A fairly classical (and pretty dated at that) example is the study of solitons which was to a large extent triggered by slovingsolving numerically the so-called Fermi--Pasta--Ulam chain and then of its continuous limit, the Korteweg--de Vries equation. It would not be much of exaggeration to say that the whole modern theory of integrable systems grew out of this. For more details see e.g. the Wikipedia entryWikipedia entry on solitons and the first chapter of the book Solitons in Mathematics and PhysicsSolitons in Mathematics and Physics by Alan Newell.