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Didier Pilod

Publications and source records attributed to Didier Pilod.

At least 19 recordsLinked to original sources
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

Improved existence time for the Whitham equation and a Whitham-Boussinesq system

In this paper, we investigate the time of existence of the solutions to two full dispersion models derived from the water waves equations in the shallow water regime: the Whitham equation and a Whitham-Boussinesq system in dimension one and two. The regime is characterized by the nonlinearity parameter $\epsilon\in(0,1]$ and the shallow water parameter $\mu\in(0,1]$. We extend the lifespan of the solution beyond the hyperbolic time $\epsilon^{-1}$. More precisely, we establish well-posedness on the timescale of order $\mu^{\frac{1}{4}^-}\epsilon^{(-\frac{5}{4})^+}$ in the one-dimensional case, and of order $\mu^{\frac{1}{4}^-}\epsilon^{(-\frac{3}{2})^+}$ in dimension two. We emphasize that for the two-dimensional case, we obtain a time of existence of order $\epsilon^{-\frac54}$ in the long wave regime $\mu \sim \epsilon$. This kind of result seems to be new, even for the Boussinesq systems. The proofs combine energy methods with Strichartz estimates. Here, a key ingredient is to obtain new refined Strichartz estimates that include the small parameter $\mu$. These techniques are robust and could be adapted to improve the lifespan of solutions for other equations and systems of the same form.

math.AP↗
arXiv2025

Low regularity well-posedness of nonlocal dispersive perturbations of Burgers' equation

We consider the Cauchy problem associated to a class of dispersive perturbations of Burgers' equations, which contains the low dispersion Benjamin-Ono equation, (also known as low dispersion fractional KdV equation), $$ \partial_tu-D_x^{\alpha}\partial_xu=\partial_x(u^2) \, ,$$ and prove that it is locally well-posed in $H^s(\mathbb K)$, $\mathbb K=\mathbb R$ or $\mathbb T$, for $s>s_{\alpha}$, where \begin{equation*} s_\alpha=\begin{cases} 1-\frac{3\alpha}4 & \text{for} \quad \frac23 \le \alpha \le 1; \frac 32(1-\alpha) & \text{for} \quad \frac13 \le \alpha \le \frac23; \frac 32-\frac{\alpha}{1-\alpha} & \text{for} \quad 0 < \alpha \le \frac13 . \end{cases} \end{equation*} The uniqueness is unconditional in $H^s(\mathbb K)$ for $s>\max\{\frac12,s_{\alpha}\}$. Moreover, we obtain \emph{a priori} estimates for the solutions at the lower regularity threshold $s>\widetilde{s}_\alpha$ where \begin{equation*} \widetilde{s}_\alpha=\begin{cases} \frac 12-\frac \alpha 4 & \text{for} \quad \frac23 \le \alpha \le 1; 1-\alpha & \text{for} \quad \frac12 \le \alpha \le \frac23; \frac 32-\frac{\alpha}{1-\alpha} & \text{for} \quad 0 < \alpha \le \frac12 . \end{cases} \end{equation*} As a consequence of these results and of the Hamiltonian structure of the equation, we deduce global well-posedness in $H^s(\mathbb K)$ for $s>s_{\alpha}$ when $\alpha>\frac23$, and in the energy space $H^{\frac{\alpha}2}(\mathbb K)$ when $\alpha>\frac45$.

math.AP↗
arXiv2024

Intermediate long wave equation in negative Sobolev spaces

We study the intermediate long wave equation (ILW) in negative Sobolev spaces. In particular, despite the lack of scaling invariance, we identify the regularity $s = -\frac 12$ as the critical regularity for ILW with any depth parameter, by establishing the following two results. (i) By viewing ILW as a perturbation of the Benjamin-Ono equation (BO) and exploiting the complete integrability of BO, we establish a global-in-time a priori bound on the $H^s$-norm of a solution to ILW for $ - \frac 12 < s < 0$. (ii) By making use of explicit solutions, we prove that ILW is ill-posed in $H^s$ for $s < - \frac 12$. Our results apply to both the real line case and the periodic case.

math.AP↗
arXiv2024

Dynamics of the collision of two nearly equal solitary waves for the Zakharov-Kuznetsov equation

