Quasiperiodic solutions of an extended MKdV hierarchy

An extended MKdV hierarchy associated with a $3\times3$ matrix spectral problem is derived by resorting to the Lenard recursion series and zero-curvature equation. The three-sheeted Riemann surface $\mathcal K_{m-1}$ for the extended MKdV hierarchy is defined by the zeros of the characteristic polynomial of the Lax matrix together with two points at infinity. On $\mathcal K_{m-1}$, we introduce the Baker–Akhiezer function and a meromorphic function, and then obtain their explicit representations in terms of the Riemann theta function with the aid of algebraic geometry tools. The asymptotic expansions of the meromorphic function give rise to quasiperiodic solutions for the entire extended MKdV hierarchy.