Yvan Martel

Stability and Asymptotic Stability for Subcritical gKdV Equations

Abstract: We prove in this paper the stability and asymptotic stability in H1 of a decoupled sum of N solitons for the subcritical generalized KdV equations The proof of the stability result is based on energy arguments and monotonicity of the local L2 norm. Note that the result is new even for p=2 (the KdV equation). The asymptotic stability result then follows directly from a rigidity theorem in [16].

A Liouville theorem for the critical generalized Korteweg-de Vries equation

Abstract We prove in this paper a rigidity theorem on the flow of the critical generalized Korteweg–de Vries equation close to a soliton up to scaling and translation. To prove this result we introduce new tools to understand nonlinear phenomenon. This will give a result of asymptotic completeness.

Asymptotic stability of solitons of the subcritical gKdV equations revisited

We consider the generalized Korteweg–de Vries (gKdV) equations in the subcritical cases p = 2, 3 or 4. The first objective of this paper is to present a direct, simplified proof of the asymptotic stability of solitons in the energy space H1, which was proved by the same authors in Martel and Merle (2001 Arch. Ration. Mech. Anal. 157 219–54).Then, in the case of the KdV equation, we show the optimality of the result by constructing a solution which behaves asymptotically as a soliton located on a curve which is a logarithmic perturbation of a line. This example justifies the apparently weak control on the location of the soliton in the asymptotic stability result.

Blow up in finite time and dynamics of blow up solutions for the ²–critical generalized KdV equation

In this paper, we are interested in the phenomenon of blow up in finite time (or formation of singularity in finite time) of solutions of the critical generalized KdV equation. Few results are known in the context of partial differential equations with a Hamiltonian structure. For the semilinear wave equation, or more generally for hyperbolic systems, the finite speed of propagation allows one to build blowing up solutions by reducing the problem to an ordinary differential equation. For the nonlinear Schrödinger equation, iut = −∆u− |u|p−1u, where u : R×R → C, (1)

Stability of blow-up profile and lower bounds for blow-up rate for the critical generalized KdV equation

The generalized Korteweg-de Vries equations are a class of Hamiltonian systems in infinite dimension derived from the KdV equation where the quadratic term is replaced by a higher order power term. These equations have two conservation laws in the energy space H 1 (L 2 norm and energy). We consider in this paper the critical generalized KdV equation, which corresponds to the smallest power of the nonlinearity such that the two conservation laws do not imply a bound in H 1 uniform in time for all H 1 solutions (and thus global existence). From [15], there do exist for this equation solutions u(t) such that |u(t)| H1 → +∞ as t ↑ T, where T < +∞ (we call them blow-up solutions). The question is to describe, in a qualitative way, how blow up occurs. For solutions with L 2 mass close to the minimal mass allowing blow up and with decay in L 2 at the right, we prove after rescaling and translation which leave invariant the L 2 norm that the solution converges to a universal profile locally in space at the blow-up time T. From the nature of this profile, we improve the standard lower bound on the blow-up rate for finite time blow-up solutions.

Instability of solitons for the critical generalized Korteweg—de Vries equation

Abstract. We prove in this paper the instability of the solitons for the critical generalized Korteweg—de Vries equation. We obtain this result by a qualitative study of the solution close to the soliton and obtain a large set of initial data giving the instability. This study is a step towards the existence and description of blow up solutions.

Stability in

In this article we consider nonlinear Schrodinger (NLS) equations in R for d = 1, 2, and 3. We consider nonlinearities satisfying a flatness condition at zero and such that solitary waves are stable. Let Rk(t, x) be K solitary wave solutions of the equation with different speeds v1, v2, . . . , vK . Provided that the relative speeds of the solitary waves vk − vk−1 are large enough and that no interaction of two solitary waves takes place for positive time, we prove that the sum of the Rk(t) is stable for t 0 in some suitable sense in H 1. To prove this result, we use an energy method and a new monotonicity property on quantities related to momentum for solutions of the nonlinear Schrodinger equation. This property is similar to the L2 monotonicity property that has been proved by Martel and Merle for the generalized Korteweg–de Vries (gKdV) equations (see [12, Lem. 16, proof of Prop. 6]) and that was used to prove the stability of the sum of K solitons of the gKdV equations by the authors of the present article (see [15, Th. 1(i)]).

H^1

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

of the sum of

t\to +\infty

K

, 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.