# Studium

## Seminář z diferenciální geometrie a jejich aplikací – vedoucí doc. Sergyeyev

### Program v akademickém roce 2014/2015

22.05.2015

Roman Popovych, D.Sc. (Wolfgang Pauli Institute, Rakousko),
Noether theorems

Abstrakt:

After recalling classical Noether's first and second theorems on conservation laws
of differential equations, we enhance and generalize Noether's second theorem.

More specifically, we prove that a system of differential equations is abnormal,
i.e., it has an identically vanishing differential consequence, if and only if
it possesses a trivial conserved vector corresponding to a nontrivial
characteristic.

Moreover, the above properties are also equivalent to that the system admits a family
of characteristics that is parameterized by an arbitrary function of all independent
variables in a local fashion and whose Frechet derivative with respect to the parameter-
function does not vanish on solutions of the system for a value of the parameter-function.
The theorem is illustrated by physically relevant examples.

14.05.2015

Doc. RNDr. Artur Sergyeyev, Ph.D.
Noncommutative integrability of non-Hamiltonian systems

07.05.2015

Mgr. Jiří Jahn
Úvod do teorie komplexních variet

Abstrakt:

Cílem přednášky je seznámit posluchače se základními pojmy a výsledky z
oblasti komplexních variet. V průběhu přednášky se pokusíme vysvětlit, v čem
odpovídající protějšky elementárních konstrukci dobře známých z kategorie
hladkých variet (vektorová pole, diferenciální formy), a pokud čas dovolí,
skončíme komplexní obdobou Riemannovy variety (tou je tzv. Hermiteovska
varieta), resp. jejím asi nejdůležitějším speciálním případem (tzv. Kählerova varieta).

30.04.2015

Doc. RNDr. Artur Sergyeyev, Ph.D.
Integrabilita S-deformovatelných ploch

Abstrakt:

viz arXiv:1501.07171 (společná práce s prof. I.S. Krasil'shchikem)

23.04.2015

Mgr. Diana Barseghyan, Ph.D. (Ostravská univerzita a Ústav jaderné fyziky AV ČR),
Schrödinger operators exhibiting spectral transition

Abstrakt:

In the first part of the talk we analyse two-dimensional Schrödinger operators
with the potential unbounded from below which demonstrate spectral tran-
sition. More precisely, we consider two-dimensional Schrödinger operators
with the potential |xy| p − λ(x 2 + y 2) p/(p+2) where p ≥ 1 and λ ≥ 0. We
show that there is a critical value of λ such that the spectrum for λ < λ crit
is bounded below and purely discrete, while for λ > λ crit it is unbounded
from below. In the subcritical case we prove upper and lower bounds for
the eigenvalue sums. As the second example we discuss a modification of
Smilansky model in which a singular potential ‘channel’ is replaced by a reg-
ular, below unbounded potential which shrinks as it becomes deeper. We
demonstrate that, similarly to the original model, such a system exhibits a
spectral transition with respect to the coupling constant, and determine the
critical value above which a new spectral branch opens.

The goal of the second part of the talk is to derive estimates of eigenvalue
moments for Dirichlet Laplacians and Schrödinger operators in regions having
infinite cusps which are geometrically nontrivial being curved; we are going
to show how those geometric properties enter the eigenvalue bounds.

In the third part of the talk we investigate Dirichlet Laplacian in a straight
twisted tube of a non-circular cross section, in particular, its discrete spec-
trum coming from a local slowdown of the twist. We prove a Lieb-Thirring-
type estimate for the spectral moments and present two examples illustrating
how the bound depends on the tube cross section.

16.04.2015

RNDr. Petr Vojčák, Ph.D.,
Jets: efektivní cesta od kosymetrie k zákonu zachování.

Abstrakt:

Hledání zákonů zachování systému diferenciálních rovnic v n nezávislých
proměnných lze rozdělit do dvou víceméně samostatných kroků.
Nejdříve hledáme řešení adjungovaného linearizovaného systému, kterými
jsou tzv. charakteristiky zákonů zachování (kosymetrie).

