  Recently, the author and collaborators have developed a systematic program
for proving the existence of homoclinic orbits in partial differential
equations. Two typical forms of homoclinic orbits thus obtained are: (1).
transversal homoclinic orbits, (2). Silnikov homoclinic orbits. Around the
transversal homoclinic orbits in infinite dimensional autonomous systems, the
author was able to prove the existence of chaos through a shadowing lemma.
Around the Silnikov homoclinic orbits, the author was able to prove the
existence of chaos through a horseshoe construction.
  Very recently, there has been a breakthrough by the author in finding Lax
pairs for Euler equations of incompressible inviscid fluids. Further results
have been obtained by the author and collaborators.
