Explicit theta function representations of soliton hierarchies in solutions to the completely integrable models (the work pioneered by Dubrovin, Matveev, Novikov, Krichever, McKean...). For instance
In the simplest case of the KdV equation, solitons are obtained from a singularization of the corresponding hyperelliptic curve. The kdV equation was originally derived to describe waves in shallow water and presumably had nothing algebro-geometric to do with hyperelliptic curves.
Edit. And eventually the connection worked in its naturethe opposite direction as well: a solution of Shottky's problem which exploited the integrability of the Kadomtsev–Petviashvili equation.