Algebro-geometric solutions to the Manakov hierarchy

The Manakov hierarchy associated with a [GRAPHICS] matrix spectral problem is proposed with the aid of Lenard recursion equations. By using the characteristic polynomial of Lax matrix for the Manakov hierarchy, we introduce a trigonal curve [GRAPHICS] of arithmetic genus [GRAPHICS] , from which we construct the related Baker-Akhiezer function, two algebraic functions carrying the data of the divisor and Dubrovin-type equations. Based on the theory of trigonal curves, the explicit theta function representations of the Baker-Akhiezer function, the two algebraic functions, and in particular, that of solutions for the entire Manakov hierarchy are obtained.