A co-publication with EDP Sciences, France, licensed for sale in all countries outside of France. Sold and distributed within France by EDP Sciences, 17, av. du Hoggar F-91944 Les Ulis, France EDP Sciences ISBN 978-2-7598-0009-4

The nonlinear Schrödinger equation has several families of quasi-periodic travelling 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 [12] 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 [15,16], and requires a detailed analysis of the Hamiltonian system satisfied by the wave profile. Running head: Periodic waves in the NLS equationCorresponding author: Thierry Gallay, Thierry.Gallay@ujf-grenoble.fr

The nonlinear Schrödinger equation possesses three distinct six-parameter families of complex-valued quasiperiodic traveling waves, one in the defocusing case and two in the focusing case. All these solutions have the property that their modulus is a periodic function of x − ct for some c ∈ R. In this paper we investigate the stability of the small amplitude traveling waves, both in the defocusing and the focusing case. Our first result shows 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. Next, we consider general bounded perturbations and focus on spectral stability. We show that the small amplitude traveling waves are stable in the defocusing case, but unstable in the focusing case. The instability is of side-band type, and therefore cannot be detected in the periodic set-up used for the analysis of orbital stability.

We study existence and stability of interfaces in reaction-diffusion systems which are asymptotically planar. The problem of existence of corners is reduced to an ordinary differential equation that can be viewed as the travelling-wave equation to a viscous conservation law or variants of the Kuramoto-Sivashinsky equation. The corner typically, but not always, points in the direction opposite to the direction of propagation. For the existence and stability problem, we rely on a spatial dynamics formulation with an appropriate equivariant parameterization for relative equilibria.  2005 Elsevier SAS. All rights reserved. RésuméNous étudions l'existence et la stabilité des interfaces asymptotiquement planes dans des systèmes de réaction-diffusion. Le problème de l'existence des défauts est réduit à l'étude d'une équation différentielle ordinaire qui est, selon le cas, approchée par l'équation stationnaire d'une loi de conservation scalaire ou d'une variante de l'équation de Kuramoto-Sivashinsky. Typiquement, les défauts pointent dans la direction opposée à la direction de propagation. Pour l'analyse des problèmes d'existence et de stabilité, nous utilisons une formulation de type dynamique spatiale combinée avec une paramétrisation adéquate d'équilibres relatifs.Hypothesis 2.1 (Existence). We assume that there exists c * > 0 and asymptotic states q ± such that there exists an x-independent planar travelling-wave solution q * (y) of (2.2)(2.5)We emphasize that we allow for the possibility of pulses, q + = q − . The second assumption in this section is concerned with stability of the above travelling wave solution. Therefore, consider the linearized operator( 2.6) Notice that under suitable decay assumptions, q * belongs to the kernel of L * due to the translation invariance in y.Hypothesis 2.2 (Zero-stability). We assume that L * − λ id is invertible for all λ < 0 and that λ = 0 is an isolated eigenvalue with algebraic multiplicity one.Although this might not seem obvious, Hypothesis 2.2 is intimately related to stability properties of the travelling wave q * (·). Consider the linearized operatorand its Fourier conjugates(2.8) Hypothesis 2.3 (Transverse asymptotic stability). Assume that the travelling wave is asymptotically stable in one space dimension, that is, the essential spectrum of M 0 is strictly contained in the left complex half plane and zero is the only eigenvalue in the closed right half plane, with algebraic multiplicity one. Furthermore, assume that the spectra of M k , for k = 0 are strictly contained in the left half plane and that the unique eigenvalue λ d (k), k ∼ 0 with λ d (0) = λ d (0) = 0 satisfies λ d (0) < 0; see Fig. 1. Remark 2.4.(i) This hypothesis and, in particular, the quadratic tangency of the dispersion relation λ d (0) < 0 implies asymptotic stability of the travelling wave with respect to perturbations that are sufficiently localized in the transverse, x-direction [28,35,36]. Fig. 1. To the left the spectrum of M 0 and to the right the critical spectra of the M k parameterized by k.

This article presents a rigorous existence theory for small-amplitude three-dimensional travelling water waves. The hydrodynamic problem is formulated as an infinite-dimensional Hamiltonian system in which an arbitrary horizontal spatial direction is the time-like variable. Wave motions which are periodic in a second, different horizontal direction are detected using a centre-manifold reduction technique by which the problem is reduced to a locally equivalent Hamiltonian system with a finite number of degrees of freedom.A catalogue of bifurcation scenarios is compiled by means of a geometric argument based upon the classical dispersion relation for travelling water waves. Taking all parameters into account, one finds that this catalogue includes virtually any bifurcation or resonance known in Hamiltonian systems theory. Nonlinear bifurcation theory is carried out for a representative selection of bifurcation scenarios; solutions of the reduced Hamiltonian system are found by applying results from the well-developed theory of finite-dimensional Hamiltonian systems such as the Lyapunov centre theorem and the Birkhoff normal form.We find oblique line waves which depend only upon one spatial direction which is not aligned with the direction of wave propagation; the waves have periodic, solitary-wave or generalised solitary-wave profiles in this distinguished direction. Truly three-dimensional waves are also found which have periodic, solitary-wave or generalised solitary-wave profiles in one direction and are periodic in another. In particular, we recover doubly periodic waves with arbitrary fundamental domains and oblique versions of the results on threedimensional travelling waves already in the literature.

We prove the existence of domain walls for the Bénard-Rayleigh convection problem. Our approach relies upon a spatial dynamics formulation of the hydrodynamic problem, a center manifold reduction, and a normal form analysis of a reduced system. Domain walls are constructed as heteroclinic orbits of this reduced system.

We study the existence of grain boundaries in the Swift-Hohenberg equation. The analysis relies on a spatial dynamics formulation of the existence problem and a centre-manifold reduction. In this setting, the grain boundaries are found as heteroclinic orbits of a reduced system of ODEs in normal form. We show persistence of the leading-order approximation using transversality induced by wavenumber selection.

In the present work, we aim at taking a step towards the spectral stability analysis of Peregrine solitons, i.e., wave structures that are used to emulate extreme wave events. Given the space-time localized nature of Peregrine solitons, this is a priori a nontrivial task. Our main tool in this effort will be the study of the spectral stability of the periodic generalization of the Peregrine soliton in the evolution variable, namely the Kuznetsov-Ma breather. Given the periodic structure of the latter, we compute the corresponding Floquet multipliers, and examine them in the limit where the period of the orbit tends to infinity. This way, we extrapolate towards the stability of the limiting structure, namely the Peregrine soliton. We find that multiple unstable modes of the background are enhanced, yet no additional unstable eigenmodes arise as the Peregrine limit is approached. We explore the instability evolution also in direct numerical simulations.