Mathematics Area - SISSA - Integrable systems, Frobenius manifolds and nonlinear waves

Integrable structures and bi-hamiltonian geometry

Thomas Nicosanti — Tue, 12 May 2026 12:00:10 +0000

Lecturer:

Guido Carlet

Course Type:

PhD Course

Academic Year:

2025-2026

Period:

May

Duration:

3 h

Description:

Contents:

These two lectures will cover the following topics:

• KdV hierarchy and its bi-hamiltonian formulation
• Local multivector fields, delta and theta formalisms,
• Deformations of Poisson brackets and Poisson cohomology,
• Deformations and cohomology in bi-Hamiltonian geometry.

Research Group:

Geometry and Mathematical Physics

Location:

A-136

Next Lectures:

Integrable systems and generalized hydrodynamic

Ariel Surya Boiardi — Thu, 11 Sep 2025 09:05:00 +0000

Lecturer:

Tamara Grava

Course Type:

PhD Course

Master Course

Academic Year:

2025-2026

Period:

April - May

Duration:

40 h

Description:

Prerequisites: some knowledge of classical mechanics will be useful.

Contents:
Integrable Systems: 2 cycles 40 hours

The first part of the course is an introduction to the modern theory of integrable systems and its fundamental concepts.
• 1.1 Short review of the classical theory of finite-dimensional integrable systems
• 1.2 Bi-Hamiltonian Geometry and Lax pair
• 1.3 The Toda system
• 1.4 The Korteweg de Vries equation: direct and inverse scattering on the line and soliton gas
This part of the course is classical, based on my lecture notes and the following book
• Novikov, S.; Manakov, S. V.; Pitaevski, L. P.; Zakharov, V. E. Theory of solitons. The inverse scattering method. Translated from the Russian. Contemporary Soviet Mathematics. New York, 1984. xi+276 pp. ISBN: 0-306-10977-8

The second part of the course is an exploration of the generalized hydrodynamic equations. Generalized Hydrodynamics extends the usual Euler hydrodynamics to integrable systems. These equations have been derived for several integrable equations, from two different perspectives. Zakharov obtained the equations in his formulation of the kinetic theory of solitons 1971. The same equations have emerged in 2016 in the statistical mechanics of integrable systems in the thermodynamic limit. A rigorous proof of these equations exists only for the box-ball model (discrete in space and time) and the hard rod model and there is a road-map for a proof for the Toda lattice. We will explore what has been done for integrable partial differential equations.

References:

• Soliton gas: theory, numerics, and experiments P.Suret, S.Randoux, A.Gelash, D.Agafontsev, B.Doyon, G.El, Phys. Rev. E 109 (2024), no. 6, Paper No. 061001, 35 pp.
• Generalized hydrodynamics of the KdV soliton gas. T. Bonnemain, B. Doyon, G. A. El https://arxiv.org/pdf/2203.08551
• Law of Large Numbers and Central Limit Theorem for random sets of solitons of the focusing nonlinear Schr¨odinger equation. M. Girotti, T.Grava, K. D. T-R McLaughlin, J. Najnudel, Duke Math Journal 2026. Preprint https://arxiv.org/pdf/2411.17036

Research Group:

Geometry and Mathematical Physics

Location:

A-136

Next Lectures:

Frobenius manifolds: analytic theory

Guglielmo Padula — Thu, 26 Sep 2024 08:30:17 +0000

Lecturer:

Davide Guzzetti

Course Type:

PhD Course

Academic Year:

2024-2025

Period:

November - January

Duration:

60 h

Description:

The aim of the course is to introduce the audience to the analytic theory of Dubrovin-Frobenius manifolds.

The theory of Frobenius manifolds was constructed by B. Dubrovin to formulate in geometrical terms the WDVV equations of associativity of 2D topological field theories.

It has links to many branches of mathematics, like singularity theory and reflection groups, algebraic and enumerative geometry, quantum cohomology, theory of isomonodromic deformations, boundary value problems and Painlevé equations, integrable systems and non-linear waves.

In this course, we will give the general setting of Dubrovin-Frobenius manifolds, and then we will concentrate on the study of their moduli, namely the monodromy data of a flat connection defined on the manifold. We will see some examples and applications.

Prerequisites. Basic differential and Riemannian geometry, complex functions, theory of differential equations in the complex domain and isomonodromy deformations.

Research Group:

Geometry and Mathematical Physics

Next Lectures:

Excursion in Integrability

Guglielmo Padula — Thu, 26 Sep 2024 08:29:12 +0000

Lecturer:

Tamara Grava

Course Type:

PhD Course

Academic Year:

2024-2025

Period:

February - March

Duration:

20 h

Description:

The goal of the course is an excursion on algebraic aspects of integrable systems. The topics are

KP equation and Sato-Segal-Wilson Grassmannian.
Tau function and Hirota bilinear relation.
Plücker coordinates.
Schur function expansion.
Hermitian random matrix models.
Orthogonal polynomials, partition function and Schur expansion.
Combinatorial interpretation of correlator expansion of classical matrix.
Integrals: ribbon graphs and Hurwitz numbers.

