Searcharxiv⌕ Search

arXiv subjects

Yvan Martel

Publications and source records attributed to Yvan Martel.

At least 19 recordsLinked to original sources
arXiv2026

Classification of multi-solitons in one sense of time for the 5D energy-critical wave equation

We classify the multi-solitons, in the positive sense of time, of the focusing energy-critical wave equation in space dimension $5$, more precisely the solutions $u(t,x)$ of the equation \[ \partial_t^2 u - \Delta u - |u|^{\frac 4{3}} u = 0, \quad (t,x)\in [T_0,\infty)\times {\mathbb R}^5, \] which satisfy the estimate \[ \left\|\nabla_{t,x} \left(u(t) - \sum_{k} W_k(t) \right) \right\|_{L^2} \lesssim t^{-\frac12^+} \quad \hbox{for all $t\gg 1$}. \] Here, $\{W_k\}_k$ is any given finite family of traveling waves, based on the explicit ground state solution \[ W(x) = \left(1+ \frac{|x|^2}{15}\right)^{-\frac32} \] with collinear two-by-two different velocities.

math.AP↗
arXiv2026

Continuum of finite point blowup rates for the critical generalized Korteweg-de Vries equation

For any $\nu\in(\frac 37,\frac12)$, we prove the existence of an $H^1$ solution $u$ of the mass critical generalized Korteweg-de Vries equation on the time interval $(0,T_0]$, for some $T_0>0$, which blows up at the time $t=0$ and at the point $x=0$ with the rate $\|\partial_x u (t,x)\|_{L^2} \approx t^{-\nu}$. Such a blowup rate is associated to a blowup residue of the form $r_\alpha(x)= x^{\alpha -\frac 12}$ for $x>0$ close to the blowup point, where $\alpha=\frac{3\nu-1}{2-4\nu}$. The condition $\nu\in(\frac37,\frac12)$ is equivalent to $\alpha>1$, which corresponds to the full range for which the residue $r_\alpha$ belongs to $H^1$. Such blowup at a finite point is in contrast with all the blowup solutions constructed for this equation, except the one constructed previously by the authors corresponding to the special value $\nu=\frac 25$. Finally, we present some open problems regarding the blowup phenomenon for the mass critical gKdV equation.

math.AP↗
arXiv2025

Asymptotic analysis of small energy breathers for the nonlinear Klein-Gordon equation

For a class of nonlinear Klein-Gordon equations, we prove that in the small energy limit, any sequence of breathers decomposes into a finite sum of decoupled, periodically modulated canonical solitons. Each of these solitons is asymptotically equal to an explicit sine-Gordon breather and the distance between them grows to infinity as the energy decreases to 0. Finally we prove that none of these breathers is centered in a bounded set provided that a certain non resonance condition holds.

math.AP↗
arXiv2025

Existence of multi-solitons with any parameters for the 5D energy critical wave equation

For the focusing, energy critical wave equation in dimension 5, we construct multi-solitons with any number of solitons, any choice of signs, speeds, scaling parameters and translation parameters. This requires to revisit in depth previous constructions of multi-solitons based on a unidirectional approach, to fully take into account the dimension of the space and the possibility for solitons to move in any direction. Then, as a consequence of this more general construction and of the arguments developed in a previous article, the inelastic nature of any collision of solitons is proved under a non-cancellation assumption on the parameters.

math.AP↗
arXiv2025

A new proof of a Liouville theorem for the one dimensional Gross-Pitaevskii equation

The asymptotic stability of the black and dark solitons of the one-dimensional Gross-Pitaevskii equation was proved by B\'ethuel, Gravejat and Smets (Ann. Sci. \'Ec. Norm. Sup\'er. 48 (2015)) and Gravejat and Smets (Proc. Lond. Math. Soc. 111 (2015)), using a rigidity property in the vicinity of solitons. We provide an alternate proof of the Liouville theorems in the above articles using a factorization identity for the linearized operator which trivializes the spectral analysis.

math.AP↗
arXiv2024

Asymptotic stability of small standing solitary waves of the one-dimensional cubic-quintic Schrödinger equation

For the Schrödinger equation with a cubic-quintic, focusing-focusing nonlinearity in one space dimension, this article proves the local asymptotic completeness of the family of small standing solitary waves under even perturbations in the energy space. For this model, perturbative of the integrable cubic Schrödinger equation for small solutions, the linearized equation around a small solitary wave has an internal mode, whose contribution to the dynamics is handled by the Fermi golden rule.

math.AP↗
arXiv2022

Non-flat conformal blow-up profiles for the 1D critical nonlinear Schrödinger equation

For the critical one-dimensional nonlinear Schrödinger equation, we construct blow-up solutions that concentrate a soliton at the origin at the conformal blow-up rate, with a non-flat blow-up profile. More precisely, we obtain a blow-up profile that equals $|x|+iκx^2$ near the origin, where $κ$ is a universal real constant. Such profile differs from the flat profiles obtained in the same context by Bourgain and Wang [Construction of blowup solutions for the nonlinear Schrödinger equation with critical nonlinearity. Ann. Sc. Norm. Super. Pisa Cl. Sci. 25 (1997)].

