Abstract
S. Roy Choudhury
A Novel Analytic Technique for the Disintegration of Soliton Solutions
of Integrable NLPDEs and Coherent Structure Solutions of Non-integrable
NLPDEs
We consider a technique for deriving exact analytic coherent structure
(pulse/front/
domain wall) solutions of general NLPDEs via the use of truncated invariant
Painleve
expansions. We shall also discuss a recent proof[1] that such analytic
solutions
satisfy the corresponding traveling wave reduced ODE. Thus, they provide
`partial
integrability' in Painleve's original sense and hence a parametrization of the
homoclinic or heteroclinic structures of the the traveling wave reduced ODEs.
Coupling this to Melnikov theory, we then consider the breakdown to chaos
of such
analytic coherent structure solutions under forcing. We also demonstrate
that similar treatments are possible for integrable systems where the
soliton/kink
solutions represent the homoclinic/heteroclinic structures of the reduced
ODEs. In addition,
we treat the dynamics of the system prior to the onset of chaos
by the use of multiscale and direct soliton perturbation theory. It is
conceivable
that resummation of these perturbation series via the use of Pade approximants
or other techniques may enable one to analytically follow the homoclinic
or heteroclinic tangling beyond the first transversal intersection of the
stable and unstable manifolds and into the chaotic regime.
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