V druhém kroku pomocí nalezených kosymetrií sestavíme lagrangiány, pro
které hledáme jejich rozklad na totální divergenci nějaké n-tice
diferenciálních funkcí.

V této přednášce představíme proceduru naprogramovanou v softwaru Maple s
užitím balíčku Jets, která za jistých předpokladů poskytuje velmi
efektivní nástroj pro nalezení zákonů zachování příslušných k daným
kosymetriím. Představíme ovšem také několik příkladů, ve kterých uvedená
procedura selhává a prodiskutujeme příčiny, proč tomu tak je.

Jedná se o společnou práci s Hynkem Baranem.

09.04.2015

On multisoliton solutions of the constant astigmatism equation

Abstrakt:

26.03.2015

RNDr. Jiřina Jahnová, Ph.D.,
On the proof of Yoshida's conjecture

Abstrakt:

19.03.2015

Doc. RNDr. Artur Sergyeyev, Ph.D.
Bezdispersní integrabilní systémy: od dimenze (2+1) k (3+1)

15.01.2015

Prof. Maciej Blaszak (Uniwersytet im. Adama Mickiewicza, Poznan, Polsko)
Hamiltonian systems: from classical to quantum models IV: Quantum separability

08.01.2015

Prof. Maciej Blaszak (Uniwersytet im. Adama Mickiewicza, Poznan, Polsko)
Hamiltonian systems: from classical to quantum models II: Riemannian
representation of quantum Hamiltonian dynamics

18.12.2014

Prof. I.S. Krasil'shchik, DrSc.,
Symmetry reduction of Lax-integrable 3D systems and their properties

Abstrakt:

We present a complete description of 2-dimensional equations that arise as symmetry reductions of four 3-dimensional Lax-integrable equations:

(1) the universal hierarchy equation uyy=uzuxy-uyuxz;
(2) the 3D rdDym equation uty=uxuxy-uyuxx;
(3) the equation uty=utuxy-uyutx, which we call the modified Veronese web equation;
(4) Pavlov's equation uyy=utx+uyuxx-uxuxy.

For the reductions possessing finite-dimensional symmetry algebras, we describe nonlinear coverings and infinite hierarchies of nonlocal conservation laws.

This is joint work with H. Baran, O. Morozov, and P. Vojčák.

11.12.2014

Andrey Mironov, D.Sc. (Sobolev Institute of Mathematics,  Novosibirsk, Rusko) Algebro-geometric solutions of the equations of minimal Lagrangian surfaces in CP2

Abstrakt:

We suggest a method for constructing minimal Lagrangian immersions of R2 in CP2 with induced diagonal metric in terms of Baker–Akhiezer functions of algebraic curves.

04.12.2014

Andrey Mironov, D.Sc. (Sobolev Institute of Mathematics,  Novosibirsk, Rusko)
Integrable geodesic flows on 2-torus

Abstrakt:

We discuss classical question of existence of polynomial in momenta integrals for geodesic flows on the 2-torus. For the quasi-linear system on the coefficients of the polynomial integral we study the region (the so-called elliptic region) where there are complex-conjugate eigenvalues. We show that for quartic integrals the other two eigenvalues are real and necessarily genuinely nonlinear. This observation together with the property of the system to be rich (semi-Hamiltonian) enables us to classify elliptic regions completely. The case of complex-conjugate eigenvalues for the system corresponding to the integral of degree 3 is done similarly. These results show that if new integrable examples exist they could be found only within the region of hyperbolicity of the quasi-linear system.

01.12.2014 (mimořádně v pondělí v zasedací místnosti rektorátu SU v 14.45)

Prof. Raffaele Vitolo (Universita del Salento, Itálie)
On the geometry of homogeneous third-order Hamiltonian operators

Abstrakt:

We will present old and new results on the affine and projective geometry of homogeneous third order Hamiltonian operators. We give a classification of such operators for a number of components less than or equal to 3. If such operators are part of a bi-Hamiltonian formulation of a given system of PDEs, then the structure of the operator can be expressed in terms of a sequence of homogeneous conservation law densities in involution coming from the Magri scheme. This fact can be used in opposite direction to reconstruct the third-order homogeneous operator and find new bi-Hamiltonian systems. The WDVV hydrodynamic type systems are considered as old and new examples of PDEs that can be presented as Hamiltonian systems using third-order homogeneous Hamiltonian operators.

