Xavier Friederich’s research while affiliated with Institut de Recherche Mathématique Avancée and other places

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Publications (10)


Spatial decay of multi-solitons of the generalized Korteweg-de Vries and nonlinear Schrödinger equations
  • Article
  • Publisher preview available

October 2022

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10 Reads

Mathematische Annalen

Raphaël Côte

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Xavier Friederich

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 Schrödinger equations, that is, exponential decay in the solitons region and rapid decay outside.

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On existence and uniqueness of asymptotic N-soliton-like solutions of the nonlinear Klein–Gordon equation

September 2022

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6 Reads

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3 Citations

Mathematische Zeitschrift

We are interested in solutions of the nonlinear Klein–Gordon equation (NLKG) in R1+dR1+d\mathbb {R}^{1+d}, d≥1d1d\ge 1, 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 Schrödinger equations, we obtain an N-parameter family of solutions of (NLKG) which converges exponentially fast to a sum of N given (unstable) solitons. For N=1N=1, this family completely describes the set of solutions converging to the soliton considered; for N≥2N2N\ge 2, we prove uniqueness in a class with explicit algebraic rate of convergence.


Spatial decay of the multi-solitons of the generalized Korteweg-de Vries equation

March 2022

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7 Reads

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. The behavior in this last region is based on a new argument involving a triangular induction process.


On smoothness and uniqueness of multi-solitons of the non-linear Schrödinger equations

July 2021

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13 Reads

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24 Citations

Communications in Partial Differential Equations

In this paper, we study some properties of multi-solitons for the non-linear Schrödinger equations in Rd with general non-linearities. Multi-solitons have already been constructed in H1(Rd) in papers by Merle (1990), Martel and Merle (2006), and Côte, Martel and Merle (2011). 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.


On qualitative properties of multi-solitons

June 2021

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11 Reads

This thesis is devoted to the qualitative properties of the multi-solitons of three nonlinear focusing dispersive partial differential equations. The first new results concern the nonlinear Schrödinger equation (NLS). We construct smooth multi-solitons and prove a conditional uniqueness result for the multi-solitons of (NLS) in the stable and L2-critical cases. Moreover, we state a Liouville property in the neighborhood of the multi-solitons of the generalized Korteweg-de Vries equation (gKdV). It consists of a rigidity result which relies on the concept of non dispersion. In the integrable cases, this concept allows us to characterize the multi-solitons. We also study the pointwise behavior of the multi-solitons of (gKdV). Lastly we consider the nonlinear Klein-Gordon equation (NLKG). We construct an N-parameter family of N-solitons with exponential decrease in time which is unique in a class with polynomial decay. In the case of one solitary wave, the classification is obtained in a general way.


PROPRIETES QUALITATIVES DES MULTI-SOLITONS

June 2021

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21 Reads

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2 Citations

Cette thèse est consacrée aux propriétés qualitatives des multi-solitons de trois équations aux dérivées partielles non-linéaires dispersives focalisantes.Les premiers résultats nouveaux concernent l'équation de Schrödinger non-linéaire (NLS). Nous construisons des multi-solitons réguliers, à valeurs dans des espaces de Sobolev HsH^s, où s>1s>1 est directement relié à la régularité de la non-linéarité. Nous démontrons également un résultat d'unicité sous condition pour les multi-solitons de (NLS) dans les cas stable et L2L^2-critique.Par ailleurs, nous énonçons une propriété de Liouville au voisinage des multi-solitons de l'équation de Korteweg-de Vries généralisée (gKdV). Il s'agit d'un résultat de rigidité qui repose sur le concept de non-dispersion. En particulier dans les cas intégrables, ce concept permet de caractériser les multi-solitons. Dans une autre mesure, nous étudions le comportement ponctuel des multi-solitons de (gKdV).Enfin, nous nous intéressons à l'équation de Klein-Gordon non-linéaire (NLKG). Pour cette dernière, nous construisons une famille à N paramètres de N-solitons à décroissance exponentielle en temps. En ce qui concerne la question de l'unicité, nous classifions les multi-solitons dans une certaine classe à décroissance polynomiale et obtenons la classification générale dans le cas où l'on considère une seule onde solitaire.


On Existence and Uniqueness of Asymptotic N-Soliton-Like Solutions of the Nonlinear Klein-Gordon Equation

June 2021

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4 Reads

We are interested in solutions of the nonlinear Klein-Gordon equation (NLKG) in R1+d\mathbb{R}^{1+d}, d1d\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{\"o}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=1N = 1, this family completely describes the set of solutions converging to the soliton considered; for N2N\ge 2, we prove uniqueness in a class with explicit algebraic rate of convergence.


Non dispersive solutions of the generalized Korteweg-de Vries equations are typically multi-solitons

January 2021

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8 Reads

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4 Citations

Annales de l Institut Henri Poincaré C Analyse Non Linéaire

We consider solutions of the generalized Korteweg-de Vries equations (gKdV) which are non dispersive in some sense 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.


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

July 2020

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12 Reads

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.


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

June 2020

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15 Reads

In this paper, we study some properties of multi-solitons for the non-linear Schr{\"o}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{\^o}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.

Citations (3)


... Again in this paper, specific monotonicity formulae are key tools. In the works by Côte-Friederich [16] and Friederich [19], assuming certain algebraic decay rates in time, they are able to prove uniqueness of multi-solitons in those classes for various models. In our recent work, Chen-Jendrej [12], instead of the weak convergence argument, we used a fixed point argument, which naturally results in the uniqueness, to construct pure multikink (soliton) solutions for 1 + 1 scalar field models but we have to restrict ourselves onto the class of exponential multi-kink solution, i.e., solutions converging to multi-kink exponentially. ...

Reference:

Asymptotic Stability and classification of multi-solitons for Klein-Gordon equations
On existence and uniqueness of asymptotic N-soliton-like solutions of the nonlinear Klein–Gordon equation

Mathematische Zeitschrift

... arisen from population genetics [3,22], and the important double-power nonlinearity f (u) = −u − λ|u| q−1 u + µ|u| p−1 u, 1 < p = q < n + 2 n − 2 , λ, µ > 0, (1.10) often arisen from the corresponding nonlinear Schrödinger equations [16,31,36,58]. ...

On smoothness and uniqueness of multi-solitons of the non-linear Schrödinger equations
  • Citing Article
  • July 2021

Communications in Partial Differential Equations

... In [7] and [8, Section 3.1.2, (3.6) and (3.9)], the second author defines non dispersive solution of (gKdV) u at +∞ by the property that for some ρ > 0, x≤ρt |u(t, x)| 2 dx → 0 as t → +∞. ...

Non dispersive solutions of the generalized Korteweg-de Vries equations are typically multi-solitons
  • Citing Article
  • January 2021

Annales de l Institut Henri Poincaré C Analyse Non Linéaire