Numerical and analytical approaches for nonlinear dispersive PDEs

1 déc. 2025, 09:45 → 3 déc. 2025, 17:30 Europe/Paris

A 318 (IMB)

Welcome to the webpage of the conference "Numerical and analytical approaches for nonlinear dispersive PDEs"

List of participants

S. Abenda (Bologna)

 

M. Akrititidis (Dijon)

 

E. de Leon (Munich)

 

M. Fasondini (Leicester) 

 

L. Gassot (Rennes)

 

S. Gavrilyuk (Marseille) 

 

T. Gaudry (Dijon)

 

P. Gérard (Orsay) 

 

J. He (Orsay) 

 

O. Maehlen (Orsay) 

 

A. Ostermann (Innsbruck) 

 

P. Perry (Lexington) 

 

S. Roudenko (Miami)

 

J.-C. Saut (Orsay)

 

J. Sjoestrand (Dijon)

 

D. Sulz (Munich)

 

M. Zejly (Dijon)

 

 

 

Organizing committee

Christian Klein, and Nikola Stoilov

Local organizer

Magali Crochot

 

book_of_abstracts_25.pdf

lun. 1 décembre

09:45 → 10:45 P. Gérard

A proof of the soliton resolution conjecture for the Benjamin-Ono equation
The soliton resolution conjecture for the Benjamin-Ono equation states that every solution on the line with a sufficiently smooth and decaying initial datum expands as time tends to infinity as a finite sum of decoupled soliton solutions added to a radiation term.
I will state a precise version of this conjecture and I will describe the main steps of it proof. This is a joint work in collaboration with Louise Gassot and Peter Miller.

11:00 → 12:00 J. He

Explicit Formulas and the Zero-Dispersion Limit for the Benjamin–Ono Hierarchy on line

I will show how the Lax pair structure leads to the explicit formulas for the Benjamin–Ono Hierarchy on the line. Applications to the zero-dispersion limit for third order equation will be discussed. This work is in collaboration with Patrick Gérard.

14:00 → 15:00 S. Roudenko

15:15 → 16:15 O. Maehlen

The zero-dispersion limit for the Benjamin–Ono equation on the circle

Using the explicit formula of P. Gérard, we characterize the zero-dispersion limit for
solutions of the Benjamin–Ono equation on the circle with bounded initial dat. The result generalizes the work of L. Gassot, who focused on periodic bell-shaped data,
and complements the work of Gérard and X. Chen who identified the zero-dispersion limit on the line. Here, as well as in the mentioned cases, the characterization agrees with the
one first obtained by Miller–Xu for bell-shaped data on the line: The zero-dispersion limit is given as
an alternating sum of the characteristics appearing in the multivalued solution of Burgers’ equation.

16:15 → 16:45 T. Gaudry

Numerical study of solutions to the Kaup-Broer-Kupershmidt systems

We will discuss recent numerical results on the Kaup–Broer–Kupershmidt (KBK) systems with a particular focus on line solitary waves and their transverse stability. We numerically construct a stationary solution and study its (in)stability. We introduce a dynamical rescaling to discuss a blow-up of the solutions.

mar. 2 décembre

09:45 → 10:45 L Gassot

Global well-posedness for perturbations of the Benjamin-Ono equation on the torus

We prove global-wellposedness in the Sobolev spaces H^s for s>-1/2 for the Benjamin-Ono equation on the torus, perturbed by a class of zero-order Fourier multipliers. Examples of such equations include the periodic intermediate long wave equation. The method consists in using the Birkhoff map, that sends the Benjamin-Ono equation into an infinite system of linear ODEs, and which is used as a nonlinear Fourier transform for the perturbed equation. Then, a-priori relative compactness of trajectories is obtained by showing a-priori estimates on quantities that are conserved by the Benjamin-Ono flow. This work is in collaboration with Thierry Laurens.

11:00 → 12:00 A. Ostermann

A conservative dynamical low-rank algorithm

Kinetic equations present computational challenges due to their (up to) six-dimensional phase space dependence, resulting in high memory requirements and high computational cost. Traditional particle methods alleviate this burden, but generally suffer from noise. Dynamical low-rank approximation is a novel approach that significantly reduces the computational burden, but often fails to preserve physical invariants such as mass, momentum, and energy. This destroys the inherent physical structure of the problem. In this talk, we present a modified low-rank algorithm that conserves mass and momentum while improving energy conservation. The resulting method is applied to the Vlasov-Poisson equations, though the same strategy can be applied to other equations. Numerical results for phenomena such as Landau damping and two-stream instability demonstrate the effectiveness of the new method.

