384 research outputs found
Exit problems related to the persistence of solitons for the Korteweg-de Vries equation with small noise
We consider two exit problems for the Korteweg-de Vries equation perturbed by
an additive white in time and colored in space noise of amplitude a. The
initial datum gives rise to a soliton when a=0. It has been proved recently
that the solution remains in a neighborhood of a randomly modulated soliton for
times at least of the order of a^{-2}. We prove exponential upper and lower
bounds for the small noise limit of the probability that the exit time from a
neighborhood of this randomly modulated soliton is less than T, of the same
order in a and T. We obtain that the time scale is exactly the right one. We
also study the similar probability for the exit from a neighborhood of the
deterministic soliton solution. We are able to quantify the gain of eliminating
the secular modes to better describe the persistence of the soliton
Nonlinear PDEs with modulated dispersion II: Korteweg--de Vries equation
We continue the study of various nonlinear PDEs under the effect of a
time--inhomogeneous and irregular modulation of the dispersive term. In this
paper we consider the modulated versions of the 1d periodic or non-periodic
Korteweg--de Vries (KdV) equation and of the modified KdV equation. For that we
use a deterministic notion of "irregularity" for the modulation and obtain
local and global results similar to those valid without modulation. In some
cases the irregularity of the modulation improves the well-posedness theory of
the equations. Our approach is based on estimates for the regularising effect
of the modulated dispersion on the non-linear term using the theory of
controlled paths and estimates stemming from Young's theory of integration.Comment: 37 page
Localization and Coherence in Nonintegrable Systems
We study the irreversible dynamics of nonlinear, nonintegrable Hamiltonian
oscillator chains approaching their statistical asympotic states. In systems
constrained by more than one conserved quantity, the partitioning of the
conserved quantities leads naturally to localized and coherent structures. If
the phase space is compact, the final equilibrium state is governed by entropy
maximization and the final coherent structures are stable lumps. In systems
where the phase space is not compact, the coherent structures can be collapses
represented in phase space by a heteroclinic connection to infinity.Comment: 41 pages, 15 figure
White noise for KdV and mKdV on the circle
We survey different approaches to study the invariance of the white noise for
the periodic KdV. We mainly discuss the following two methods. First, we
discuss the PDE method, following Bourgain \cite{BO4}, in a general framework.
Then, we show how it can be applied to the low regularity setting of the white
noise for KdV by introducing the Besov-type space \hat{b}^s_{p, \infty}, sp<
-1. Secondly, we describe the probabilistic method by Quastel, Valk\'o, and the
author \cite{OQV}. We also use this probabilistic approach to study the white
noise for mKdV.Comment: 26 pages. To appear in RIMS Kokyuroku Bessats
Global dynamics for the stochastic KdV equation with white noise as initial data
We study the stochastic Korteweg-de Vries equation (SKdV) with an additive
space-time white noise forcing, posed on the one-dimensional torus. In
particular, we construct global-in-time solutions to SKdV with spatial white
noise initial data. Due to the lack of an invariant measure, Bourgain's
invariant measure argument is not applicable to this problem. In order to
overcome this difficulty, we implement a variant of Bourgain's argument in the
context of an evolution system of measures and construct global-in-time
dynamics. Moreover, we show that the white noise measure with variance is
an evolution system of measures for SKdV with the white noise initial data.Comment: 41 pages. Minor typos corrected. To appear in Trans. Amer. Math. So
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