48 research outputs found
Curvature and Chaos in the Defocusing Parameteric Nonlinear Schrodinger System
The parametric nonlinear Schrodinger equation models a variety of
parametrically forced and damped dispersive waves. For the defocusing regime,
we derive a normal velocity for the evolution of curved dark-soliton fronts
that represent a -phase shift across a thin interface. We establish that
depending upon the strength of parametric term the normal velocity evolution
can transition from a curvature driven flow to motion against curvature
regularized by surface diffusion of curvature. In the former case interfacial
length shrinks, while in the later the interface length generically grows until
self-intersection followed by a transition to chaotic motion.Comment: 15 pages and 1 figur
Adiabatic stability under semi-strong interactions: The weakly damped regime
We rigorously derive multi-pulse interaction laws for the semi-strong
interactions in a family of singularly-perturbed and weakly-damped
reaction-diffusion systems in one space dimension. Most significantly, we show
the existence of a manifold of quasi-steady N-pulse solutions and identify a
"normal-hyperbolicity" condition which balances the asymptotic weakness of the
linear damping against the algebraic evolution rate of the multi-pulses. Our
main result is the adiabatic stability of the manifolds subject to this normal
hyperbolicity condition. More specifically, the spectrum of the linearization
about a fixed N-pulse configuration contains essential spectrum that is
asymptotically close to the origin as well as semi-strong eigenvalues which
move at leading order as the pulse positions evolve. We characterize the
semi-strong eigenvalues in terms of the spectrum of an explicit N by N matrix,
and rigorously bound the error between the N-pulse manifold and the evolution
of the full system, in a polynomially weighted space, so long as the
semi-strong spectrum remains strictly in the left-half complex plane, and the
essential spectrum is not too close to the origin