14:00 → 15:00 P. Perry

Large-Data Global Well-Posedness for the modified Novikov-Veselov (mNV) Equation

This is joint work with Adrian Nachman and Daniel Tataru. The  mNV equation is a nonlinear dispersive equation in two space dimensions, related to the Novikov-Veselov equation by a Miura-type map. Drawing on Nachman, Regev, and Tataru's work on global well-posedness for the Davey-Stewartson equation, we prove large-data global well-posedness for the mNV equation in $L^2$ using the inverse scattering method. The mNV equation is $L^2$-critical so this result should be regarded as optimal. A key ingredient in our proof is a new nonlinear Gagliardo-Nirenberg estimate for the scattering transform.

Using the Miura map, we are also able to prove large-data global well-posedness for a spectrally determined class of initial conditions in the Novikov-Veselov (NV) equation.  Along the way we obtain a sharp, scale-invariant version of the Agmon-Allegretto-Piepenbrink theorem for Schrödinger operators in the critical two-dimensional case.

15:15 → 16:15 S. Gavrilyuk

About the speed and amplitude of the leading edge of a dispersive shock wave

In 1850, at the age of 33, Ivan Aivazovsky, a Russian painter of Armenian origin, presented his major work, The Ninth Wave, to the Russian public. The painting depicts the sea after a cruel storm. The shipwrecked sailors, clinging to the mast of their destroyed ship, continue to fight for their lives. But the sea is not yet calm, and a ninth wave is already forming, ready to strike. According to popular sailors’ superstition, the ninth wave is the most violent and dan-
gerous of a storm (Figure 1). The objective of my presentation is to qualitatively describe the solitary wave of largest amplitude appearing in the solution of Riemann problems for the Serre-Green-Naghdi equations describing non-linear long dispersive waves. Such a large-amplitude
wave is the leading wave (i.e., it is the first wave, and not the ninth) of the corresponding dispersive shock. Its speed and amplitude are defined analytically through the solitary limit of the corresponding Whitham modulation equations. The numerical results are in accordance
with the analytical prediction.

References
[1] 2025 T. Congy, Gennady El, S. Gavrilyuk, M. Hoefer and K. -M. Shyue, Solitary wave-mean
flow interaction in strongly nonlinear dispersive shallow water waves, J. Nonlinear Waves, v. 1.

16:15 → 17:15 D. Sulz

Time Integration in Quantum Dynamics using Tree Tensor Networks


This talk studies the numerical solution of high-dimensional tensor differential equations. The prohibitive computational cost and memory requirements of numerically simulating such equations are often referred to as the curse of dimensionality. Such prohibitively large differential equations arise in many fields of application such as plasma physics, machine learning, radiation transport, or quantum physics. Dynamical low-rank approximation offers a promising approach to overcome the curse by representing the high-dimensional tensors in a low-rank format and solving a projected differential equation on a low-rank manifold. The low-rank manifold considered in this talk is the manifold of tree tensor networks.
The time integration of tree tensor networks requires the update of each low-rank factor. Several numerical schemes to compute this time integration are proposed in this thesis. All of these methods fall into the class of Basis Update and Galerkin (BUG) integrators, where all basis matrices are evolved through a small matrix differential equation and all core tensors by a Galerkin step. We present a rigorous error analysis of all integration schemes, demonstrating robustness with respect to small singular values. This is important since small singular values can lead to numerical instabilities as they correspond to high curvatures in the corresponding low-rank manifold. Remarkable properties like rank-adaptivity, parallelism, norm and energy preservation, and the diminishing of energy in gradient systems are discussed. Further, the representation of the right-hand side of a differential equation in tree tensor network format is discussed for a class of long-range interacting Hamiltonians. Efficient constructions of these tree tensor network operators are provided, and bounds on the maximal tree rank for an exact and approximated representation of the operator are discussed.
Numerical experiments for problems from quantum physics verify theoretical results and investigate the applicability of dynamical low-rank approximation to many-body quantum systems in detail.

