Algebro-geometric solutions to a new hierarchy of soliton equations

With the help of the Lenard recursion equations, we derive a new hierarchy of soliton equations associated with a $3\times3$ matrix spectral problem and establish Dubrovin type equations in terms of the introduced trigonal curve $\mathcal{K}_{m-1}$ of arithmetic genus $m-1$. Basing on the theory of algebraic curve, we construct the corresponding Baker–Akhiezer functions and meromorphic functions on $\mathcal{K}_{m-1}$. The known zeros and poles for the Baker–Akhiezer function and meromorphic functions allow us to find their theta function representations, from which algebro-geometric constructions and theta function representations of the entire hierarchy of soliton equations are obtained.