27.11.2014

prof. Mirjana Djoric (University of Belgrade, Srbsko)
Submanifolds of complex projective space

Abstrakt:

If for a real submanifold M of a complex manifold (M',J), the holomorphic tangent space Hx(M)=JTx(M)Tx(M) has constant dimension with respect to x ε M,  the submanifold M is called the CR submanifold and the constant complex dimension is called the CR dimension of M.

In  M. Djoric, M. Okumura, CR submanifolds of complex projective space, Dev. in Math. 19, Springer, 2009 we collected  the elementary facts about complex manifolds and their submanifolds and introduced the reader to the study of CR submanifolds of complex manifolds, especially complex projective space.

Since in the case of maximal CR dimension there are two geometric structures: an almost contact structure F, induced from the complex structure J of the ambient space M', and a submanifold structure, represented by the second fundamental tensor h of M in M', it is interesting to study certain conditions on F and h, and we present some of these results.

In this lecture we also discuss submanifolds of real codimension two of a complex manifold, especially of a complex space form. An n-dimensional complex hypersurface, which is a CR submanifold of CR dimension (n-2)/2, is a submanifold of real codimension two, but there also exist submanifolds of real  codimension two which are not CR submanifolds. We  present some new classification theorems for  submanifolds of real codimension two of a complex space form under the algebraic condition on the second fundamental form of the submanifold and the endomorphism induced from the almost complex structure on the tangent bundle of the submanifold.

This talk is based on joint research with M. Okumura.

20.11.2014

Pasha Zusmanovich, PhD. (Ostravská univerzita)
Cohomology and deformations of current Lie algebras

Abstrakt:

I will discuss cohomology and deformations of current Lie algebras, i.e.
Lie algebras formed as a tensor product of a Lie algebra and an
associative commutative algebra, as well as a more general class of Lie
algebras arising from any pair of algebras over Koszul dual operads. I
will try to explain why these algebras are interesting, and touch upon
their numerous applications in mathematics and physics.

10.11.2014 (mimořádně v pondělí v zasedací místnosti MÚ v 15.35)

O.I. Morozov, D.Sc. (AGH, Krakow, Polsko)
Integrable dispersionless PDEs in 4D, their symmetry pseudogroups and deformations

Abstrakt:

We study integrable non-degenerate Monge-Ampere equations of Hirota type in 4D and demonstrate that their symmetry pseudogroups have a distinguished graded structure, uniquely determining the equations. This is used to deform the equations into new integrable equations of Monge-Ampere type having large symmetry pseudogroups. We classify these integrable symmetric deformations, and we discuss their geometry and integrability.

06.11.2014

Prof. V.V. Lychagin (Universitetet i Tromsø, Norsko)
Invariants or observables in relativity theory II

Abstrakt:

See the abstract of the lecture of 03.11.2014.

04.11.2014 (mimořádně v úterý v místnosti R2 v 14.45)

Mgr. Vojtěch Pravda, Ph.D. (Matematický ústav AV ČR, Praha)
Universal spacetimes

Abstrakt:

Typical vacuum solutions of the Einstein equations do not solve the
vacuum equations of modified gravities (such as Lovelock gravity, quadratic
gravity etc). However, there is one exceptional class of Einstein spacetimes that
are immune to all corrections to Einstein equations following from the modified
gravities. These, so called universal spacetimes, solve vacuum equations of all
gravitational theories with the Lagrangian being a polynomial curvature invariant
constructed from the metric, the Riemann tensor and its derivatives of arbitrary
order. It is known since 1990s that pp-waves are universal. In this talk, we will
present our recent results on various more general classes of universal
spacetimes.

