# Integrable Systems 2012

Faculty of Mathematics and Computer Science, ul. Słoneczna 54

University of Warmia nad Mazury in Olsztyn 21 - 22 June 2012

LECTURE ROOM A2/15

**LIST OF SPEAKERS**

- Marco Bertola (Concordia University, Montreal)
- Mariusz Białecki (Institute of Geophysics PAN, Warsaw)
- Maciej Błaszak (A. Mickiewicz University, Poznan)
- Jan L. Cieśliński (University of Białystok)
- Marek Czachor (Gdańsk University of Technology)
- Adam Doliwa (University of Warmia and Mazury, Olsztyn)
- Ziemowit Domański (A. Mickiewicz University, Poznan)
- Grzegorz Kwiatkowski (Gdansk University of Technology)
- Sergey Leble (Gdańsk University of Technology)
- Andrzej J. Maciejewski (University of Zielona Gora)
- Michal Marvan (Silesian University in Opava)
- Maciej Nieszporski (University of Warsaw)
- Andriy Panasyuk (University of Warmia and Mazury, Olsztyn)
- Maria Przybylska (University of Zielona Gora)
- Tomasz Stachowiak (Copernicus Center, Krakow)
- Błażej Szablikowski (A. Mickiewicz University, Poznan)
- Adam Szereszewski (University of Warsaw)
- Wojciech Szumiński (University of Zielona Gora)
- Karol Życzkowski (Institute of Physics, Jagiellonian University, Cracow, and Center for Theoretical Physics, Polish Academy of Science, Warsaw)

Antoni Sym (University of Warsaw), XXX-lecie powierzchni solitonowych

**SCHEDULE**

**Thursday 21.06 (10 lectures)**

09:00 - 10:00 M. Bertola

10:00 - 10:30 B. Szablikowski

10:30 - 11:00 coffee break

11:00 - 12:00 K. Życzkowski (jointly with Geometry and Dynamical Systems Seminar)

12:00 - 12:30 A. Panasyuk

12:30 - 14:00 lunch

14:00 - 15:30 Z. Domański, M. Błaszak, M. Czachor

15:30 - 16:00 coffee break

16:00 - 17:30 G. Kwiatkowski, S. Leble, T. Stachowiak

**Friday 22.06 (9 lectures)**

09:00 - 10:30 M. Białecki, J. Cieśliński, M. Nieszporski

10:30 - 11:00 coffee break

11:00 - 12:30 M. Przybylska, W. Szumiński. J. Maciejewski

12:30 - 14:00 lunch

14:00 - 15:30 A. Szereszewski, M. Marvan, A. Doliwa

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**NEXT MEETING WILL BE ORGANIZED BY JAN CIEŚLIŃSKI IN BIAŁYSTOK**

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**TITLES AND ABSTRACT OF TALKS**

**Marco Bertola (Concordia University, Montreal)**

Title: The Riemann--Hilbert Method; a Swiss Army knife in integrable systems

Abstract: Random Matrix models, nonlinear integrable waves, Painleve' transcendents, determinantal random point processes seem very unrelated topics. They have, however, a common point in that they can be formulated or related to a Riemann-Hilbert problem, which then enters prominently as a very versatile tool. Its importance is not only in providing a common framework, but also in that it opens the way to rigorous asymptotic analysis using the nonlinear steepest descent method. I will briefly sketch and review some results in the above mentioned areas.

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**Mariusz Białecki (Institute of Geophysics PAN, Warsaw)**

Title: Many faces of Random Domino Automaton

Abstract: Random Domino Automaton is a toy-model of earthquakes. It is a slowly driven system exhibiting avalanches. Depending on parameters of the model, it produces wide range of distributions: from exponential, through inverse-power, up to quasi-periodic. This allows to explain within the uniform framework various distributions of real earthquakes. Random Domino Automaton has also other meanings. It is a generalisation of prominent Drossel-Schwabl forest-fires model. It is also a test model for reconstruction of Ito equation from time-series. Moreover, suitably specified equations describing stationary state of the automaton take a form Motzkin numbers recurrence, and are solvable.

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**Maciej Błaszak, Ziemowit Domański (A. Mickiewicz University, Poznan)**

Title: Canonical transformations of coordinates in quantum mechanics I. II

Abstract: The talk aims at introducing the general theory of canonical transformations of phase space coordinates in quantum mechanics. We develop the theory of transformations within a context of phase space quantum mechanics which seems to be the most natural approach to quantization, allowing a straightforward development of transformations of coordinates in a complete analogy with classical theory. Later we pass the developed formalism to ordinary description of quantum mechanics. Moreover, we show that the presented theory of transformations gives a consistent way of quantizing classical Hamiltonian systems after a canonical transformation of coordinates.

