We examine the motion of the free surface of a body of fluid with a periodically varying bottom. We consider the water wave system linearized near a stationary state and develop a Bloch theory. The analysis takes the form of a spectral problem for the Dirichlet– Neumann operator in a fluid domain with a periodic bottom and a flat surface elevation. We find that, generically, the presence of...
This talk is a survey of a recent joint work with Thomas Kappeler about the construction of Birkhoff coordinates for real valued, spatially periodic and square integrable solutions of the Benjamin-Ono equation.
In this talk I report on joint work with Patrick Gérard and Peter Topalov concerning properties of the flow map of the Benjamin-Ono equation on the torus. The main result says that the flow map, introduced in our previous work on the space $L^2_{r,0}$ of real valued, $2\pi-$periodic $L^2-$functions with mean $0$,
can be extended to the Sobolev spaces $H^{-s}_{r,0}$ for $0 < s < 1/2$. The key...
Hydrodynamics is a powerful framework for large-wavelength phenomena in many-body systems. It was extended recently to include integrable models, giving ``generalised hydrodynamics”. In this talk, I will review fundamental aspects of the hydrodynamic of integrable systems, with the simple examples of the quantum Lieb-Liniger and the classical Toda models. I will then show some of the exact...
I will review the propreties and recent results for conformal fishnet theory (FCFT) which was proposed by O.Gurdogan and myself as a special double scaling limit of gamma-twisted N=4 SYM theory. FCFT, in its simplest, bi-scalar version, is a UV finite strongly coupled 4-dimensionl logarithmic CFT dominated by planar fishnet Feynman graphs (of the shape of regular square lattice). FCFT...
Quantum beating may nowadays refer to many, often quite different phenomena studied in various domains of quantum physics. A paradigmatic example is the inversion in the ammonia molecule, observed experimentally in 1935.
A theoretical explanation of the quantum beating was obtained by modelling the nitrogen atom as a quantum particle in a double well potential. The quantum environment of this...
We consider 1D scattering problems related to quantum transport in diodes. We discuss the efficient numerical integration of ODEs like $\epsilon^2*u"+a(x)*u=0$ for $0<\epsilon<<1$ on coarse grids, but still yielding accurate solutions; including oscillatory (for given $a(x)>0$) and evanescent regimes (for $a(x)<0$), partly including turning points. In the oscillatory case we use a marching...
We study a gas of solitons in the limit when the number of solitons goes to infinity.
We characterize the asymptotic behaviour and the long time behaviour of the soliton gas.
Fredholm determinants associated to deformations of the Airy kernel are known to be closely connected to the solution to the Khardar-Parisi-Zhang (KPZ) equation with narrow wedge initial data, and they also appear as largest particle distribution in models of positive-temperature free fermions. It is of particular importance in these models to understand the lower tail of the Fredholm...
Non-ultralocality is a long-standing open problem in classical integrable field theory which has precluded the first-principle quantisation of many important models. I will review a recent proposal for dealing with this issue, which relates it to the study of Gaudin models associated with affine Kac-Moody algebras. I will then go on to discuss its relation to another very recent proposal of...
In this talk, we will discuss realisations of affine Gaudin models and their application to integrable sigma-models. After reviewing the properties of affine Gaudin models and their relation with integrable sigma-models (see also B. Vicedo's talk), we will explain a systematic procedure allowing to assemble two affine Gaudin models into a unique one. As an application of this formalism, we...
Boiti-Pempinelli-Pogrebkov's inverse scattering theories on the KPII equation provide an integrable approach to solve the Cauchy problem of the perturbed KPII multi-line soliton solutions and the stability problem of KPII multi-line solitons.
In this talk, we will present rigorous analysis for the direct scattering theory of perturbed KPII multi line solitons, illustrated by perturbations...
This talk reports on joint work with Joel Klipfel (University of Kentucky) and Yilun Wu (University of Oklahoma). The intermediate long wave equation (ILW) is a model of weakly nonlinear wave propagation in a fluid of finite depth. It interpolates between the Benjamin-Ono equation (infinite depth) and the Korteweg-de Vries equation (shallow water). Ablowitz and Kodama showed that ILW is...
This is part of a joint work with
Christian Klein and Nikola Stoilov. We consider a $2\times 2$ system of $\partial ,\overline{\partial }$ type in the large parameter limit, appearing in the study of the Davey-Stuartson II equation.
When a certain potential is smooth, we show that the solution is given by a convergent perturbation series.
When it is the characteristic function of a strictly...
The D-bar method, introduced by Beals and Coifman in the 1980’s, provides a solution method for Calderón’s inverse conductivity problem in dimension two, as was shown by Nachman in 1996. This presentation shows how Nachman’s proof can be developed further to yield the D-bar method, a practical imaging algorithm for Electrical Impedance Tomography (EIT). Furthermore, demonstrated is how machine...
We survey the relation between the relativistic sine-Gordon model and (a special case of) the hyperbolic relativistic integrable $N$-particle systems of Calogero-Moser type. More specifically, we review the intimate link between the classical version of the latter and the particle-like sine-Gordon solutions, and present compelling evidence that this soliton-particle correspondence turns into...
Tilings of polygonal regions with non-convexities (cuts) lead to a new kernel and a new statistics for the asymptotic fluctuations of the tiles, when the size of the region and the cuts gets large under an appropriate scaling. The limiting statistics has been observed in very different circumstances.
This talk is dedicated to the description of the modulation instabilities, leading to the appearance of phenomena of multiple rogue waves generation in a frame of integrable hierarchies. It is mainly concentrated on the AKNS hierarchy and some related systems.
An essential part of the results is exposed in our recent article: V.B. Matveev, A.O. Smirnov "AKNS and NLS hierarchies, MRW...
We consider Zakharov-Kuznetsov (ZK) equation, a higher-dimensional generalization of the well-known KdV equation. We discuss the behavior of solutions close to the solitary wave given by Q(x-t,y,z) with Q being the standard ground state.
We discuss the stability of solitary waves in the 3d quadratic (subcritical) ZK equation, proving that solutions in the energy space that are orbitally stable...
We present recent results and open issues on KP-type equations arising in the modeling of internal waves