03.11.2014 (mimořádně v pondělí v zasedací místnosti rektorátu SU v 14.45)

Prof. V.V. Lychagin (Universitetet i Tromsø, Norsko)
Invariants or observables in relativity theory I

Abstrakt:

In these lectures, based on joint research with Alexei Kotov and Valerii Yumaguzhin, I will discuss differenatial invariants of the diffeomorphism groups in the following three cases:
1. Riemann manifolds
2. Einstein manifolds, i.e. solutions of the vacuum Einstein equations.
3. Einstein-Maxwell manifolds, i.e. solutions of the Einstein-Maxwell equations.
In all three cases it will be shown how to compute numbers of independent differential invariants and find Hilbert and Poincare functions of the corresponding differential invariant algebras.
It is also will be discussed methods of finding basic differential invariants and the corresponding factor-equations.

30.10.2014

RNDr. Petr Vojčák, Ph.D.,
Coverings and nonlocal symmetries of some 3-dimensional PDEs

Abstrakt:

In this talk we will consider four 3-dimensional Lax-integrable partial
differential equations, namely
(1) the 3D rdDym equation uty=uxyux-uxxuy,
(2) the universal hierarchy equation uyy=uxyuz-uxzuy,
(3) the modified Veronese web equation uty=uxyut-utxuy,
(4) Pavlov's equation uyy=utx+uxxuy-uxyux.
For the equations in question, we will construct some coverings and also
This is a joint work with H. Baran, I.S. Krasil'shchik and O.I. Morozov.

23.10.2014

RNDr. Jiřina Jahnová, Ph.D.,
On symmetries and conservation laws of the Majda–Biello system

Abstrakt:

In 2003, A.J. Majda and J.A. Biello derived and studied the so-called
reduced equations for equatorial baroclinic–barotropic waves, to which we
refer as to the Majda–Biello system. The equations in question describe
the nonlinear interaction of long-wavelength equatorial Rossby waves and
barotropic Rossby waves with a significant midlatitude projection in the
presence of suitable horizontally and vertically sheared zonal mean flows.

In this talk we present a Hamiltonian structure for the Majda–Biello
system and describe all generalized symmetries and conservation laws for
the latter. It turns out that there are only three symmetries
corresponding to x-translations, t-translations and to a scaling of t, x,
u and v, and four conservation laws, one of which is associated with the
conservation of energy, the second conserved quantity is just the
Hamiltonian functional and the other two are Casimir functionals of the
Hamiltonian operator admitted by our system. Our result provides inter
alia a rigorous proof of the fact that the Majda–Biello system has just
the conservation laws mentioned in the paper by Majda and Biello.

16.10.2014

Roman Popovych, D.Sc. (Wolfgang Pauli Institute, Rakousko),
Linear potential framework of linear evolution equations

Abstrakt:

We study the linear potential framework for (1+1)-dimensional linear evolution equations of order not less than two. Related objects include linear potential systems and associated high-level potential equations as well as their conservations laws, cosymmetries, Lie symmetries and generalized symmetries. The main tool of the study is the use of multiple dual Darboux transformations, which map modified potential equations to the corresponding initial equations and establish the inverse maps between the associated adjoint equations. In particular, every potential conservation law of any (1+1)-dimensional linear evolution equation of even order proves to be linear and induced by a local conservation law of the same equation.

09.10.2014

Roman Popovych, D.Sc. (Wolfgang Pauli Institute, Rakousko),
Algebraic methods of group analysis of differential equations

Abstrakt:

Direct group classification is a tool for selection of modeling differential equations from classes of such equations parameterized by arbitrary constants of functions. The following criterion is used for the selection: Modeling differential equations have to admit the most extensive symmetry groups from the possible ones. The study of group classification problems is interesting not only from the purely mathematical point of view, but is also important for physical applications. The complexity of group classification led to the development of a great variety of specialized techniques, which are conventionally partitioned into two approaches. The first approach is based on the direct compatibility analysis and integration of the corresponding determining equations up to a relevant equivalence relation. We discuss the other approach, which is of algebraic nature. Any version of the algebraic method of group classification involves, in some way, the classification of algebras of vector fields up to certain equivalence induced by point transformations. The key question is what set of vector fields should be classified and what kind of equivalence should be used. Depending on this and completeness of solution, one can talk about partial preliminary group classification, complete preliminary group classification and complete group classification.

