Modeling binary black hole systems with helically symmetric spacetime

• El Mehdi Zejly

Spacetimes with helical symmetry provide a model for the quasi-stationary regime of widely separated orbiting binary black holes. The problem is formulated using the Ernst formalism, in which the vacuum Einstein equations are reduced, via the quotient by a helical Killing vector field, to equations for a complex Ernst potential and a three-dimensional metric. To encode the presence of two distinct Killing horizons, bispherical coordinates are introduced as a natural coordinate system: constant-coordinate surfaces describe separated spherical horizons, while spatial infinity is compactified to a boundary point. The equal-mass double Schwarzschild solution is then used as an example and expressed in bispherical coordinates through an explicit conformal map from Weyl coordinates involving elliptic functions. Its numerical reconstruction using multi-domain spectral methods shows that the regular part of the solution can be recovered with high accuracy, providing a test case for helically symmetric binary black hole configurations.

Integrability in Asymptotic Symmetries of Spacetime : the bms3 scenario

• Corentin Vitel

4D gravity is not fully integrable. Nevertheless, one may hope that certain sectors are. In this spirit, we consider gravitational asymptotic symmetries, namely transformations of spacetime coordinates that leave the large-scale behavior of the gravitational field invariant. Such transformations are described by the so-called BMS group. While subtleties do persist in four dimensions, the three-dimensional case offers an appealing playground. There, we uncover the presence of an integrable hierarchy.

Numerical study of the 2D Kaup-Broer-Kuperschmidt system

• Théo Gaurdry

The Kaup-Broer-Kupershmidt (KBK) system is a particular case of the abcd Boussinesq systems for surface water waves. It has two versions, a “bad” one (linearly ill-posed) and a “good” one; we focus on the latter. In one dimension this system is completely formally integrable and its solitons are stable. The theory of the two dimensional KBK system is much less understood and no rigorous results are available. We will discuss recent conjectures based on numerical results on the KBK system in two dimensions 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.

Bihamiltonian Structures of the Genus-Zero Whitham Hierarchy

• Dimitrios Makris

The genus-zero Whitham hierarchy, introduced in the 1990s by I. M. Krichever, is a family of evolutionary quasi-linear PDEs describing the slow modulation of nonlinear waves. It includes, as special cases, many well-known dispersionless integrable systems, such as the dispersionless KP hierarchy and the two-dimensional Toda hierarchy. In this talk, we explain how to derive a bihamiltonian formulation of the hierarchy using the method of R-matrices. More precisely, we construct a Poisson pencil on the loop space of holomorphic functions defined on disjoint circles in the Riemann sphere, and then apply Dirac reduction to obtain a bihamiltonian structure for the genus-zero Whitham hierarchy. Time permitting, we will also discuss the relationship between this work and the theory of Frobenius manifolds, and propose a definition for a dispersive deformation of the Whitham hierarchy.

Keldysh analysis of non-modal transient growth

• Jérémy Besson

In this talk we show how to apply the Keldysh decomposition scheme in order to study a conceptual problem regarding time-domain non-modal transients.

Towards enhanced reconstruction quality in electrical impedance tomography

• Michalis Akritidis

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.

IMB colloquium : Bernard Deconinck [https://indico.math.cnrs.fr/event/16545/][1] [1]: https://indico.math.cnrs.fr/event/16545/