1,954 research outputs found
Numerical simulations for the energy-supercritical nonlinear wave equation
We carry out numerical simulations of the defocusing energy-supercritical
nonlinear wave equation for a range of spherically-symmetric initial
conditions. We demonstrate numerically that the critical Sobolev norm of
solutions remains bounded in time. This lends support to conditional scattering
results that have been recently established for nonlinear wave equations.Comment: 28 pages, 13 figures. New references and new cases adde
A new proof of scattering below the ground state for the non-radial focusing NLS
We revisit the scattering result of Duyckaerts, Holmer, and Roudenko for the
non-radial -critical focusing NLS. By proving an interaction
Morawetz inequality, we give a simple proof of scattering below the ground
state in dimensions that avoids the use of concentration compactness.Comment: 14 page
The defocusing energy-supercritical NLS in four space dimensions
We consider a class of defocusing energy-supercritical nonlinear
Schr\"odinger equations in four space dimensions. Following a
concentration-compactness approach, we show that for , any solution
that remains bounded in the critical Sobolev space must
be global and scatter. Key ingredients in the proof include a long-time
Strichartz estimate and a frequency-localized interaction Morawetz inequality.Comment: 52 page
The final-state problem for the cubic-quintic NLS with non-vanishing boundary conditions
We construct solutions with prescribed scattering state to the cubic-quintic
NLS in three spatial dimensions in the class
of solutions with as . This models
disturbances in an infinite expanse of (quantum) fluid in its quiescent state
--- the limiting modulus corresponds to a local minimum in the energy
density.
Our arguments build on work of Gustafson, Nakanishi, and Tsai on the
(defocusing) Gross--Pitaevskii equation. The presence of an energy-critical
nonlinearity and changes in the geometry of the energy functional add several
new complexities. One new ingredient in our argument is a demonstration that
solutions of such (perturbed) energy-critical equations exhibit continuous
dependence on the initial data with respect to the \emph{weak} topology on
.Comment: 46 page
Invariance of white noise for KdV on the line
We consider the Korteweg--de Vries equation with white noise initial data,
posed on the whole real line, and prove the almost sure existence of solutions.
Moreover, we show that the solutions obey the group property and follow a white
noise law at all times, past or future.
As an offshoot of our methods, we also obtain a new proof of the existence of
solutions and the invariance of white noise measure in the torus setting
- …