Within the framework of group classification, an important role is played by the notion of normalized classes of differential equations. Thus, for a weakly normalized class, complete preliminary group classification and complete group classification coincide. If the class is semi-normalized, the group classification up to equivalence generated by the associated equivalence group coincides with the group classification up to general point equivalence. As normalized classes are both semi-normalized and weakly normalized, it is especially convenient to carry out group classification in such classes by the algebraic method. This is why the normalization property can be used as a criterion for selecting classes of differential equations to be classified or for splitting of such classes into subclasses which are appropriate for group classification.

To illustrate the approach developed, we present, in particular, the complete solution of the group classification problem for the class of nonlinear wave equations arising in the theory of elasticity. The symmetry analysis of such equations had been initiated in twenty years ago, but only partial preliminary group classification of this class was carried out. We also briefly overview group classifications of various classes of linear and nonlinear Schrödinger equations.

### Program v akademickém roce 2013/2014

22.05.2014

dr hab. Katarzyna Grabowska (Uniwersytet Warszawski, Polsko),
Variational calculus with constraints on general algebroids II

Abstrakt:

After a necessary introduction in the first part of the seminar we can use the Tulczyjew triple for general algebroids to describe mechanical systems with constraints. In the second part of the seminar constraints of different kinds (holonomic, nonholonomic and vaconomic) will be discussed.
The algebroid formulation makes the distinction between different types of constraints more clear than it is for systems defined on a tangent bundle. We shall illustrate our theory with a number of important examples.

The second part of the seminar will be based on the paper: K. Grabowska, J. Grabowski: Variational calculus with constraints on general algebroids, J. Phys. A 41 (2008), 175204, 25pp.

24.04.2014

RNDr. Jiřina Jahnová, Ph.D.,
The Painlevé property of differential equations III

Abstrakt:

In the third part of the talk we continue the description of the so-called Painlevé test for ODEs. We also briefly discuss the relationship between the Painlevé property and different approaches to integrability of dynamical systems. Finally, we present fragments of Painlevé analysis for partial differential equations.

17.04.2014

Prof. V.V. Lychagin (Universitetet i Tromsø, Norsko)
Applications: higher symmetries and differential constraints

Abstrakt:

Different applications, especially higher symmetries and differential constrains, will be discussed.

10.04. 2014

Prof. D.V. Alekseevsky (MU Brno),
Cohomogeneity one Kaehler and Kaehler-Einstein Manifolds

Abstrakt:

A Riemannian manifold M is called to be a cohomogeneity one if there is an isometry group G with a codimension one orbit. We give a description of cohomogeneity one Kaehler manifolds of a compact semisimple Lie group G in terms of painted Dynkin diagrams and parametrized intervals in a T-Weyl chamber. We describe when such manifolds admit an invariant Kaehler-Einstein metric. In this case the Einstein equation is reduced to a second order ODE for function f(t) which describe the parametrization of the interval.

08.04.2014 (mimořádně v úterý v místnosti R2 v 13.05)

Prof. V.V. Lychagin (Universitetet i Tromsø, Norsko)
Obstructions for integrability: Spencer cohomology and Weyl tensors

Abstrakt:

The classical Spencer-Goldshmidt approach to Integrability (Cartan-Kahler theorem) together with an alternative approach based on Weyl tensors will be discussed.

03.04.2014

RNDr. Jiřina Jahnová, Ph.D.,
The Painlevé property of differential equations II

Abstrakt:

In the second part of the talk we describe the collection of necessary conditions of the Painlevé property for ODEs - the so-called Painlevé test. We also briefly discuss the methods that can be used to prove their sufficiency, and the relationship between the Painlevé property and different approaches to integrability. Finally we present fragments of Painlevé analysis for partial differential equations.

