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Patrick Gérard

Publications and source records attributed to Patrick Gérard.

At least 19 recordsLinked to original sources
arXiv2026

An explicit formula for the solution of the Benjamin-Ono equation and its applications

We summarize recent developments in the theory of the Benjamin-Ono equation, a well-known long-wave asymptotic model for internal waves in deep water, focusing on results obtained by means of an explicit formula for the solution of the Cauchy problem in terms of given initial data. We give a full description of this formula, and provide simple examples, with the aim of bringing it into the realm of applied mathematics. We also show how the formula simplifies further upon restriction to rational initial data. We then show how the formula and its rational restriction explain in great detail, and in a strikingly simple fashion, the asymptotic behavior of solutions of the Benjamin-Ono equation in the small-dispersion and long-time limits. In the setting of long-time asymptotics, we prove some new results related to soliton resolution yielding pointwise convergence with a concrete decay rate under the assumption of rational initial data. The first half of the paper is a stand-alone general survey and user's guide, while the appendices constituting the second half are intended for readers who want to explore the topics in greater depth, and it includes the proofs of our new results.

math.AP↗
arXiv2026

A proof of the soliton resolution conjecture for the Benjamin--Ono equation

We give a proof of the soliton resolution conjecture for the Benjamin--Ono equation, namely every solution with sufficiently regular and decaying initial data can be written as a finite sum of soliton solutions with different velocities up to a radiative remainder term in the long--time asymptotics. We provide a detailed correspondence between the spectral theory of the Lax operator associated to the initial data and the different terms of the soliton resolution expansion. The proof is based on a new use of a representation formula of the solution due to the second author, and on a detailed analysis of the distorted Fourier transform associated to the Lax operator.

math.AP↗
arXiv2026

The half-wave maps equation on $\mathbb{T}$: Global well-posedness in $H^{1/2}$ and almost periodicity

We consider the half-wave maps equation $$ \partial_t \mathbf{u} = \mathbf{u} \times |D| \mathbf{u} $$ for $\mathbf{u} : \mathbb{R} \times \mathbb{T} \to \mathbb{S}^2$, where $\mathbb{T}=\mathbb{R}/2 π\mathbb{Z}$ is the one-dimensional torus and $\mathbb{S}^2 \subset \mathbb{R}^3$ denotes the unit sphere. By extension from rational initial data, we construct a unique and continuous flow map for data in the critical energy space $H^{1/2}(\mathbb{T}; \mathbb{S}^2)$. Moreover, we show almost periodicity in time of these solutions. For the dense subset of rational initial data, we establish quasi-periodicity in time and a-priori bounds on $\| \mathbf{u}(t) \|_{H^s(\mathbb{T})}$ for any $s >0$. Our analysis relies crucially on an explicit formula arising from the Lax pair structure acting on a Hardy space of vector-valued holomorphic functions on the unit disk. As a central ingredient, we develop a general {\em stability principle} for explicit formulae associated with completely integrable PDEs possessing a Lax pair structure on Hardy spaces, including the Benjamin--Ono equation, Calogero--Sutherland DNLS, and the half-wave-maps equation posed on $\mathbb{T}$. Our results extend to the matrix-valued half-wave maps equation $$ \partial_t \mathbf{U} = -\frac{i}{2} [ \mathbf{U}, |D| \mathbf{U} ] $$ with target manifold given by the complex Grassmannians $\mathsf{Gr}_k(\mathbb{C}^d)$, thereby generalizing the special case $\mathbb{S}^2 \cong \mathbb{CP}^1 \cong \mathsf{Gr}_1(\mathbb{C}^2)$. In a companion work, we prove global well-posedness for the half-wave maps equation posed on $\mathbb{R}$ in the scaling-critical energy space $\dot{H}^{1/2}$, by establishing a stability principle for explicit formulae on Hardy spaces in the complex half-plane $\mathbb{C}_+$.

math.AP↗
arXiv2026

The Benjamin-Ono Equation in the Long-Time Limit: Linearized Self-Similar Universality

We obtain the leading term in the solution of the Cauchy problem for the Benjamin-Ono equation in the limit $t\to+\infty$ with $x=O(t^{1/2})$. We show that the rate of decay exceeds that of self-similar solutions and obtain an explicit universal profile for the decaying solution, relating it to the linearization of the profile equation for self-similar solutions. The proof assumes a class of rational initial data $u_0$ in $L^2(\mathbb{R})\cap L^1(\mathbb{R})$ that exhibit generic behavior of the reflection coefficient at the origin.

math.AP↗
arXiv2026

An Explicit Formula for the Benjamin-Ono Hierarchy with Applications to Traveling Waves and Zero-Dispersion Limits

