Theta Function Solutions of Coupled Korteweg-de Vries Hierarchy

Abstract

Coupled Korteweg-de Vries hierarchy associated with a 3 × 3 matrix spectral problem is derived via a stationary zero-curvature equation and Lenard recursion equations. Resorting to the characteristic polynomial of the Lax matrix for coupled Kortewegde Vries hierarchy, we introduce a trigonal curve $\mathscr{K}_g$ with three infinite points and establish the corresponding Baker-Akhiezer function and a meromorphic function on $\mathscr{K}_g$. Coupled Korteweg-de Vries equations are decomposed into systems of ordinary differential equations of Dubrovin-type. Analytic Riemann theta function solutions are obtained by using asymptotic expansions of the Baker-Akhiezer function and a meromorphic function and their Riemann theta function representations.