Hyperelliptic curves play a basic role in the construction of solutions to completely integrable systems (soliton equations), e.g. KdV. But it should be noted that these solutions are of non-soliton type.
The basic idea is, that such a system can be written as a Lax pair: $$ \dot{L} = [P_j, L] $$ for some $j$. Different $j$ correspond to different members of the hierachy. Now to construct such an algebro-geometric solution. Consider some $\ell > j$, and look for an $L$ such that $$ [L, P_{\ell}] = 0 $$ then some general theory implies that this solution satisfies a polynomial equation, and can thus be written in terms of data on a curve.
