80 research outputs found
Talbot effect for the cubic nonlinear Schr\"odinger equation on the torus
We study the evolution of the one dimensional periodic cubic Schr\"odinger
equation (NLS) with bounded variation data. For the linear evolution, it is
known that for irrational times the solution is a continuous, nowhere
differentiable fractal-like curve. For rational times the solution is a linear
combination of finitely many translates of the initial data. Such a dichotomy
was first observed by Talbot in an optical experiment performed in 1836. In
this paper we prove that a similar phenomenon occurs in the case of the NLS
equation.Comment: 12 page
Improved interaction Morawetz inequalities for the cubic nonlinear Schr\"odinger equation on
We prove global well-posedness for low regularity data for the
defocusing nonlinear
Schr\"odinger equation (NLS) in 2d. More precisely we show that a global
solution exists for initial data in the Sobolev space and
any . This improves the previous result of Fang and Grillakis where
global well-posedness was established for any . We use the
-method to take advantage of the conservation laws of the equation. The new
ingredient is an interaction Morawetz estimate similar to one that has been
used to obtain global well-posedness and scattering for the cubic NLS in 3d.
The derivation of the estimate in our case is technical since the smoothed out
version of the solution introduces error terms in the interaction Morawetz
inequality. A byproduct of the method is that the norm of the solution
obeys polynomial-in-time bounds.Comment: 21 page
Existence and Uniqueness theory for the fractional Schr\"odinger equation on the torus
We study the Cauchy problem for the -d periodic fractional Schr\"odinger
equation with cubic nonlinearity. In particular we prove local well-posedness
in Sobolev spaces, for solutions evolving from rough initial data. In addition
we show the existence of global-in-time infinite energy solutions. Our tools
include a new Strichartz estimate on the torus along with ideas that Bourgain
developed in studying the periodic cubic NLS.Comment: 19 page
Remarks on global a priori estimates for the nonlinear Schr\"odinger equation
We present a unified approach for obtaining global a priori estimates for
solutions of nonlinear defocusing Schr\"odinger equations with defocusing
nonlinearities. The estimates are produced by contracting the local momentum
conservation law with appropriate vector fields. The corresponding law is
written for defocusing equations of tensored solutions. In particular, we
obtain a new estimate in two dimensions. We bound the restricted
Strichartz norm of the solution on any curve in
. For the specific case of a straight line we upgrade this
estimate to a weighted Strichartz estimate valid in the full plane
High frequency perturbation of cnoidal waves in KdV
The Korteweg-de Vries (KdV) equation with periodic boundary conditions is
considered. The interaction of a periodic solitary wave (cnoidal wave) with
high frequency radiation of finite energy (-norm) is studied. It is proved
that the interaction of low frequency component (cnoidal wave) and high
frequency radiation is weak for finite time in the following sense: the
radiation approximately satisfies Airy equation.Comment: 23 page
Near-linear Dynamics for Shallow Water Waves
It is shown that spatially periodic one-dimensional surface waves in shallow
water behave almost linearly, provided large part of the energy is contained in
sufficiently high frequencies. The amplitude is not required to be small (apart
from the shallow water approximation assumption) and the near-linear behavior
occurs on a much longer time scale than might be anticipated based on the
amplitude size. Heuristically speaking, this effect is due to the nonlinearity
getting averaged by the dispersive action.
This result is obtained by an averaging procedure, which is briefly outlined,
and is also confirmed by numerical simulations
The derivative nonlinear Schr\"odinger equation on the half line
We study the initial-boundary value problem for the derivative nonlinear
Schr\"odinger (DNLS) equation. More precisely we study the wellposedness theory
and the regularity properties of the DNLS equation on the half line. We prove
almost sharp local wellposedness, nonlinear smoothing, and small data global
wellposedness in the energy space. One of the obstructions is that the crucial
gauge transformation we use replaces the boundary condition with a nonlocal
one. We resolve this issue by running an additional fixed point argument.
Our method also implies almost sharp local and small energy global
wellposedness, and an improved smoothing estimate for the quintic Schr\"odinger
equation on the half line. In the last part of the paper we consider the DNLS
equation on and prove smoothing estimates by combining the restricted norm
method with a normal form transformation.Comment: 36 page
Smoothing for the fractional Schrodinger equation on the torus and the real line
In this paper we study the cubic fractional nonlinear Schrodinger equation
(NLS) on the torus and on the real line. Combining the normal form and the
restricted norm methods we prove that the nonlinear part of the solution is
smoother than the initial data. Our method applies to both focusing and
defocusing nonlinearities. In the case of full dispersion (NLS) and on the
torus, the gain is a full derivative, while on the real line we get a
derivative smoothing with an loss. Our result lowers the regularity
requirement of a recent theorem of Kappeler et al. on the periodic defocusing
cubic NLS, and extends it to the focusing case and to the real line. We also
obtain estimates on the higher order Sobolev norms of the global smooth
solutions in the defocusing case.Comment: 22 page
The Fifth Order KP--II Equation on the Upper Half--plane
In this paper we study the fifth order Kadomtsev--Petviashvili II (KP--II)
equation on the upper half-plane . In particular we
obtain low regularity local well-posedness using the restricted norm method of
Bourgain and the Fourier-Laplace method of solving initial and boundary value
problems. Moreover we prove that the nonlinear part of the solution is in a
smoother space than the initial data.Comment: 39 page
Low-regularity global well-posedness for the Klein-Gordon-Schr\"odinger system on
In this paper we establish an almost optimal well-posedness and regularity
theory for the Klein-Gordon-Schr\"odinger system on the half line. In
particular we prove local-in-time well-posedness for rough initial data in
Sobolev spaces of negative indices. Our results are consistent with the sharp
well-posedness results that exist in the full line case and in this sense
appear to be sharp. Finally we prove a global well-posedness result by
combining the conservation law of the Schr\"odinger part with a careful
iteration of the rough wave part in lower order Sobolev norms.Comment: 34 page
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