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Chaotic evolution equations sometimes display regular patterns,
which often correspond to closed form analytic solutions.
In the dissipative-dispersive Kuramoto and Sivashinsky
traveling wave reduction u(x-ct), νu'''+ bu''+μu'+u2/2+A=0,ν≠0,
with (ν,b,μ,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 will review the search for the most general analytic solution
admissible by this chaotic differential equation.
Several investigations,
both analytic by the Painlevé test (Thual and Frisch, 1986)
and numerical by Padé approximants (Yee, Conte, Musette, 2003)
conclude to its quite probable singlevaluedness for any (ν,b,μ,A).
Moreover,
Nevanlinna theory on the growth of solutions
rules out (Eremenko, 2005)
the possibility for this unknown close form singlevalued
expression to be generically meromorphic.
We reduce the search for this solution to the search for an entire function which is a deformation, yet to be found, of the entire function σ(z,g2,g3) of Weierstrass violating the odd parity of the latter. The validity of this feature for the one-dimensional complex Ginzburg-Landau equation, whether cubic or quintic, will be discussed. |
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