88 research outputs found
Local ill-posedness of the 1D Zakharov system
Ginibre-Tsutsumi-Velo (1997) proved local well-posedness for the Zakharov
system for any dimension , in the inhomogeneous Sobolev spaces for a range of exponents ,
depending on . Here we restrict to dimension and present a few results
establishing local ill-posedness for exponent pairs outside of the
well-posedness regime. The techniques employed are rooted in the work of
Bourgain (1993), Birnir-Kenig-Ponce-Svanstedt-Vega (1996), and
Christ-Colliander-Tao (2003) applied to the nonlinear Schroedinger equation
Fast soliton scattering by attractive delta impurities
We study the Gross-Pitaevskii equation with an attractive delta function
potential and show that in the high velocity limit an incident soliton is split
into reflected and transmitted soliton components plus a small amount of
dispersion. We give explicit analytic formulas for the reflected and
transmitted portions, while the remainder takes the form of an error. Although
the existence of a bound state for this potential introduces difficulties not
present in the case of a repulsive potential, we show that the proportion of
the soliton which is trapped at the origin vanishes in the limit
Phase-driven interaction of widely separated nonlinear Schr\"odinger solitons
We show that, for the 1d cubic NLS equation, widely separated equal amplitude
in-phase solitons attract and opposite-phase solitons repel. Our result gives
an exact description of the evolution of the two solitons valid until the
solitons have moved a distance comparable to the logarithm of the initial
separation. Our method does not use the inverse scattering theory and should be
applicable to nonintegrable equations with local nonlinearities that support
solitons with exponentially decaying tails. The result is presented as a
special case of a general framework which also addresses, for example, the
dynamics of single solitons subject to external forces
The Rigorous Derivation of the 2D Cubic Focusing NLS from Quantum Many-body Evolution
We consider a 2D time-dependent quantum system of -bosons with harmonic
external confining and \emph{attractive} interparticle interaction in the
Gross-Pitaevskii scaling. We derive stability of matter type estimates showing
that the -th power of the energy controls the Sobolev norm of the
solution over -particles. This estimate is new and more difficult for
attractive interactions than repulsive interactions. For the proof, we use a
version of the finite-dimensional quantum di Finetti theorem from [49]. A high
particle-number averaging effect is at play in the proof, which is not needed
for the corresponding estimate in the repulsive case. This a priori bound
allows us to prove that the corresponding BBGKY hierarchy converges to the GP
limit as was done in many previous works treating the case of repulsive
interactions. As a result, we obtain that the \emph{focusing} nonlinear
Schr\"{o}dinger equation is the mean-field limit of the 2D time-dependent
quantum many-body system with attractive interatomic interaction and
asymptotically factorized initial data. An assumption on the size of the
-norm of the interatomic interaction potential is needed that
corresponds to the sharp constant in the 2D Gagliardo-Nirenberg inequality
though the inequality is not directly relevant because we are dealing with a
trace instead of a power
A class of solutions to the 3d cubic nonlinear Schroedinger equation that blow-up on a circle
We consider the 3d cubic focusing nonlinear Schroedinger equation (NLS)
i\partial_t u + \Delta u + |u|^2 u=0, which appears as a model in condensed
matter theory and plasma physics. We construct a family of axially symmetric
solutions, corresponding to an open set in H^1_{axial}(R^3) of initial data,
that blow-up in finite time with singular set a circle in xy plane. Our
construction is modeled on Rapha\"el's construction \cite{R} of a family of
solutions to the 2d quintic focusing NLS, i\partial_t u + \Delta u + |u|^4 u=0,
that blow-up on a circle.Comment: updated introduction, expanded Section 21, added reference
Divergence of infinite-variance nonradial solutions to the 3d NLS equation
We consider solutions to the 3d NLS equation such that and is nonradial.
Denoting by and , the mass and energy, respectively, of a solution
, and by the ground state solution to , we
prove the following: if and , then either blows-up in
finite positive time or exists globally for all positive time and there
exists a sequence of times such that . Similar statements hold for negative time
- β¦