In this paper, we first extend the explicit formula \cite{gerard2023explicit} for the classical Benjamin-Ono equation to each flow of the Benjamin-Ono hierarchy on line. We then use this representation to derive two main applications. First, we obtain a complete classification of traveling wave solutions for all higher-order flows in the hierarchy. Second, we analyze the zero-dispersion limit for the corresponding small-dispersion flows. For every fixed time $t\in\mathbb R$, we prove that, at any time, the solution converges weakly in $L^2(\mathbb R)$ as the dispersion parameter tends to $0$, and we provide a geometric characterization of the limit in terms of an alternating sum, which yields the higher-order analogue of the formula obtained in \cite{miller2011zero}, \cite{Gerard2025small} for the Benjamin-Ono equation.

math.AP↗
arXiv2025

Global Well-Posedness and Soliton Resolution for the Half-Wave Maps Equation with Rational Data

We study the energy-critical half-wave maps equation: \[ \partial_t \mathbf{u} = \mathbf{u} \times |D| \mathbf{u} \] for $\mathbf{u} : [0, T) \times \mathbb{R} \to \mathbb{S}^2$. Our main result establishes the global existence and uniqueness of solutions for all rational initial data $\mathbf{u}_0 : \mathbb{R} \to \mathbb{S}^2$. This demonstrates global well-posedness for a dense subset within the scaling-critical energy space $\dot{H}^{1/2}(\mathbb{R}; \mathbb{S}^2)$. Furthermore, we prove soliton resolution for a dense subset of initial data in the energy space, with uniform bounds for all higher Sobolev norms $\dot{H}^s$ for $s > 0$. Our analysis utilizes the Lax pair structure of the half-wave maps equation on Hardy spaces in combination with an explicit flow formula. Extending these results, we establish global well-posedness for rational initial data (along with a soliton resolution result) for a generalized class of matrix-valued half-wave maps equations with target spaces in the complex Grassmannians $\mathbf{Gr}_k(\mathbb{C}^d)$. Notably, this includes the complex projective spaces $ \mathbb{CP}^{d-1} \cong \mathbf{Gr}_1(\mathbb{C}^d)$ thereby extending the classical case of the target $\mathbb{S}^2 \cong \mathbb{CP}^1$.

math.AP↗
arXiv2025

Energy cascades and condensation via coherent dynamics in Hamiltonian systems

This work makes analytic progress in the deterministic study of turbulence in Hamiltonian systems by identifying two types of energy cascade solutions and the corresponding large- and small-scale structures they generate. The first cascade represents condensate formation via a highly coherent process recently uncovered, while the second cascade, which has not been previously observed, leads to the formation of other large-scale structures. The concentration of energy at small scales is characterized in both cases by the development of a power-law spectrum in finite time, causing the blow-up of Sobolev norms and the formation of coherent structures at small scales. These structures approach two different types of singularities: a point discontinuity in one case and a cusp in the other. The results are fully analytic and explicit, based on two solvable families of Hamiltonian systems identified in this study.

math-ph↗
arXiv2025

The Benjamin-Ono Initial-Value Problem for Rational Data with Application to Long-Time Asymptotics and Scattering

We show that the initial-value problem for the Benjamin-Ono equation on $\mathbb{R}$ with $L^2(\mathbb{R})$ rational initial data with only simple poles can be solved in closed form via a determinant formula involving contour integrals. The dimension of the determinant depends on the number of simple poles of the rational initial data only and the matrix elements depend explicitly on the independent variables $(t,x)$ and the dispersion coefficient $ε$. This allows for various interesting asymptotic limits to be resolved quite efficiently. As an example, and as a first step towards establishing the soliton resolution conjecture, we prove that the solution with initial datum equal to minus a soliton exhibits scattering.

math.AP↗
arXiv2024

The Benjamin-Ono equation in the zero-dispersion limit for rational initial data: generation of dispersive shock waves

The leading-order asymptotic behavior of the solution of the Cauchy initial-value problem for the Benjamin-Ono equation in $L^2(\mathbb{R})$ is obtained explicitly for generic rational initial data $u_0$. An explicit asymptotic wave profile $u^\mathrm{ZD}(t,x;ε)$ is given, in terms of the branches of the multivalued solution of the inviscid Burgers equation with initial data $u_0$, such that the solution $u(t,x;ε)$ of the Benjamin-Ono equation with dispersion parameter $ε>0$ and initial data $u_0$ satisfies $u(t,x;ε)-u^\mathrm{ZD}(t,x;ε)\to 0$ in the locally uniform sense as $ε\to 0$, provided a discriminant inequality holds implying that certain caustic curves in the $(t,x)$-plane are avoided. In some cases this convergence implies strong $L^2(\mathbb{R})$ convergence. The asymptotic profile $u^\mathrm{ZD}(t,x;ε)$ is consistent with the modulated multi-phase wave solutions described by Dobrokhotov and Krichever.