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**Jan L. Cieśliński (University of Białystok)**

Title: The cross ratio and its applications

Abstract: The cross ratio (or anharmonic ratio) is a fundamental invariant in projective geometry and Moebius conformal geometry. As an example of a beautiful application of the cross ratio there will be presented the van der Pauw method (1958) of measuring the resistivity of thin samples (assumed to be simply connected, i.e., without holes). The method consists in appropriate measurements of currents and voltages at four contacts (placed anywhere on the circumference of the sample). One does not need to know neither a shape of the sample nor positions of the contacts. Some new results (2012) will be announced.

The classical cross ratio is a function of four (ordered) points lying on a real or complex line. In order to show important links between the cross ratio and the soliton theory I will recall my results (1997) extending this notion on points of a Euclidean or pseudo-Euclidean space. The main idea consists in embedding this space in a Clifford algebra (then the cross ratio takes values in the corresponding Spin group). The generalized cross ratio is useful in dealig with discretizations of integrable systems associated with orthogonal and conjugate coordinates (2008).

**Marek Czachor (Gdańsk University of Technology)**

Title: Application of density matrices beyond quantum mechanics (a variation on von Neumann equations)

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**Adam Doliwa (University of Warmia and Mazury, Olsztyn)**

Title: Minimal surfaces in the soliton surfaces approach

Abstract: The soliton surfaces approach (A. Sym, Soliton Surfaces, Lett. Nuovo Cimento 33 (1982) 394-400 ) provides a tool to extract the geometric content of integrable PDEs. However, for a long time it was believed that it works for S-integrable systems only. In my talk I will present an application of the method to the Liouville equation, which is genuine example of a C-integrable system. In this case a suitable application of the soliton surface approach gives the classical Weierstrass representation of minimal surfaces.

Concerning history of the soliton surfaces approach see also an essay (in polish) by Antoni Sym: XXX-lecie powierzchni solitonowych

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**Grzegorz Kwiatkowski, Sergey Leble**

(Gdansk University of Technology)

Title: Green functions by Darboux transformations for a class of heat equations

Abstract: Darboux transformations theory is applied to construction of Green fumctions for heat equation with variable coefficients. Some classes of explicit expressions of the Green function in the case of finite-gap potential coefficient of the heat equation are constructed. An algorithm and program for Mathematica are presented for a subclass directly linked to hyperelliptic functions. A construction of the heat kernel diagonal is considered as element of generalized Zeta function theory, that, being meromorfic function, its gradient at the origin defines determinant of a differential operator in a technique for regularizing quadratic path integral.

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**Sergey Leble (Gdańsk University of Technology)**

Title: Planar zero-range potentials via Moutard transformations and differential geometry

Abstract: Darboux-like (Moutard) and generalized Moutard transformations in two dimensions are applied to construct families of zero range potentials for Schrodinger and Dirac problems. Its applications to differential geometry of surfaces is discussed. Darboux-like (Moutard) and generalized Moutard transformations in two dimensions are applied to construct families of zero range potentials for Schrodinger and Dirac problems. Its applications to differential geometry of surfaces is discussed.

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**Andrzej J. Maciejewski (University of Zielona Gora)**

Title: Reduction and relative equilibria in the Kinoshita problem

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**Michal Marvan (Silesian University in Opava)**

Title: Some results concerning the constant astigmatism equation

Authors: A. Hlaváč and M. Marvan.

Abstract: The talk concerns the constant astigmatism equation $z_{yy} + (1/z)_{xx} + 2 = 0$. We newly interpret its solutions as describing spherical orthogonal equiareal patterns, with relevance to two-dimensional plasticity. We show how the classical Bianchi superposition principle for the sine-Gordon equation can be extended to generate an arbitrary number of solutions of the constant astigmatism equation by algebraic manipulations. As a by-product, we show that sine-Gordon solutions give slip line fields on the sphere. Finally, we compute the solutions corresponding to classical Lipschitz surfaces of constant astigmatism via the corresponding equiareal patterns.

