arXiv subjects
Publications and source records attributed to Pierre Raphaël.
We derive a family of degenerate vortices with non trivial swirls and fat tails which can be used to bifurcate both stationary and self similar solutions to the three dimensional axi-symmetric incompressible Euler equations.
Based on recent ideas, stemming from the use of bubbles, we discuss an algorithm for the numerical simulation of the cubic nonlinear Schr{ö}dinger equation with harmonic potential in any dimension, which could be easily extended to other polynomial nonlinearities. For the linear part of the equation, the algorithm consists in discretizing the initial function as a sum of modulated complex functions, each one having its own set of parameters, and then updating the parameters exactly so that the modulated function remains a solution to the equation. When cubic interactions are introduced, the Dirac-Frenkel-MacLachlan principle is used to approximate the time evolution of parameters. We then obtain a grid-free algorithm in any dimension, and it is compared to a spectral method on numerical examples.
We construct radially symmetric self-similar blow-up profiles for the mass supercritical nonlinear Schrödinger equation $i\partial_t u + Δu + |u|^{p-1}u=0$ on $\mathbf{R}^d$, close to the mass critical case and for any space dimension $d\ge 1$. These profiles bifurcate from the ground state solitary wave. The argument relies on the classical matched asymptotics method suggested in [Sulem, C.; Sulem, P.-L., The nonlinear Schrödinger equation. Self-focusing and wave collapse. Applied Mathematical Sciences, 139. Springer-Verlag, New York, 1999] which needs to be applied in a degenerate case due to the presence of exponentially small terms in the bifurcation equation related to the log-log blow-up law observed in the mass critical case.
We consider the mass critical fractional (NLS). We show the existence of travelling waves for all mass below the ground state mass, and give a complete description of the associated profiles in the small mass limit. We therefore recover a situation similar to the one discovered in [Gerard P.; Lenzmann E.; Pocovnicu O.; Raphaël, P., A two soliton with transient turbulent regime for the one dimensional cubic half wave, submitted] for the critical case s = 1, but with a completely different asymptotic profile when the mass vanishes.
We consider the focusing cubic half-wave equation on the real line $$i \partial_t u + |D| u = |u|^2 u, \ \ \widehat{|D|u}(ξ)=|ξ|\hat{u}(ξ), \ \ (t,x)\in \Bbb R_+\times \Bbb R.$$ We construct an asymptotic global-in-time compact two-soliton solution with arbitrarily small $L^2$-norm which exhibits the following two regimes: (i) a transient turbulent regime characterized by a dramatic and explicit growth of its $H^1$-norm on a finite time interval, followed by (ii) a saturation regime in which the $H^1$-norm remains stationary large forever in time.
We consider the nonlinear Schrödinger equation $i\partial_tu+Δu+u|u|^{p-1}=0$ in dimension $N\geq 2$ and in the mass super critical and energy subcritical range $1+\frac 4N<p<\min\{\frac{N+2}{N-2},5\}.$ For initial data $u_0\in H^1$ with radial symmetry, we prove a universal upper bound on the blow up speed. We then prove that this bound is sharp and attained on a family of collapsing ring blow up solutions first formally predicted by Gavish, Fibich and Wang.
We consider the energy critical Schrödinger 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.
We exhibit $\mathcal C^{\infty}$ type II blow up solutions to the focusing energy critical wave equation in dimension $N=4$. These solutions admit near blow up time a decomposiiton $u(t,x)=1/l(t)(Q+e(t))(x/l(t)}}$ with $|e(t),\pa_t e(t)|_{\dot{H}^1\times L^2}<<1$ where $Q$ is the extremizing profile of the Sobolev embedding $\dot{H}^1\subset L^{2^*}$, and a blow up speed $l(t)=(T-t)e^{-\sqrt{|\log (T-t)|}(1+o(1))}$ as $t\to T.$