arXiv subjects

Xavier Friederich

Publications and source records attributed to Xavier Friederich.

4 recordsLinked to original sources

arXiv2022

Spatial decay of multi-solitons of the generalized Korteweg-de Vries and nonlinear Schr{ö}dinger equations

We study pointwise spatial decay of multi-solitons of the generalized Korteweg-de Vries equations. We obtain that, uniformly in time, these solutions and their derivatives decay exponentially in space on the left of and in the solitons region, and prove rapid decay on the right of the solitons. We also prove the corresponding result for multi-solitons of the nonlinear Schr{ö}dinger equations, that is, exponential decay in the solitons region and rapid decay outside.

math.AP↗

arXiv2021

On existence and uniqueness of asymptotic $N$-soliton-like solutions of the nonlinear klein-gordon equation

We are interested in solutions of the nonlinear Klein-Gordon equation (NLKG) in $\mathbb{R}^{1+d}$, $d\ge1$, which behave as a soliton or a sum of solitons in large time. In the spirit of other articles focusing on the supercritical generalized Korteweg-de Vries equations and on the nonlinear Schr{ö}dinger equations, we obtain an $N$-parameter family of solutions of (NLKG) which converges exponentially fast to a sum of given (unstable) solitons. For $N = 1$, this family completely describes the set of solutions converging to the soliton considered; for $N\ge 2$, we prove uniqueness in a class with explicit algebraic rate of convergence.

math.AP↗

arXiv2021

On smoothness and uniqueness of multi-solitons of the non-linear Schr{ö}dinger equations

In this paper, we study some properties of multi-solitons for the non-linear Schr{ö}dinger equations in R^d with general non-linearities. Multi-solitons have already been constructed in H^1, successively by Merle, by Martel and Merle, and by C{ô}te, Martel and Merle. We show here that multi-solitons are smooth, depending on the regularity of the non-linearity. We obtain also a result of uniqueness in some class, either when the ground states are all stable, or in the mass-critical case.

math.AP↗

arXiv2020

Non dispersive solutions of the generalized KdV equations are typically multi-solitons

We consider solutions of the generalized Korteweg-de Vries equations (gKdV) which are non dispersive in some sense (in the spirit of [18]) and which remain close to multi-solitons. We show that these solutions are necessarily pure multi-solitons. For the Korteweg-de Vries equation (KdV) and the modified Korteweg-de Vries equation (mKdV) in particular, we obtain a characterization of multi-solitons and multi-breathers in terms of non-dispersion.

math.AP↗