Abstract:
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.
Key words and phrases:trigonal curve; Baker–Akhiezer function; algebro-geometric solutions.
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