17:15 → 17:45 M. Akritidis

Towards enhanced reconstruction quality in electrical impedance tomography

Electrical impedance tomography (EIT) is a non-invasive imaging modality that reconstructs the spatial distribution of electrical conductivity or impedance within a body (e.g., lungs or brain) using boundary measurements from surface electrodes. Physically, EIT can be formulated as an inverse scattering problem, where one seeks to recover the conductivity distribution of a domain from boundary data represented by the Dirichlet-to-Neumann (DtN) map. From a mathematical standpoint, this inverse problem can be reformulated as a D-bar equation.

We outline the fundamental theory underlying conductivity recovery within the D-bar formulation and identify key factors whose improvement may enhance reconstruction quality beyond that of existing methods.

mer. 3 décembre

09:45 → 10:45 S. Abenda (online)

Tropical curves and KP solitons: the case of banana graphs
The Kadomtsev–Petviashvili (KP) II equation is the first member of an integrable hierarchy in 2+1-variables, it contains as reductions other interesting integrable systems (Korteweg de Vries, Boussinesq, ...), and it possesses two distinguished classes of solutions:
1 KP-finite-gap solutions are parametrized by non special divisors on algebraic curves and are expressed in terms of Riemann theta functions;
2 KP solitons are parametrized by points in finite-dimensional Grassmannians and may be obtained from rational degenerations of finite-gap solutions.
The relation between these two classes has been the subject of intensive study in recent years and shows interesting connections with tropical geometry. Since,  in the tropical limit, the Riemann theta function becomes a finite sum of exponentials, one expects that the combinatorial structure of the soliton solutions is encoded by tropical curves and their Jacobians.
In this talk based on a paper in progress with T.O. Çelik, C. Fevola and Y. Mandelshtam,  I will explain this correspondence for tropical degenerations of genus g hyperelliptic curves associated with banana graphs (metric graphs with two vertices connected by g+1 edges).

11:00 → 12:00 E. de Leon

Numerical solutions of a class of evolution PDEs

In this talk I will present two different approaches to construct the numerical solution of a class of evolution PDEs, such as the Schrödinger equation. We consider a combination of parametrization (time-dependent complex parameters) and time discretization processes. The main differences regarding the order in which this combination is carried out will be discussed throughout the talk.

14:00 → 15:00 J.C. Saut

On higher order nonlinear Schrödinger equations

We will survey results and open questions on various nonlinear Schrödinger equations occuring in water wave theory and nonlinear optics.

15:15 → 16:15 M. Fasondini

A sparse spectral method on domains bounded by planar algebraic curves

We develop a sparse spectral method for solving partial differential equations on a class of two-dimensional geometries bounded by algebraic curves.  The numerical method uses generalised bivariate Koornwinder polynomials, which form a basis that is orthogonal but not ordered by degree.  The generalised Koornwinder polynomials are built from new families of univariate semiclassical orthogonal polynomials whose associated operator matrices are computed with linear complexity in the number of basis functions.  The generalised Koornwinder basis allows for sparse matrix representations of conversion (change-of-basis), multiplication and differentiation operators, while also enabling fast transforms.  The efficiency and accuracy of the spectral method are illustrated through a series of numerical experiments. (This is joint work with Jiajie Yao and Sheehan Olver.)

16:15 → 16:45 M. Zejli

Modeling binary black hole systems using a spacetime with helical symmetry

Binary black hole systems can be approximated as a single circular orbit at fixed separation, from large distances down to the innermost stable circular orbit (ISCO), where the gravitational energy lost through gravitational waves is exactly compensated by an incoming flux of gravitational waves from infinity. In this approximation, one looks for a spacetime that is stationary in a corotating frame and admits a helical Killing vector, capturing the idea of an orbit that is “frozen” when viewed from a frame rotating with the binary. The Ernst formalism with respect to this Killing field reduces the study of a four-dimensional spacetime to the study of a three-dimensional quotient space. Within this framework, bispherical coordinates appear as a natural choice: they are well adapted to geometries with two spherical objects and offer the possibility to represent each black hole horizon as a surface {η= const.}, which is exactly what is needed to impose boundary conditions on two separate horizons. The strategy is to start from a simpler “toy model”, the double Schwarzschild metric describing two non-rotating black holes in equilibrium, held apart by a massless strut, originally expressed in Weyl coordinates. The main challenge is to find a suitable gauge, i.e. a bispherical coordinate system that
captures all the symmetries and simplifies the metric as much as possible. In such a gauge, the double Schwarzschild solution would help to explore how bispherical coordinates can be used to describe the
situation with helical symmetry and might provide insight into the different numerical issues that may be encountered, particularly near the horizons and at spatial infinity.