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 $T$ in a universal regime with speed $1/(T-t)$; (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 $L^2$ 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 $S(t)$. We also completely describe the blow up behavior of $S(t)$. Second, we prove that $S(t)$ is the universal attractor in the (EXIT) case, i.e. any solution as above in the (EXIT) case is close to $S$ (up to invariances) in $L^2$ at the exit time. In particular, assuming scattering for $S(t)$ (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 $1/t$. 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 $t^{-\nu}$ where $\nu>11/13,$ 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 $k=1$. 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