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...
In many interesting cases, distribution functions of random matrix theory and correlation functions of integrable models of statistical mechanics and quantum C-field theory are given by tau functions of Painlevé equations. I will discuss an extension of the Jimbo-Miwa-Ueno differential to the space of monodromy data and explain how this construction can be used to compute constant terms in the...
Some results on non-generic isomonodormy deformations, already presented in other conferences and seminars is some detail, will be here reviewed in 25 minutes, stressing their relevance for computations involving Painleve' equations and Dubrovin-Frobenius Manifolds. From joint works with G. Cotti and B. Dubrovin
In the past years, in a joint project with A. Buryak, we have developed a general framework to construct classical and quantum field systems (in one space and one time dimensions) from intersection theory of the moduli space of stable curves. Recently a particularly interesting example came under our attention, which actually produces a integrable system of Hamiltonian PDEs in two space and...
We study the defocusing Non-Linear Schrödinger (NLS) equation written in hydrodynamic form through the Madelung transform. From the mathematical point of view, the hydrodynamic form can be seen as the Euler-Lagrange equations for a Lagrangian submitted to a differential constraint corresponding to the mass conservation law. The dispersive nature of the NLS equation poses some major numerical...
In this talk I will overview recent developments concerning a version of the Kadomtsev-Petviashvili (KP) equation for surface gravity waves related to elliptic-cylindrical geometry, a system of coupled Ostrovsky equations derived for strongly interacting internal waves in the presence of background rotation and a shear flow, and 2+1-dimensional cylindrical Korteweg-de Vries (cKdV)-type model...
We consider a linear two-dimensional hyperbolic equation of second order, whose coefficients are polynomials with respect to one of independent variables.
We show that this equation possesses infinitely many particular solutions, determined by solutions of ordinary differential equations.
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...
A surprisingly large number of physically relevant dispersive partial differential equations are integrable. Using the connection between the spectrum and the eigenfunctions of the associated Lax pair and the linear stability problem, we investigate the stability of the spatially periodic traveling wave solutions of such equations, extending the results to orbital stability in those case where...
In this talk, I shall report on some recent results obtained in collaboration with P.G. Grinevich (LITP,RAS). We construct edge vectors on planar networks obtained by gluing the positive part of copies of Gr(1,3) and Gr(2,3) and apply such construction to associate real regular KP divisors on rational degenerations of M-curves.
We will present multi-component generalizations of derivative nonlinear Schrödinger (DNLS) type of equations having quadratic bundle Lax pairs related to $\mathbb{Z}_2$-graded Lie algebras and A.III symmetric spaces. The Jost solutions and the minimal set of scattering data for the case of local and nonlocal reductions are constructed.
Furthermore, the fundamental analytic solutions (FAS)...
In this work we will look at the focusing Davey-Stewartson equation from two different angles, using advanced numerical tools.
As a nonlinear dispersive PDE and a generalisation of the non-linear Schrödinger equation, DS possesses solutions that develop a singularity in finite time. We numerically study the long time behaviour and potential blow-up of solutions to the focusing Davey-Stewartson...
The "tropical limit'' of a matrix (KdV, Boussinesq, or the like) soliton solution in two-dimensional space-time consists of a piecewise linear graph in space-time, together with values of the dependent variable along its segments. In two space-time dimensions, it associates with such waves a point particle picture, in which free "particles'' with internal degrees of freedom interact at...
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...
The classical Muira maps transforms the second Hamiltonian structure of the KdV hierarchy into constant, or Darboux, form. In this talk multi-component versions of this map are constructed, for Hamiltonian structures defined by Novikov algebras.
We consider first-order Lagrangians whose Euler-Lagrange equations belong to the class of 3D dispersionless integrable systems. Our main results can be summarised as follows:
(1) A link between integrable Lagrangians and Picard curves/Picard modular forms studied by E. Picard as far back as in 1883 is established.
(2) A parametrisation of integrable Lagrangian densities by generalised...
We present recent advances in solution of rational spin chains that build on results from representation theory of Yangian and analytic Bethe Ansatz. First we show how to find spectrum of supersymmetric GL(N|M) chains using Q-system on Young diagrams. This approach can be more efficient than conventional nested Bethe equations, and also it provides one with means for explicitly counting...
We investigate the statistically stationary state of spontaneous noise-driven modulation instability of a plane wave (condensate) background. As a model we use the integrable focusing one-dimensional nonlinear Schrodinger equation (NLSE). The fundamental statistical characteristics of the stationary state of the modulation instability obtained numerically by Agafontsev and Zakharov in 2015 [1]...
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
