368 research outputs found
LECTURES ON NONLINEAR DISPERSIVE EQUATIONS I
CONTENTS
J. Bona
Derivation and some fundamental properties of nonlinear dispersive waves equations
F. Planchon
Schr\"odinger equations with variable coecients
P. Rapha\"el
On the blow up phenomenon for the L^2 critical non linear Schrodinger Equatio
Construction of two-bubble solutions for energy-critical wave equations
We construct pure two-bubbles for some energy-critical wave equations, that
is solutions which in one time direction approach a superposition of two
stationary states both centered at the origin, but asymptotically decoupled in
scale. Our solution exists globally, with one bubble at a fixed scale and the
other concentrating in infinite time, with an error tending to 0 in the energy
space. We treat the cases of the power nonlinearity in space dimension 6, the
radial Yang-Mills equation and the equivariant wave map equation with
equivariance class k > 2. The concentrating speed of the second bubble is
exponential for the first two models and a power function in the last case.Comment: 44 pages; the new version fixes an error in the proof of Theorem
Systems of Points with Coulomb Interactions
Large ensembles of points with Coulomb interactions arise in various settings
of condensed matter physics, classical and quantum mechanics, statistical
mechanics, random matrices and even approximation theory, and give rise to a
variety of questions pertaining to calculus of variations, Partial Differential
Equations and probability. We will review these as well as "the mean-field
limit" results that allow to derive effective models and equations describing
the system at the macroscopic scale. We then explain how to analyze the next
order beyond the mean-field limit, giving information on the system at the
microscopic level. In the setting of statistical mechanics, this allows for
instance to observe the effect of the temperature and to connect with
crystallization questions.Comment: 30 pages, to appear as Proceedings of the ICM201
Blow-up dynamics for smooth finite energy radial data solutions to the self-dual Chern-Simons-Schr\"odinger equation
We consider the finite-time blow-up dynamics of solutions to the self-dual
Chern-Simons-Schr\"odinger (CSS) equation (also referred to as the Jackiw-Pi
model) near the radial soliton with the least -norm (ground state).
While a formal application of pseudoconformal symmetry to gives rise to an
-continuous curve of initial data sets whose solutions blow up in finite
time, they all have infinite energy due to the slow spatial decay of . In
this paper, we exhibit initial data sets that are smooth finite energy radial
perturbations of , whose solutions blow up in finite time. Interestingly,
their blow-up rate differs from the pseudoconformal rate by a power of
logarithm. Applying pseudoconformal symmetry in reverse, this also yields a
first example of an infinite-time blow-up solution, whose blow-up profile
contracts at a logarithmic rate.
Our analysis builds upon the ideas of previous works of the first two authors
on (CSS) [21,22], as well as the celebrated works on energy-critical geometric
equations by Merle, Rapha\"el, and Rodnianski [33,38]. A notable feature of
this paper is a systematic use of nonlinear covariant conjugations by the
covariant Cauchy-Riemann operators in all parts of the argument. This not only
overcomes the nonlocality of the problem, which is the principal challenge for
(CSS), but also simplifies the structure of nonlinearity arising in the proof.Comment: 80 page
Fourth order real space solver for the time-dependent Schr\"odinger equation with singular Coulomb potential
We present a novel numerical method and algorithm for the solution of the 3D
axially symmetric time-dependent Schr\"odinger equation in cylindrical
coordinates, involving singular Coulomb potential terms besides a smooth
time-dependent potential. We use fourth order finite difference real space
discretization, with special formulae for the arising Neumann and Robin
boundary conditions along the symmetry axis. Our propagation algorithm is based
on merging the method of the split-operator approximation of the exponential
operator with the implicit equations of second order cylindrical 2D
Crank-Nicolson scheme. We call this method hybrid splitting scheme because it
inherits both the speed of the split step finite difference schemes and the
robustness of the full Crank-Nicolson scheme. Based on a thorough error
analysis, we verified both the fourth order accuracy of the spatial
discretization in the optimal spatial step size range, and the fourth order
scaling with the time step in the case of proper high order expressions of the
split-operator. We demonstrate the performance and high accuracy of our hybrid
splitting scheme by simulating optical tunneling from a hydrogen atom due to a
few-cycle laser pulse with linear polarization
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