We study the dynamics of the collision of two solitary waves for the Zakharov-Kuznetsov equation in dimension $2$ and $3$. We describe the evolution of the solution behaving as a sum of $2$-solitary waves of nearly equal speeds at time $t=-\infty$ up to time $t=+\infty$. We show that this solution behaves as the sum of two modulated solitary waves and an error term which is small in $H^1$ for all time $t \in \mathbb R$. Finally, we also prove the stability of this solution for large times around the collision. The proofs are a non-trivial extension of the ones of Martel and Merle for the quartic generalized Korteweg-de Vries equation to higher dimensions. First, despite the non-explicit nature of the solitary wave, we construct an approximate solution in an intrinsic way by canceling the error to the equation only in the natural directions of scaling and translation. Then, to control the difference between a solution and the approximate solution, we use a modified energy functional and a refined modulation estimate in the transverse variable. Moreover, we rely on the hamiltonian structure of the ODE governing the distance between the waves, which cannot be approximated by explicit solutions, to close the bootstrap estimates on the parameters. We hope that the techniques introduced here are robust and will prove useful in studying the collision phenomena for other focusing non-linear dispersive equations with non-explicit solitary waves.

math.AP↗
arXiv2023

Asymptotic stability of a finite sum of solitary waves for the Zakharov-Kuznetsov equation

We prove the asymptotic stability of a finite sum of well-ordered solitary waves for the Zakharov-Kuznetsov equation in dimensions two and three. Moreover, we derive a qualitative version of the orbital stability result which turns out to be useful for the study of the collision of two solitary waves. The proof extends the ideas of Martel, Merle and Tsai for the sub-critical gKdV equation in dimension one to the higher-dimensional case. It relies on monotonicity properties on oblique half-spaces and rigidity properties around one solitary wave introduced by C\^ote, Mu\~noz, Pilod and Simpson in dimension two, and by Farah, Holmer, Roudenko and Yang in dimension three.

math.AP↗
arXiv2023

Deep-water limit of the intermediate long wave equation in $L^2$

We study the well-posedness issue of the intermediate long wave equation (ILW) on both the real line and the circle. By applying the gauge transform for the Benjamin-Ono equation (BO) and adapting the $L^2$ well-posedness argument for BO by Molinet and the fourth author (2012), we prove global well-posedness of ILW in $L^2$ on both the real line and the circle. In the periodic setting, this provides the first low regularity well-posedness of ILW. We then establish convergence of the ILW dynamics to the BO dynamics in the deep-water limit at the $L^2$-level.

math.AP↗
arXiv2023

Well-posedness for the extended Schrödinger-Benjamin-Ono system

In this work we prove that the initial value problem associated to the Schrödinger-Benjamin-Ono type system \begin{equation*} \left\{ \begin{array}{ll} \mathrm{i}\partial_{t}u+ \partial_{x}^{2} u= uv+ βu|u|^{2}, \partial_{t}v-\mathcal{H}_{x}\partial_{x}^{2}v+ ρv\partial_{x}v=\partial_{x}\left(|u|^{2}\right) u(x,0)=u_{0}(x), \quad v(x,0)=v_{0}(x), \end{array} \right. \end{equation*} with $β,ρ\in \mathbb{R}$ is locally well-posed for initial data $(u_{0},v_{0})\in H^{s+\frac12}(\mathbb{R})\times H^{s}(\mathbb{R})$ for $s>\frac54$. Our method of proof relies on energy methods and compactness arguments. However, due to the lack of symmetry of the nonlinearity, the usual energy has to be modified to cancel out some bad terms appearing in the estimates. Finally, in order to lower the regularity below the Sobolev threshold $s=\frac32$, we employ a refined Strichartz estimate introduced in the Benjamin-Ono setting by Koch and Tzvetkov, and further developed by Kenig and Koenig.

math.AP↗
arXiv2023

Unconditional uniqueness for the Benjamin-Ono equation

We study the unconditional uniqueness of solutions to the Benjamin-Ono equation with initial data in $H^{s}$, both on the real line and on the torus. We use the gauge transformation of Tao and two iterations of normal form reductions via integration by parts in time. By employing a refined Strichartz estimate we establish the result below the regularity threshold $s=1/6$. As a by-product of our proof, we also obtain a nonlinear smoothing property on the gauge variable at the same level of regularity.

math.AP↗
arXiv2022

Breather Solutions to the Cubic Whitham Equation

We are concerned with numerical approximations of breather solutions for the cubic Whitham equation which arises as a water-wave model for interfacial waves. The model combines strong nonlinearity with the non-local character of the water-wave problem. The equation is non-integrable as suggested by the inelastic interaction of solitary waves. As a non local model, it generalizes, in the low frequency limit, the well known modified KdV (mKdV) equation which is a completely-integrable model. The mKdV equation has breather solutions, i.e. periodic in time and localized in space biparametric solutions. It was recently shown that these breather solutions appear naturally as ground states of invariant integrals, suggesting that such structures may also exist in non-integrable models, at least in an approximate sense. In this work, we present numerical evidence that in the non-integrable case of the cubic Whitham equation, breather solutions may also exist.

nlin.PS↗
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↗
arXiv2020

Global well-posedness and scattering for the Dysthe equation in $L^2(\mathbb R^2)$

