37 research outputs found
Continuations of the nonlinear Schr\"odinger equation beyond the singularity
We present four continuations of the critical nonlinear \schro equation (NLS)
beyond the singularity: 1) a sub-threshold power continuation, 2) a
shrinking-hole continuation for ring-type solutions, 3) a vanishing
nonlinear-damping continuation, and 4) a complex Ginzburg-Landau (CGL)
continuation. Using asymptotic analysis, we explicitly calculate the limiting
solutions beyond the singularity. These calculations show that for generic
initial data that leads to a loglog collapse, the sub-threshold power limit is
a Bourgain-Wang solution, both before and after the singularity, and the
vanishing nonlinear-damping and CGL limits are a loglog solution before the
singularity, and have an infinite-velocity{\rev{expanding core}} after the
singularity. Our results suggest that all NLS continuations share the universal
feature that after the singularity time , the phase of the singular core
is only determined up to multiplication by . As a result,
interactions between post-collapse beams (filaments) become chaotic. We also
show that when the continuation model leads to a point singularity and
preserves the NLS invariance under the transformation and
, the singular core of the weak solution is symmetric
with respect to . Therefore, the sub-threshold power and
the{\rev{shrinking}}-hole continuations are symmetric with respect to ,
but continuations which are based on perturbations of the NLS equation are
generically asymmetric
Dimension reduction for systems with slow relaxation
We develop reduced, stochastic models for high dimensional, dissipative
dynamical systems that relax very slowly to equilibrium and can encode long
term memory. We present a variety of empirical and first principles approaches
for model reduction, and build a mathematical framework for analyzing the
reduced models. We introduce the notions of universal and asymptotic filters to
characterize `optimal' model reductions for sloppy linear models. We illustrate
our methods by applying them to the practically important problem of modeling
evaporation in oil spills.Comment: 48 Pages, 13 figures. Paper dedicated to the memory of Leo Kadanof
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Vascular Biology and Interventio
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Numerical scaling analysis of the small-scale structure in turbulence
We show how to use numerical methods within the framework of successive
scaling to analyse the microstructure of turbulence, in particular to find
inertial range exponents and structure functions. The methods are first
calibrated on the Burgers problem and are then applied to the 3D Euler
equations. Known properties of low order structure functions appear with a
relatively small computational outlay; however, more sensitive properties
cannot yet be resolved with this approach well enough to settle ongoing
controversies