A fairly classical (and pretty old at that) example is the study of solitons which was to a large extent triggered by the numerical solving of 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 entry][1] on solitons and the first chapter of the book [Solitons in Mathematics and Physics][2] by Alan Newell. 


  [1]: http://en.wikipedia.org/wiki/Soliton
  [2]: http://books.google.com/books?id=gUxsnMcSDFAC