2 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
Ring-type singular solutions of the biharmonic nonlinear Schrodinger equation
We present new singular solutions of the biharmonic nonlinear Schrodinger
equation in dimension d and nonlinearity exponent 2\sigma+1. These solutions
collapse with the quasi self-similar ring profile, with ring width L(t) that
vanishes at singularity, and radius proportional to L^\alpha, where
\alpha=(4-\sigma)/(\sigma(d-1)). The blowup rate of these solutions is
1/(3+\alpha) for 4/d\le\sigma<4, and slightly faster than 1/4 for \sigma=4.
These solutions are analogous to the ring-type solutions of the nonlinear
Schrodinger equation.Comment: 21 pages, 13 figures, research articl