References:

Tau functions and their applications, Cambridge University press 2021, F. Balogh, J. Harnad.
Free fermions and tau-functions, Alexander Alexandrov, Anton Zabrodin.
Combinatorics and random matrix theory, Baik, Jinho; Deift, Percy; Suidan, Toufic, Grad. Stud. Math., 172, American Mathematical Society, Providence, RI, 2016.

Research Group:

Geometry and Mathematical Physics

Next Lectures:

Integrable systems

Anantha Krishnan Orunnukaran Mani — Fri, 22 Sep 2023 18:23:05 +0000

Lecturer:

Tamara Grava

Course Type:

PhD Course

Anno (LM):

Second Year

Academic Year:

2023-2024

Period:

February-May

Duration:

50 h

Description:

Integrable systems are special systems which can be solved exactly in some sense. They arise in a variety of settings, ranging from Hamiltonian systems, nonlinear wave equations and probability. This course covers the origins of the subject as well as modern topics in nonlinear waves and integrable probability.

1. The Korteweg de Vries equation (KdV)

Scattering and inverse scattering of the Schrodinger equation on the line and solution of KdV equation;
as a Fredholm determinant. Tau-function and Fredholm determinants;
Periodic problem of KdV and finite gap solutions on Riemann surfaces;
Modulation theory of nonlinear waves and Whitham equations.

2. The Toda lattice: an example of a non linear integrable system of particles

The Lax matrix and the Adler-Kostant-Syme scheme of integration;
Factorisation theorem and asymptotic behaviour of solutions;
Gibbs measure initial data and random matrices.

3. Random matrices

Hermitian random matrices. Partition function as an example of tau-function of the Toda lattice;
Partitions and Schur function expansion of tau functions;
Topological expansion, enumeration of maps and Hurwitz numbers.

Exam

The evaluation in this course will consist of homework assignments, and a final exam, in the form of a seminar.

References

O. Babelon, D. Bernard and M. Talon, Introduction to Classical Integrable Systems, Cambridge Monographs on Mathematical Physics, Cambridge University Press, 2003.
M.J. Ablowitz and P.A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering, London Mathematical Society Lecture Note Series 149, Cambridge University Press, 1992.
J. Harnad and F. Balogh, Tau functions and their applications. Cambridge University Press Online publication date: January 2021 Online ISBN: 9781108610902 DOI: https://doi.org/10.1017/9781108610902
G. Segal "Integrable systems and inverse scattering” in the book "Integrable Systems"; Oxford Sci. Publ. 1999
J. Moser "Various aspects of integrable Hamiltonian systems." Progr. Math. 8, Birkhauser, Boston, Mass.,1980, pp. 233-289
Greg W. Anderson, Alice Guionnet, Ofer Zeitouni An Introduction to Random Matrices, Cambridge University Press Print publication year: 2009/ https://doi.org/10.1017/CBO9780511801334

Rooms

For Febraury

Always A-132
21/02 A-004
28/02 A-138

From March 19

Morning A-134
Afternoon A-005

Research Group:

Geometry and Mathematical Physics

Location:

A-132

Next Lectures:

On Hirota's discrete KP equation - old and new

Song Jin Ri — Thu, 20 Jan 2022 14:28:52 +0000

Research Group:

Geometry and Mathematical Physics

Adam Doliwa

Schedule:

Monday, January 24, 2022 - 16:00 to 17:00

Abstract:

In the first part of the talk I would like to recall basic information on the non-abelian Hirota-Miwa equation and on the corresponding map satisfying Zamolodchikov's tetrahedron condition. This includes the projective geometric interpretation of the Hirota map and of its multidimensional consistency, which points out towards a generalization of the map allowing for the quantum reduction. In the second part I will show that the Hermite-Pad\'e type I approximation problem leads in a natural way to Hirota's discrete KP system subject to an integrable constraint. Our result explains the appearence of various ingredients of the integrable systems theory in application to multiple orthogonal polynomials, numerical algorithms, random matrices, and in other branches of mathematical physics and applied mathematics where the Hermite-Pad\'e approximation problem is relevant. If time permits I will show how generalize this connection to the non-commutative level.

