861 research outputs found
Blow up for the critical gKdV equation II: minimal mass dynamics
We fully revisit the near soliton dynamics for the mass critical (gKdV)
equation.
In Part I, for a class of initial data close to the soliton, we prove that
only three scenario can occur:
(BLOW UP) the solution blows up in finite time in a universal regime with
speed ;
(SOLITON) the solution is global and converges to a soliton in large time;
(EXIT) the solution leaves any small neighborhood of the modulated family of
solitons in the scale invariant norm.
Regimes (BLOW UP) and (EXIT) are proved to be stable. We also show in this
class that any nonpositive energy initial data (except solitons) yields finite
time blow up, thus obtaining the classification of the solitary wave at zero
energy.
In Part II, we classify minimal mass blow up by proving existence and
uniqueness (up to invariances of the equation) of a minimal mass blow up
solution . We also completely describe the blow up behavior of .
Second, we prove that is the universal attractor in the (EXIT) case,
i.e. any solution as above in the (EXIT) case is close to (up to
invariances) in at the exit time. In particular, assuming scattering for
(in large positive time), we obtain that any solution in the (EXIT)
scenario also scatters, thus achieving the description of the near soliton
dynamics
Blow up for the critical gKdV equation III: exotic regimes
We consider the blow up problem in the energy space for the critical (gKdV)
equation in the continuation of part I and part II.
We know from part I that the unique and stable blow up rate for solutions
close to the solitons with strong decay on the right is . In this paper,
we construct non-generic blow up regimes in the energy space by considering
initial data with explicit slow decay on the right in space. We obtain finite
time blow up solutions with speed where as well as
global in time growing up solutions with both exponential growth or power
growth. These solutions can be taken with initial data arbitrarily close to the
ground state solitary wave
Blow up dynamics for smooth equivariant solutions to the energy critical Schr\"odinger map
We consider the energy critical Schr\"odinger map problem with the 2-sphere
target for equivariant initial data of homotopy index . We show the
existence of a codimension one set of smooth well localized initial data
arbitrarily close to the ground state harmonic map in the energy critical norm,
which generates finite time blow up solutions. We give a sharp description of
the corresponding singularity formation which occurs by concentration of a
universal bubble of energy
Resampling: an improvement of Importance Sampling in varying population size models
Sequential importance sampling algorithms have been defined to estimate
likelihoods in models of ancestral population processes. However, these
algorithms are based on features of the models with constant population size,
and become inefficient when the population size varies in time, making
likelihood-based inferences difficult in many demographic situations. In this
work, we modify a previous sequential importance sampling algorithm to improve
the efficiency of the likelihood estimation. Our procedure is still based on
features of the model with constant size, but uses a resampling technique with
a new resampling probability distribution depending on the pairwise composite
likelihood. We tested our algorithm, called sequential importance sampling with
resampling (SISR) on simulated data sets under different demographic cases. In
most cases, we divided the computational cost by two for the same accuracy of
inference, in some cases even by one hundred. This study provides the first
assessment of the impact of such resampling techniques on parameter inference
using sequential importance sampling, and extends the range of situations where
likelihood inferences can be easily performed
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