Results 21 to 30 of about 63 (30)

Transverse nonlinear instability of solitary waves for some Hamiltonian PDE's [PDF]

arXiv, 2008
We present a general result of transverse nonlinear instability of 1-d solitary waves for Hamiltonian PDE's for both periodic or localized transverse perturbations. Our main structural assumption is that the linear part of the 1d model and the transverse perturbation "have the same sign".
arxiv  

Random data Cauchy problem for supercritical Schrödinger equations [PDF]

arXiv, 2009
In this paper we consider the Schr\"odinger equation with power-like nonlinearity and confining potential or without potential. This equation is known to be well-posed with data in a Sobolev space $\H^{s}$ if $s$ is large enough and strongly ill-posed is $s$ is below some critical threshold $s_{c}$.
arxiv  

Long time dynamics for the one dimensional non linear Schrödinger equation [PDF]

arXiv, 2010
In this article, we first present the construction of Gibbs measures associated to nonlinear Schr\"odinger equations with harmonic potential. Then we show that the corresponding Cauchy problem is globally well-posed for rough initial conditions in a statistical set (the support of the measures).
arxiv  

Space-time resonances [PDF]

arXiv, 2011
This article is a short exposition of the space-time resonances method. It was introduced by Masmoudi, Shatah, and the author, in order to understand global existence for nonlinear dispersive equations, set in the whole space, and with small data. The idea is to combine the classical concept of resonances, with the feature of dispersive equations: wave
arxiv  

Global existence for the Euler-Maxwell system [PDF]

arXiv, 2011
The Euler-Maxwell system describes the evolution of a plasma when the collisions are important enough that each species is in a hydrodynamic equilibrium. In this paper we prove global existence of small solutions to this system set in the whole three-dimensional space, by combining the space-time resonance method, dispersive estimates, localization ...
arxiv  

Unconditional global well-posedness for the 3D Gross-Pitaevskii equation for data without finite energy [PDF]

arXiv, 2012
The Cauchy problem for the Gross-Pitaevskii equation in three space dimensions is shown to have an unconditionally unique global solution for data of the form 1 + H^s for 5/6 < s < 1, which do not have necessarily finite energy. The proof uses the I-method which is complicated by the fact that no L^2 -conservation law holds.
arxiv  

Global solutions for 3D nonlocal Gross-Pitaevskii equations with rough data [PDF]

Electron. J. Diff. Equ., Vol. 2012 (2012), No. 170, pp. 1-34, 2012
We study the Cauchy problem for the Gross-Pitaevskii equation with a nonlocal interaction potential of Hartree type in three space dimensions. If the potential is even and positive definite or a positive function and its Fourier transform decays sufficiently rapidly the problem is shown to be globally well-posed for large rough data which not ...
arxiv  

Long time dynamics for weakly damped nonlinear Klein-Gordon equations [PDF]

arXiv, 2018
We continue our study of damped nonlinear Klein-Gordon equations. In our previous work we considered fixed positive damping and proved a form of the soliton resolution conjecture for radial solutions. In contrast, here we consider damping which decreases in time to 0.
arxiv  

On Local Well-posedness of the Periodic Korteweg-de Vries Equation Below $H^{-\frac{1}{2}}(\mathbb{T})$ [PDF]

arXiv
We utilize a modulation restricted normal form approach to establish local well-posedness of the periodic Korteweg-de Vries equation in $H^s(\mathbb{T})$ for $s> -\frac23$. This work creates an analogue of the mKdV result by Nakanishi, Takaoka, and Tsutsumi for KdV, extending the currently best-known result of $s \geq -\frac12$ without utilizing the ...
arxiv  

Long-time asymptotics of the modified KdV equation in weighted Sobolev spaces [PDF]

arXiv, 2019
The long time behavior of solutions to the defocusing modified Korteweg-de vries (MKdV) equation is established for initial conditions in some weighted Sobolev spaces. Our approach is based on the nonlinear steepest descent method of Deift and Zhou and its reformulation by Dieng and McLaughlin through $\overline{\partial}$-derivatives.
arxiv