math.AP↗
arXiv2021

Finite point blowup for the critical generalized Korteweg-de Vries equation

In the last twenty years, there have been significant advances in the study of the blow-up phenomenon for the critical generalized Korteweg-de Vries equation, including the determination of sufficient conditions for blowup, the stability of blowup in a refined topology and the classification of minimal mass blowup. Exotic blow-up solutions with a continuum of blow-up rates and multi-point blow-up solutions were also constructed. However, all these results, as well as numerical simulations, involve the bubbling of a solitary wave going at infinity at the blow-up time, which means that the blow-up dynamics and the residue are eventually uncoupled. Even at the formal level, there was no indication whether blowup at a finite point could occur for this equation. In this article, we answer this question by constructing solutions that blow up in finite time under the form of a single-bubble concentrating the ground state at a finite point with an unforeseen blow-up rate. Finding a blow-up rate intermediate between the self-similar rate and other rates previously known also reopens the question of which blow-up rates are actually possible for this equation.

math.AP↗
arXiv2021

Soliton resolution for critical co-rotational wave maps and radial cubic wave equation

In this paper we prove the soliton resolution conjecture for all times, for all solutions in the energy space, of the co-rotational wave map equation. To our knowledge this is the first such result for all initial data in the energy space for a wave-type equation. We also prove the corresponding results for radial solutions, which remain bounded in the energy norm, of the cubic (energy-critical) nonlinear wave equation in space dimension 4.

math.AP↗
arXiv2021

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 C{ô}te and Mu{ñ}oz (Forum Math. Sigma 2 (2014)) which was restricted to ground states, as were most previous similar results for other nonlinear dispersive and wave models.

math.AP↗
arXiv2020

A sufficient condition for asymptotic stability of kinks in general (1+1)-scalar field models

We study stability properties of kinks for the (1+1)-dimensional nonlinear scalar field theory models \begin{equation*} \partial_t^2ϕ-\partial_x^2ϕ+ W'(ϕ) = 0, \quad (t,x)\in\mathbb{R}\times\mathbb{R}. \end{equation*} The orbital stability of kinks under general assumptions on the potential $W$ is a consequence of energy arguments. Our main result is the derivation of a simple and explicit sufficient condition on the potential $W$ for the asymptotic stability of a given kink. This condition applies to any static or moving kink, in particular no symmetry assumption is required. Last, motivated by the Physics literature, we present applications of the criterion to the $P(ϕ)_2$ theories and the double sine-Gordon theory.

math.AP↗
arXiv2020

Long-time asymptotics of the one-dimensional damped nonlinear Klein-Gordon equation

For the one-dimensional nonlinear damped Klein-Gordon equation \[ \partial_{t}^{2}u+2α\partial_{t}u-\partial_{x}^{2}u+u-|u|^{p-1}u=0 \quad \mbox{on $\mathbb{R}\times\mathbb{R}$,}\] with $α>0$ and $p>2$, we prove that any global finite energy solution either converges to $0$ or behaves asymptotically as $t\to \infty$ as the sum of $K\geq 1$ decoupled solitary waves. In the multi-soliton case $K\geq 2$, the solitary waves have alternate signs and their distances are of order $\log t$.

math.AP↗
arXiv2019

Self-similar blow-up profiles for slightly supercritical nonlinear Schrödinger equations

We construct radially symmetric self-similar blow-up profiles for the mass supercritical nonlinear Schrödinger equation $i\partial_t u + Δu + |u|^{p-1}u=0$ on $\mathbf{R}^d$, close to the mass critical case and for any space dimension $d\ge 1$. These profiles bifurcate from the ground state solitary wave. The argument relies on the classical matched asymptotics method suggested in [Sulem, C.; Sulem, P.-L., The nonlinear Schrödinger equation. Self-focusing and wave collapse. Applied Mathematical Sciences, 139. Springer-Verlag, New York, 1999] which needs to be applied in a degenerate case due to the presence of exponentially small terms in the bifurcation equation related to the log-log blow-up law observed in the mass critical case.

math.AP↗
arXiv2019

Solutions blowing up on any given compact set for the energy subcritical wave equation

We consider the focusing energy subcritical nonlinear wave equation $\partial_{tt} u - Δu= |u|^{p-1} u$ in ${\mathbb R}^N$, $N\ge 1$. Given any compact set $ E \subset {\mathbb R}^N $, we construct finite energy solutions which blow up at $t=0$ exactly on $ E$. The construction is based on an appropriate ansatz. The initial ansatz is simply $U_0(t,x) = κ(t + A(x) )^{ -\frac {2} {p-1} }$, where $A\ge 0$ vanishes exactly on $ E$, which is a solution of the ODE $h'' = h^p$. We refine this first ansatz inductively using only ODE techniques and taking advantage of the fact that (for suitably chosen $A$), space derivatives are negligible with respect to time derivatives. We complete the proof by an energy argument and a compactness method.

math.AP↗
arXiv2019

Solutions with prescribed local blow-up surface for the nonlinear wave equation

We prove that any sufficiently differentiable space-like hypersurface of ${\mathbb R}^{1+N} $ coincides locally around any of its points with the blow-up surface of a finite-energy solution of the focusing nonlinear wave equation $\partial_{tt} u - Δu=|u|^{p-1} u$ on ${\mathbb R} \times {\mathbb R} ^N$, for any $1\leq N\leq 4$ and $1 < p \le \frac {N+2} {N-2}$. We follow the strategy developed in our previous work [arXiv 1812.03949] on the construction of solutions of the nonlinear wave equation blowing up at any prescribed compact set. Here to prove blowup on a local space-like hypersurface, we first apply a change of variable to reduce the problem to blowup on a small ball at $t=0$ for a transformed equation. The construction of an appropriate approximate solution is then combined with an energy method for the existence of a solution of the transformed problem that blows up at $t=0$. To obtain a finite-energy solution of the original problem from trace arguments, we need to work with $H^2\times H^1$ solutions for the transformed problem.

math.AP↗