Asymptotic Stability of high-dimensional Zakharov-Kuznetsov solitons

We prove that solitons (or solitary waves) of the Zakharov-Kuznetsov (ZK) equation, a physically relevant high dimensional generalization of the Korteweg-de Vries (KdV) equation appearing in Plasma Physics, and having mixed KdV and nonlinear Schr\"odinger (NLS) dynamics, are strongly asymptotically stable in the energy space in the physical region. We also prove that the sum of well-arranged solitons is stable in the same space. Orbital stability of ZK solitons is well-known since the work of de Bouard. Our proofs follow the ideas by Martel and Martel and Merle, applied for generalized KdV equations in one dimension. In particular, we extend to the high dimensional case several monotonicity properties for suitable half-portions of mass and energy; we also prove a new Liouville type property that characterizes ZK solitons, and a key Virial identity for the linear and nonlinear part of the ZK dynamics, obtained independently of the mixed KdV-NLS dynamics. This last Virial identity relies on a simple sign condition, which is numerically tested for the two and three dimensional cases, with no additional spectral assumptions required. Possible extensions to higher dimensions and different nonlinearities could be obtained after a suitable local well-posedness theory in the energy space, and the verification of a corresponding sign condition.Comment: 61 pages, 10 figures, accepted version including referee comment

Scaling-sharp dispersive estimates for the Korteweg-de Vries group

We prove weighted estimates on the linear KdV group, which are scaling sharp. This kind of estimates are in the spirit of that used to prove small data scattering for the generalized KdV equations.Comment: 5 page

Multi-solitons for nonlinear Klein-Gordon equations

International audienceIn this paper we consider the existence of multi-soliton structures for the nonlinear Klein-Gordon equation (NLKG) in R^{1+d}. We prove that, independently of the unstable character of (NLKG) solitons, it is possible to construct a N-soliton family of solutions to (NLKG), of dimension 2N, globally well-defined in the energy space H^1 \times L^2 for all large positive times. The method of proof involves the generalization of previous works on supercritical NLS and gKdV equations by Martel, Merle and the first author to the wave case, where we replace the unstable mode associated to the linear NLKG operator by two generalized directions that are controlled without appealing to modulation theory. As a byproduct, we generalize the linear theory described in Grillakis-Shatah-Strauss and Duyckaerts-Merle to the case of boosted solitons, and provide new solutions to be studied using the recent Nakanishi- Schlag theory

Improved uniqueness of multi-breathers of the modified Korteweg-de Vries equation

We consider multi-breathers of (mKdV). Previously, a smooth multi-breather was constructed, and proved to be unique in two cases: first, if the class of super-polynomial convergence to the profile, and second, under the assumption that all speeds of the breathers involved are positive (without rate of convergence). The goal of this short note is to improve the second result: we show that uniqueness still holds if at most one velocity is negative or zero

Scattering below critical energy for the radial 4D Yang-Mills equation and for the 2D corotational wave map system

We describe the asymptotic behavior as time goes to infinity of solutions of the 2 dimensional corotational wave map system and of solutions to the 4 dimensional, radially symmetric Yang-Mills equation, in the critical energy space, with data of energy smaller than or equal to a harmonic map of minimal energy. An alternative holds: either the data is the harmonic map and the soltuion is constant in time, or the solution scatters in infinite time

High speed excited multi-solitons in nonlinear Schrödinger equations

Abstract We consider the nonlinear Schrödinger equation in R d i∂ t u + ∆u + f (u) = 0. For d 2, this equation admits travelling wave solutions of the form e iωt Φ(x) (up to a Galilean transformation), where Φ is a fixed profile, solution to −∆Φ + ωΦ = f (Φ), but not the ground state. This kind of profiles are called excited states. In this paper, we construct solutions to NLS behaving like a sum of N excited states which spread up quickly as time grows (which we call multi-solitons). We also show that if the flow around one of these excited states is linearly unstable, then the multi-soliton is not unique, and is unstable. Résumé On considère l'équation de Schrödinger non-linéaire dans R Pour d 2, cetteéquation admet des ondes progressives de la forme e iωt Φ(x) (à une transformation galiléenne près), où Φ est un profil fixe, solution de −∆Φ + ωΦ = f (Φ), mais pas unétat fondamental. Ces profils sont appelésétats excités. Dans cet article, nous construisons des solutions de NLS se comportant comme une somme d'états excités qui se séparent rapidement au cours du temps (nous les appelons multisolitons). Nous montrons aussi que si le flot autour d'un desétats excités est linéairement instable, alors le multi-soliton n'est pas unique et est instable