Raphaël Côte

Construction of multi-soliton solutions for the

Multi-soliton solutions, i.e. solutions behaving as the sum of N given solitons as

L^2

t\to +\infty

-supercritical gKdV and NLS equations

, were constructed in previous works for the L2 critical and subcritical (NLS) and (gKdV) equations. In this paper, we extend the construction of multi-soliton solutions to the L2 supercritical case both for (gKdV) and (NLS) equations, using a topological argument to control the direction of instability.

High-speed excited multi-solitons in nonlinear Schrödinger equations

Abstract We consider the nonlinear Schrodinger equation in R d i ∂ t u + Δ u + f ( u ) = 0 . For d ⩾ 2 , this equation admits traveling wave solutions of the form e i ω t Φ ( x ) (up to a Galilean transformation), where Φ is a fixed profile, solution to − Δ Φ + ω Φ = 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.

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 Schrödinger (NLS) dynamics, are strongly asymptotically stable in the energy space. 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 [Proc R Soc Edinburgh 126:89–112, 1996]. Our proofs follow the ideas of Martel [SIAM J Math Anal 157:759–781, 2006] and Martel and Merle [Math Ann 341:391–427, 2008], 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.

Multi-travelling waves for the nonlinear Klein-Gordon equation

For the nonlinear Klein-Gordon equation in R 1+d , we prove the existence of multi-solitary waves made of any number N of decoupled bound states. This extends the work of Cote and Munoz (Forum Math. Sigma 2 (2014)) which was restricted to ground states, as were most previous similar results for other nonlinear dispersive and wave models.

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

Abstract We prove weighted estimates on the linear KdV group, which are scaling sharp. This kind of estimates is in the spirit of that used to prove small data scattering for the generalized KdV equations. To cite this article: R. Cote, L. Vega, C. R. Acad. Sci. Paris, Ser. I 346 (2008).

Limiting Motion for the Parabolic Ginzburg–Landau Equation with Infinite Energy Data

We study a class of solutions to the parabolic Ginzburg–Landau equation in dimension 2 or higher, with ill-prepared infinite energy initial data. We show that, asymptotically, the vorticity evolves according to motion by mean curvature in Brakke’s weak formulation. Then, we prove that in the plane, point vortices do not move in the original time scale. These results extend the works of Bethuel, Orlandi and Smets (Ann Math (2) 163(1):37–163, 2006; Duke Math J 130(3):523–614, 2005) to infinite energy data; they allow us to consider point vortices on a lattice (in dimension 2), or filament vortices of infinite length (in dimension 3).