122 research outputs found
On the propagation of an optical wave in a photorefractive medium
The aim of this paper is first to review the derivation of a model describing
the propagation of an optical wave in a photorefractive medium and to present
various mathematical results on this model: Cauchy problem, solitary waves
Stability in of the sum of solitons for the Benjamin-Ono equation
This note proves the orbital stability in the energy space of the
sum of widely-spaced 1-solitons for the Benjamin-Ono equation, with speeds
arranged so as to avoid collisions
Global well-posedness for the KP-I equation on the background of a non localized solution
We prove that the Cauchy problem for the KP-I equation is globally well-posed
for initial data which are localized perturbations (of arbitrary size) of a
non-localized (i.e. not decaying in all directions) traveling wave solution
(e.g. the KdV line solitary wave or the Zaitsev solitary waves which are
localized in and periodic or conversely)
A para-differential renormalization technique for nonlinear dispersive equations
For \alpha \in (1,2) we prove that the initial-value problem \partial_t
u+D^\alpha\partial_x u+\partial_x(u^2/2)=0 on \mathbb{R}_x\times\mathbb{R}_t;
u(0)=\phi, is globally well-posed in the space of real-valued L^2-functions. We
use a frequency dependent renormalization method to control the strong low-high
frequency interactions.Comment: 42 pages, no figure
Global generalized solutions for Maxwell-alpha and Euler-alpha equations
We study initial-boundary value problems for the Lagrangian averaged alpha
models for the equations of motion for the corotational Maxwell and inviscid
fluids in 2D and 3D. We show existence of (global in time) dissipative
solutions to these problems. We also discuss the idea of dissipative solution
in an abstract Hilbert space framework.Comment: 27 pages, to appear in Nonlinearit
Global well-posedness of the KP-I initial-value problem in the energy space
We prove that the KP-I initial value problem is globally well-posed in the
natural energy space of the equation
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