Another way ergodicity can be broken is by the existence of nonlinear soliton symmetries. In 1953, Fermi, Pasta, Ulam and Tsingou conducted computer simulations of a vibrating string that included a non-linear term (quadratic in one test, cubic in another, and a piecewise linear approximation to a cubic in a third). They found that the behavior of the system was quite different from what intuition based on equipartition would have led them to expect. Instead of the energies in the modes becoming equally shared, the system exhibited a very complicated quasi-periodic behavior. This puzzling result was eventually explained by Kruskal and Zabusky in 1965 in a paper which, by connecting the simulated system to the Korteweg–de Vries equation led to the development of soliton mathematics.