Integrable two-component systems of difference equations

Song Jin Ri — Sun, 09 Jan 2022 09:47:20 +0000

Research Group:

Geometry and Mathematical Physics

Pavlos Kassotakis

Schedule:

Monday, February 7, 2022 - 16:00 to 17:00

Abstract:

We will present two lists of two-component systems of integrable difference equations defined on the edges of the $\mathbb{Z}^2$ graph. The integrability of these systems is manifested by their Lax formulation which is a consequence of the multi-dimensional compatibility of these systems. Imposing constraints consistent with the systems of difference equations, we recover known integrable quad-equations including the discrete version of the Krichever-Novikov equation. The systems of difference equations give us, in turn, Yang-Baxter maps. Some of these maps can be considered as particular reductions of non-abelian Yang-Baxter maps.

The focusing nonlinear Schrödinger equation with step-like oscillating background

Song Jin Ri — Sun, 09 Jan 2022 09:45:30 +0000

Research Group:

Geometry and Mathematical Physics

Jonatan Lenells

Schedule:

Monday, January 17, 2022 - 14:00 to 15:00

Abstract:

I will discuss joint work with Anne Boutet de Monvel and Dmitry Shepelsky where we study the asymptotic behavior of solutions of the nonlinear Schrödinger equation. More precisely, we consider the Cauchy problem for the focusing nonlinear Schrödinger equation with initial data approaching different plane waves at plus and minus infinity. Using Riemann–Hilbert techniques and Deift-Zhou steepest descent arguments, we study the long-time behavior of the solution. We show that there is a wide range of possible asymptotic scenarios. We propose a method for rigorously establishing the existence of certain higher-genus asymptotic sectors, and we compute detailed asymptotic formulas in a genus three sector, i.e., in a sector where the leading term of the asymptotics is given in terms of hyperelliptic functions attached to a Riemann surface of genus three.

Long-time asymptotics for the integrable nonlocal nonlinear Schrödinger equation

Song Jin Ri — Wed, 03 Feb 2021 08:54:24 +0000

Research Group:

Geometry and Mathematical Physics

Dmitry Shepelsky

Institution:

Institute for Low Temperature Physics and Engineering, Kharkiv, Ukraine

Schedule:

Wednesday, February 24, 2021 - 16:00 to 17:00

Abstract:

We study the initial value problem for the integrable nonlocal nonlinear Schrodinger (NNLS) equation with the initial conditions of two types: (i) decaying at infinity initial conditions;(ii) step-like initial data: Our main tool is the adaptation of the nonlinear steepest-decent method to the study of Riemann-Hilbert problems associated with the NNLS equation with the specified boundary conditions.In case (i), our main result is that, in contrast to the conventional (local) NLS equation, the power decay rate as t goes to infinity depends on the ratio x/t.For case (ii), since our equation is not translation invariant, we explore the dependence of the asymptotic scenarios on shifts of the step-like initial data.

Past Activities of the Integrable Systems, Frobenius Manifolds and Nonlineare Waves group

Daniele Agostinelli — Sun, 17 May 2020 17:24:32 +0000

(partially funded by MISGAM and ENIGMA)

International Conference Geometrical Methods in Mathematical Physics , 12-17 December 2011, Moscow, Russia
Integrable systems in pure and Applied Mathematics, a conference in honour of Boris Dubrovin's 60 birthday, 8-12 June 2010, Sardinia, Italy
Conference on Integrable Systems, Geometry, Matrix Models and Applications, Trieste, 14-18 October 2008
Frobenius Structures and Singularity Theory, Poznan, 25-29 August 2008
Conference on "Random and Integrable models in Mathematics and Physics", Solvay Institute, Brussels, 11-15 September 2007
Conference on "Algebraic aspects of integrable systems", Islay, Scotland, July 1-7, 2007
"ENIGMA Conference on Mathematical Physics", KTH, Stockholm, June 25-28, 2007
Summer School on "Aspects of Membrane Dynamics", KTH, Stockholm, June 11-23, 2007
Conference on "Tropical Geometry and Applications", Loughborough University, Leicestershire (UK), April 20-23, 2007
Workshop on "Higher Structures in Geometry and Physics", IHP, Paris, January 15-19, 2007
Conference "Random Matrices", Centre International de Rencontres Mathématiques (CIRM), Luminy, France, October 30 - November 3 2006
"Integrable Systems in Applied Mathematics", Colmenarejo (Madrid) 4-8 September 2006. A satellite Conference of the International Congress of Mathematicians (ICM 2006, Madrid 22-30 August 2006)
"Affine Hecke algebras, the Langlands Program, Conformal Field Theory and Matrix Models", Centre International de Rencontres Mathématiques (CIRM), Luminy, France, June 19 - July 14 2006
Conference on "Riemann-Hilbert Problems Integrability and Asymptotics", SISSA, Trieste, 20-25 September 2005
Workshop on "Random Matrices and Other Random Objects", FIM, ETH-Zurich, 17-21 May 2005
Workshop "Geometry and Integrable Systems", Berlin, 3-7 June 2005