It is known that the Schrödinger equation can not be solved for many body system, but in this article cited above, the wave function for the Bose-Einstein condensate was found experimentally. Does this mean that mathematics is fundamentally limited in describing nature, or that modern mathematics is not sufficiently developed to solve such equations, or that quantum systems are deterministic?
Firstly, the many-body wave function of systems can actually be found (two entangled photons is a many-body system and we can describe it's wave function). The issue is, as Willy Billy Williams' comment eludes to, that the problem of finding it scales non-polynomially (NP). So it's not that we "can't" find it, it's more that it would take (with our current methods) EXTREMELY long to find the wave function of a large many body system. This is because as you add systems together in quantum mechanics the dimensionality of the state space grows exponentially (which grows faster then any polynomial). However, often the reason why we would like to know the wave function of a many body system is so that we can calculate some interesting quantities. This paper shows how you can find some of these quantities without ever knowing the wave function (by using a BEC).