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**Maciej Nieszporski (University of Warsaw)**

Title: Integrable correspondences

Abstract: We discuss bond (or vertex-bond) lattice equations that can be rewritten as scalar vertex lattice equations or more generally as scalar vertex lattice correspondences. The scalar vertex lattice equations are well known integrable lattice equations, whereas the remaining correspondences seem to be avoided in integrable discourse. The main difficulty raised up against the correspondences is that they provide multivalued evolution. But this obstacle can be easily overcome by denying the correspondences separate existence. We show that consideration of the correspondences together with the bond lattice equations and proper reformulation of initial value problem allows us to treat all correspondences on an equal footing. We also present the B\"acklund transformations from the well known integrable lattice equations to the correspondences, consistency around the cube property and soliton solutions of the latter

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**Andriy Panasyuk (University of Warmia and Mazury, Olsztyn)**

Title: Lie-Poisson pencils related to semisimple Lie algebras: towards classification

Abstract: Compatible pairs of linear Poisson structures are on the second level of the hierarchy of algebraic mechanisms leading to integrable bihamiltonian systems: on the first level there are pairs "constant+linear" (the so-called argument shift method), on the higher levels one finds the pairs of the types "linear+quadratic", "qudratic+quadratic" etc. Examples from the second level were known in the literature but the problem of their classification remained open. In the talk an approach to calssification of such pairs (i.e. in fact pairs of compatible structures of Lie algebras) will be presented, related to simple Lie algebras. The classification leads to two classes of such pairs corresponding to Z/nZ- (n>2) and (Z/2Z)x...x(Z/2Z)-gradings on such algebras.

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**Maria Przybylska (University of Zielona Gora)**

Title: Application of residue calculus to integrability analysis of rational potentials

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**Tomasz Stachowiak (Copernicus Center, Krakow)**

Title: Existence of exact solutions of the Dirac equation

Abstract: The Dirac equation, when reducible to a second order ordinary differential equation can be effectively studied with the Galois group approach. I would like to review the basic facts of solubility in terms of Liouvillian extensions, and show a strikingly simple result for polynomial potentials. The rational case is not quite as easy, but it is still possible to recover some integrable potentials, which I show using the example of the Whittaker equation.

**Błażej Szablikowski (A. Mickiewicz University, Poznan)**

Title: Classical r-matrix like approach to Frobenius manifolds, WDVV equations and flat metrics

Abstract: The theory of Frobenius manifolds is the coordinate-free formulation of the topological field theories and corresponding Witten-Dijkgraaf-Verlinde-Verlinde (WDVV) associativity equations. These manifolds are equipped with a structure of Frobenius algebra in the tangent bundle and flat invariant metric. It is known that Frobenius manifolds are equivalent to the quasi-homogeneous flat pencils of metrics, that generate compatible pairs of hydrodynamic Poisson tensors. The bi-Hamiltonian systems of hydrodynamic type can be efficiently constructed using the theory of classical $r$-matrices. My goal was to develop an analogous scheme allowing for construction of Frobenius manifolds and corresponding solutions of WDVV equations. In the talk I will present a general scheme for the construction of Frobenius algebras, pencils of flat metrics and Frobenius manifolds. In this theory I take advantage of the Rota-Baxter identity and some relation being counterpart of the modified classical Yang-Baxter equation in the case of Poisson algebras. (The two Baxters are not related.) Some examples will be given.

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**Adam Szereszewski, Antoni Sym (Uniwersytet Warszawski)**

Title: Geometry of R-separable metrics in Schrodinger equation

Abstract: Gaston Darboux was the first to give the general treatment of R-separability in PDEq (Laplace eq. on E^3). I will discuss the geometry of 3-dimensional Riemannian metrics R-separable in Schrodinger equation and remaind the results of Darboux. I will also present the generalization of R-separability problem to n-dimensional diagonal metrics.

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**Wojciech Szumiński (University of Zielona Gora)**

Title: Dynamics of multiple pendula

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**Karol Życzkowski (Institute of Physics, Jagiellonian University, Cracow, and Center for Theoretical Physics, Polish Academy of Science, Warsaw)**

Title: Birkhoff polytope and its subset of unistochastic matrices

Abstract: A square matrix B of size N is called bistochastic (doubly stochastic) if it contains non-negative entries and their sum in each row and each column is equal to unity. A bistochastic matrix describes a Markov chain -discrete dynamics in the simplex of N-point probability vectors. The set of all bistochastic matrices, called Birkhoff polytope, is formed by the convex hull of N! permutation matrices. A bistochastic matrix B is called 'unistochastic' if there exist a unitary U such that B_{ij}=|U_{ij}|^2. Such matrices play important role in various physical problems and can be used to construct quantum analogues of classical dynamical systems. For N=2 every bistochastic matrix is unistochastic.For N=3 there exist bistochastic matrices which are not unistochastic, but the characterization of the set of unistochastic matrices is known. The question if a given matrix is unistochastic is open for N. ge. 4. Some conjectures on its structure will be presented.