This paper focuses on the Dysthe equation which is a higher order approximation of the water waves system in the modulation (Schrödinger) regime and in the infinite depth case. We first review the derivation of the Dysthe and related equations. Then we study the initial-value problem. We prove a small data global well-posedness and scattering result in the critical space $L^2(\mathbb R^2)$. This result is sharp in view of the fact that the flow map cannot be $C^3$ continuous below $L^2(\mathbb R^2)$. Our analysis relies on linear and bilinear Strichartz estimates in the context of the Fourier restriction norm method. Moreover, since we are at a critical level, we need to work in the framework of the atomic space $U^2_S$ and its dual $V^2_S $ of square bounded variation functions. We also prove that the initial-value problem is locally well-posed in $H^s(\mathbb R^2)$, $s>0$. Our results extend to the finite depth version of the Dysthe equation.

math.AP↗
arXiv2020

Dispersive estimates for full dispersion KP equations

We prove several dispersive estimates for the linear part of the Full Dispersion Kadomtsev-Petviashvili introduced by David Lannes to overcome some shortcomings of the classical Kadomtsev-Petviashvili equations. The proof of these estimates combines the stationary phase method with sharp asymptotics on asymmetric Bessel functions, which may be of independent interest. As a consequence, we prove that the initial value problem associated to the Full Dispersion Kadomtsev-Petviashvili is locally well-posed in $H^s(\mathbb R^2)$, for $s>\frac74$, in the capillary-gravity setting.

math.AP↗
arXiv2020

On the unique continuation of solutions to non-local non-linear dispersive equations

We prove unique continuation properties of solutions to a large class of nonlinear, non-local dispersive equations. The goal is to show that if $u_1,\,u_2$ are two suitable solutions of the equation defined in $\mathbb R^n\times[0,T]$ such that for some non-empty open set $Ω\subset \mathbb R^n\times[0,T]$, $u_1(x,t)=u_2(x,t)$ for $(x,t) \in Ω$, then $u_1(x,t)=u_2(x,t)$ for any $(x,t)\in\mathbb R^n\times[0,T]$. The proof is based on static arguments. More precisely, the main ingredient in the proofs will be the unique continuation properties for fractional powers of the Laplacian established by Ghosh, Salo and Ulhmann in \cite{GhSaUh}, and some extensions obtained here.

math.AP↗
arXiv2019

Full family of flattening solitary waves for the mass critical generalized KdV equation

For the mass critical generalized KdV equation $\partial_t u + \partial_x (\partial_x^2 u + u^5)=0$ on $\mathbb R$, we construct a full family of flattening solitary wave solutions. Let $Q$ be the unique even positive solution of $Q''+Q^5=Q$. For any $ν\in (0,\frac 13)$, there exist global (for $t\geq 0$) solutions of the equation with the asymptotic behavior \begin{equation*} u(t,x)= t^{-\fracν2} Q\left(t^{-ν} (x-x(t))\right)+w(t,x) \end{equation*} where, for some $c>0$, \begin{equation*} x(t)\sim c t^{1-2ν} \quad \mbox{and}\quad \|w(t)\|_{H^1(x>\frac 12 x(t))} \to 0\quad \mbox{as $t\to +\infty$.} \end{equation*} Moreover, the initial data for such solutions can be taken arbitrarily close to a solitary wave in the energy space. The long-time flattening of the solitary wave is forced by a slowly decaying tail in the initial data. This result and its proof are inspired and complement recent blow-up results for the critical generalized KdV equation. This article is also motivated by previous constructions of exotic behaviors close to solitons for other nonlinear dispersive equations such as the energy-critical wave equation.

math.AP↗
arXiv2018

On the local well-posedness for a full dispersion Boussinesq system with surface tension

In this note, we prove local-in-time well-posedness for a fully dispersive Boussinesq system arising in the context of free surface water waves in two and three spatial dimensions. Those systems can be seen as a weak nonlocal dispersive perturbation of the shallow-water system. Our method of proof relies on energy estimates and a compactness argument. However, due to the lack of symmetry of the nonlinear part, those traditional methods have to be supplemented with the use of a modified energy in order to close the a priori estimates.

math.AP↗
arXiv2018

On well-posedness for some dispersive perturbations of Burgers' equation

We show that the Cauchy problem for a class of dispersive perturbations of Burgers' equations containing the low dispersion Benjamin-Ono equation $\partial$\_t u -- D^$α$\_x $\partial$\_x u = $\partial$\_x(u^2), 0 < $α$ $\le$ 1, is locally well-posed in H^s (R) when s > 3 /2 -- 5$α$ /4. As a consequence, we obtain global well-posedness in the energy space H^{$α$/2} (R) as soon as $α$ > 6/7 .

math.AP↗