A hierarchy of generalized Drinfel’d-Sokolov equations associated with a \(4\times 4\) matrix spectral problem is proposed by using the zero-curvature equation and the Lenard recursion equation. Based on the characteristic polynomial of Lax matrix for the generalized Drinfel’d-Sokolov hierarchy, we introduce a tetragonal curve \(\mathcal {K}_g\) of genus g and study the asymptotic properties of the Baker-Akhiezer function \(\psi _2\) and the meromorphic function \(\phi \) near the infinite point on \(\mathcal {K}_g\) . The straightening out of various flows is exactly given through the Abel map and Abel-Jacobi coordinates. Using the theory of tetragonal curves and the properties of the three kinds of Abel differentials, we construct the explicit Riemann theta function representations of the Baker-Akhiezer function and the meromorphic function. Together with the asymptotic properties of the Baker-Akhiezer function, we obtain algebro-geometric quasi-periodic solutions for the entire hierarchy of generalized Drinfel’d-Sokolov equations.
