Analytic patterns for chaotic equations

Chaotic evolution equations sometimes display regular patterns (fronts, kinks, holes), which often correspond to closed form analytic solutions. In the dissipative-dispersive Kuramoto and Sivashinsky travelling wave reduction u(x - ct), vu''' + bu'' + mu u' + u(2) /2 + A = 0, v = 0, with (v, b, mu, A) constants, such analytic solutions are known for heteroclinic solutions, but one has also observed (Toh, 1987) homoclinic solutions without corresponding analytic solutions yet. We review the search for the most general analytic solution admissible by this chaotic differential equation. Several investigations, both analytic by the Painleve test (Thual and Frisch, 1986) and numerical by Pade approximants (Yee, Conte, Musette, 2003) indicate its quite probable single valuedness for any (v, b, mu, A). Moreover, Nevanlinna theory on the growth of solutions near infinity rules out (Eremenko, 2005) the possibility for the singularities of this unknown closed form single valued expression to be only poles. We propose a qualitative closed form involving a deformed elliptic function.