I have no example of hyperbolic PDE occuring in say, pure, mathematics. Perhaps one deep reason is that the notion of hyperbolic operator distinguishes a convex cone of directions which is inherently a cone of future. Therefore, there is always a distinction between time-like curves and space-like hypersurfaces ; whence the occurence of the real world.
Nevertheless, the theory of hyperbolic PDEs touches mathematics per se in some places. I have in mind the theory of hyperbolic polynomials, discovered by L. Garding. These are principal symbols of hyperbolic operators. A hyperbolic polynomial $P$ of degree $n$ is positive in the future cone $\Gamma$, and the function $P^{\frac1n}$ is concave in $\Gamma$. An example is $P=\det$ in the space of $n\times n$ symmetric matrices, with $\Gamma={\bf Sym}_n^+$. The $n$-linear $\phi$ form associated with $P$ satisfies the inequality $$P^{\frac1n}(\xi_1)\cdots P^{\frac1n}(\xi_n)\le\phi(\xi_1,\ldots,\xi_n),\qquad\forall \xi_1,\ldots,\xi_n\in\Gamma.$$ For instance, if $n=2$, this means that the quadratic form $P$ satisfies the converse of Cauchy-Schwarz in the future cone. The polynomial $x_1\cdots x_n$ is hyperbolic, its future cone is the first orthant and the corresponding $n$-linear form is nothing but the permanent of a square matrix. The so-called Van der Warden conjecture (now a theorem) is actually a special of a more general problem about hyperbolic polynomials. Through the theory of hyperbolic polynomials, one touches to Real Algebraic Geometry ; this dates back to Petrowsky's school. Actually, O. Oleinik is famous in both PDE and Algebraic Geometry communities.
I should also mention the theory of lacunae for hyperbolic PDEs, which is a problem in Algebraic Topology. See a Bourbaki seminar by M. Atiyah about that.