Showing papers by "Thierry Gallay published in 2007"
TL;DR: In this article, it was shown that these traveling waves are orbitally stable within the class of solutions having the same period and the same Floquet exponent, under a non-degeneracy condition which can be checked numerically.
Abstract: The nonlinear Schrodinger equation has several families of quasi-periodic traveling waves, each of which can be parametrized up to symmetries by two real numbers: the period of the modulus of the wave profile, and the variation of its phase over a period (Floquet exponent). In the defocusing case, we show that these travelling waves are orbitally stable within the class of solutions having the same period and the same Floquet exponent. This generalizes a previous work (Gallay and Haragus, J. Diff. Equations, 2007) where only small amplitude solutions were considered. A similar result is obtained in the focusing case, under a non-degeneracy condition which can be checked numerically. The proof relies on the general approach to orbital stability as developed by Grillakis, Shatah, and Strauss, and requires a detailed analysis of the Hamiltonian system satisfied by the wave profile.
110 citations
TL;DR: In this article, the authors investigated the stability of small amplitude traveling waves, both in the defocusing and the focusing case, and showed that these waves are orbitally stable within the class of solutions which have the same period and the same Floquet exponent as the original wave.
88 citations
Journal Article•
TL;DR: In this paper, the authors give a variational proof of global stability for bistable travelling waves of scalar reaction-diffusion equations on the real line, without any use of the maximum principle.
Abstract: We give a variational proof of global stability for bistable travelling waves of scalar reaction-diffusion equations on the real line. In particular, we recover some of the classical results by P. Fife and J.B. McLeod (1977) without any use of the maximum principle. The method that is illustrated here in the simplest possible setting has been successfully applied to more general parabolic or hyperbolic gradient-like systems.
32 citations
TL;DR: In this paper, the authors consider a weakly asymmetric strain and prove that non-axisymmetric vortices exist for all values of the Reynolds number, in the limit of large Reynolds numbers, and recover the asymptotic results of Moffatt, Kida & Ohkitani.
Abstract: Burgers vortices are stationary solutions of the three-dimensional Navier–Stokes equations in the presence of a background straining flow. These solutions are given by explicit formulas only when the strain is axisymmetric. In this paper we consider a weakly asymmetric strain and prove in that case that non-axisymmetric vortices exist for all values of the Reynolds number. In the limit of large Reynolds numbers, we recover the asymptotic results of Moffatt, Kida & Ohkitani [11]. We also show that the asymmetric vortices are stable with respect to localized two-dimensional perturbations.
29 citations
Posted Content•
TL;DR: In this article, the authors considered the damped wave equation and showed that the initial data are sufficiently close to the profile of a front for large |x|, and that the solution converges uniformly on R to a travelling front as t goes to plus infinity.
Abstract: We consider the damped wave equation \alpha u_tt + u_t = u_xx - V'(u) on the whole real line, where V is a bistable potential. This equation has travelling front solutions of the form u(x,t) = h(x-st) which describe a moving interface between two different steady states of the system, one of which being the global minimum of V. We show that, if the initial data are sufficiently close to the profile of a front for large |x|, the solution of the damped wave equation converges uniformly on R to a travelling front as t goes to plus infinity. The proof of this global stability result is inspired by a recent work of E. Risler and relies on the fact that our system has a Lyapunov function in any Galilean frame.
28 citations
TL;DR: In this paper, the large time behavior of non-negative solutions to the viscous Hamilton-Jacobi equation was investigated for the critical exponent q=(N+2)/(N+1).
Abstract: The large time behavior of non-negative solutions to the viscous Hamilton-Jacobi equation $\partial_t u - \Delta u + |
abla u|^q = 0$ is investigated for the critical exponent q=(N+2)/(N+1). Convergence towards a rescaled self-similar solution to the linear heat equation is shown, the rescaling factor being $(\ln{t})^{-(N+1)}$. The proof relies on the construction of a one-dimensional invariant manifold for a suitable truncation of the equation written in self-similar variables.
23 citations