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

A scattering operator for some nonlinear elliptic equations

We consider non linear elliptic equations of the form $\Delta u = f(u,\nabla u)$ for suitable analytic nonlinearity $f$, in the vinicity of infinity in $\mathbb{R}^d$, that is on the complement of a compact set.We show that there is a \emph{one-to-one correspondence} between the non linear solution $u$ defined there, and the linear solution $u\_L$ to the Laplace equation, such that, in an adequate space, $u - u\_L\to 0$ as $|x|\to +\infty$. This is a kind of scattering operator.Our results apply in particular for the energy critical and supercritical pure power elliptic equation and for the 2d (energy critical) harmonic maps and the $H$-system. Similar results are derived for solution defined on the neighborhood of a point in $\mathbb{R}^d$. The proofs are based on a conformal change of variables, and studied as an evolution equation (with the radial direction playing the role of time) in spaces with analytic regularity on spheres (the directions orthogonal to the radial direction)

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

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

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