math.AP↗
arXiv2023

A Nekhoroshev theorem for some perturbations of the Benjamin-Ono equation with initial data close to finite gap tori

We consider a perturbation of the Benjamin Ono equation with periodic boundary conditions on a segment. We consider the case where the perturbation is Hamiltonian and the corresponding Hamiltonian vector field is analytic as a map form energy space to itself. Let $ε$ be the size of the perturbation. We prove that for initial data close in energy norm to an $N$-gap state of the unperturbed equation all the actions of the Benjamin Ono equation remain $\cO(ε^{\frac{1}{2(N+1)}})$ close to their initial value for times exponentially long with $ε^{-\frac{1}{2(N+1)}}$.

math.AP↗
arXiv2023

An inverse spectral problem for non-compact Hankel operators with simple spectrum

We consider an inverse spectral problem for a class of non-compact Hankel operators $H$ such that the modulus of $H$ (restricted onto the orthogonal complement to its kernel) has simple spectrum. Similarly to the case of compact operators, we prove a uniqueness result, i.e. we prove that a Hankel operator from our class is uniquely determined by the spectral data. In other words, the spectral map, which maps a Hankel operator to the spectral data, is injective. Further, in contrast to the compact case, we prove the failure of surjectivity of the spectral map, i.e. we prove that not all spectral data from a certain natural set correspond to Hankel operators. We make some progress in describing the image of the spectral map. We also give applications to the cubic Szegő equation. In particular, we prove that not all solutions with initial data in BMOA are almost periodic; this is in a sharp contrast to the known result for initial data in VMOA.

math.FA↗
arXiv2023

On the low regularity phase space of the Benjamin-Ono equation

In this paper we prove that the Benjamin-Ono equation is globally in time $C^0$-well-posed in the Hilbert space $H^{-1/2,\sqrt{\log}}(\mathbb{T},\mathbb{R})$ of periodic distributions in $H^{-1/2}(\mathbb{T},\mathbb{R})$ with $\sqrt{\log}$-weights. The space $H^{-1/2,\sqrt{\log}}(\mathbb{T},\mathbb{R})$ can thus be considered as a maximal low regularity phase space for the Benjamin-Ono equation corresponding to the scale $H^s(\mathbb{T},\mathbb{R})$, $s>-1/2$.

math.AP↗
arXiv2023

The zero dispersion limit for the Benjamin--Ono equation on the line

We identify the zero dispersion limit of a solution of the Benjamin--Ono equation on the line corresponding to every initial datum in $L^2(\R)\cap L^\infty(\R )$. We infer a maximum principle and a local smoothing property for this limit. The proof is based on an explicit formula for the Benjamin--Ono equation and on the combination of calculations in the special case of rational initial data, with approximation arguments. We also investigate the special case of an initial datum equal to the characteristic function of a finite interval, and prove the lack of semigroup property for this zero dispersion limit.

math.AP↗
arXiv2023

The Calogero--Moser Derivative Nonlinear Schrödinger Equation

We study the Calogero--Moser derivative NLS equation $$ i \partial_t u +\partial_{xx} u + (D+|D|)(|u|^2) u =0 $$ posed on the Hardy-Sobolev space $H^s_+(\mathbb{R})$ with suitable $s>0$. By using a Lax pair structure for this $L^2$-critical equation, we prove global well-posedness for $s \geq 1$ and initial data with sub-critical or critical $L^2$-mass $\| u_0 \|_{L^2}^2 \leq 2 π$. Moreover, we prove uniqueness of ground states and also classify all traveling solitary waves. Finally, we study in detail the class of multi-soliton solutions $u(t)$ and we prove that they exhibit energy cascades in the following strong sense such that $\|u(t)\|_{H^s} \sim_s |t|^{2s}$ as $t \to \pm \infty$ for every $s > 0$. \end{abstract}

math.AP↗
arXiv2022

Unbounded Hankel operators and the flow of the cubic Szegő equation

We prove that, for any Hankel operator with a symbol from the Hardy class $H^2$, the maximal and minimal domains coincide. As an application, we prove that the evolution flow of the cubic Szegő equation on the unit circle can be continuously extended to the whole class $H^2$.

math.AP↗
arXiv2022

The Lax pair structure for the spin Benjamin--Ono equation

We prove that the recently introduced spin Benjamin--Ono equation admits a Lax pair, and we deduce a family of conservation laws which allow to prove global wellposedness in all Sobolev spaces $H^k$ for every integer $k\geq 2$. We also infer an additional family of matrix valued conservation laws, of which the previous family are just the traces.

math.AP↗