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.

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...

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]...

The aim of the talk is to introduce a transformation which reduces initial-value problem for one-dimensional Schroedinger equation with a non-vanishing potential to an elementary homogeneous first-order nonlinear ODE. The latter exhibits nonlinearity merely as complex conjugation and hence is very amenable to application of transform methods and further complex-analytic treatment. The obtained...