27.03.2014

RNDr. Jiřina Jahnová, Ph.D.,
The Painlevé property of differential equations I

Abstrakt:

In the first part of the talk we define the Painlevé property of ordinary differential equations and consider linear ODEs as an example of equations that have the Painlevé property. Finally we briefly present the main results that were obtained by Painlevé, Fuchs, Gambier and others within the so-called Painlevé program which was established at the beginning of the 20th century by Paul Painlevé.

20.03.2013

Doc. RNDr. Artur Sergyeyev, Ph.D.,
Formal symmetries and integrability

Abstrakt:

We give a brief survey of the formal symmetry approach to classification of integrable (1+1)-dimensional evolution equations.

13. 03. 2014

On invariant solutions of the constant astigmatism equation

Abstrakt:

In the talk some new results concerning the constant astigmatism equation (CAE) will be given.

1. We find solutions of the CAE that correspond to surfaces obtained by Lipschitz in 1887. They are invariant with respect to a suitable combination of the known Lie symmetries of the CAE.

2. We obtain a pair of reciprocal transformations for the CAE. Each of them coincides with a certain nonlocal symmetry of the CAE. A corresponding invariant solution is also computed.

06. 03. 2014

Doc. RNDr. Artur Sergyeyev, Ph.D.,
Nonlocal symmetries for the four-dimensional Martinez Alonso--Shabat equation

Abstrakt:

We construct an infinite hierarchy of commuting nonlocal symmetries (and ot just the shadows, as it is usually the case in the literature) for the Martinez Alonso--Shabat equation.

This is joint work with Oleg Morozov, see arXiv:1401.7942 for details.

27. 02. 2014

Doc. RNDr. Artur Sergyeyev, Ph.D.,
A new class of integrable (3+1)-dimensional dispersionless systems

Abstrakt:

We introduce a new class of (3+1)-dimensional dispersionless integrable systems having Lax pairs written in terms of contact vector fields. Our results show inter alia that (3+1)-dimensional integrable systems are considerably less exceptional than it was believed.

In particular, we present a new (3+1)-dimensional dispersionless integrable system with an arbitrarily large finite number of components. In the simplest special case this system yields a (3+1)-dimensional integrable generalization of the dispersionless Kadomtsev--Petviashvili equation.

For further details please see arXiv:1401.2122.

12. 12. 2013

Prof. Iosif Krasil'shchik, DrSc.,
Generalized jet spaces

Abstrakt:

I shall define generalized jet spaces and shall try to give a geometric explanation of Lax pairs that incorporate differentiations with respect to "spectral parameter". If time allows, I shall discuss other applications.

05. 12. 2013

Giovanni Moreno, Ph.D.,
Meta-symplectic geometry of 3rd order Monge-Ampère equations

Abstrakt:

In this talk I will present the geometry of a special type of scalar PDEs in one unknown function and two independent variables: those whose characteristics correspond to a 3D sub-distribution of the Cartan distribution on the Lagrangian Grassmannian bundle $M^{(1)}$ of a 5D contact manifold $M$. The main result is that any PDE of this type can be written as a linear combination of the minors of the Hankel matrix on $M^{(2)}$, with coefficients on $M^{(1)}$, i.e., they generalize the classical notion of Monge-Ampère equations, accordingly to some authors (Boillat, Ferapontov).

This is joint work with G. Manno.

28. 11. 2013

Dr. Diego Catalano Ferraioli (Federal  University of Bahia, Brazil),
Fourth order evolution equations of pseudo-spherical type

Abstrakt:

It is given a classification of pseudo-spherical evolution equations of the form u_t = u_xxxx + G(u, u_x , u_xx , u_xxx), under some suitable conditions on the associated 1-forms ω_i = f_i1 dx + f_i2 dt, 1 ≤ i ≤ 3. These equations can be equivalently described as compatibility condition (or zero-curvature representation) of an associated overdetermined linear system. The classiﬁcation provides 4  huge classes of such equations which are explicitly described.

7. 11. 2013

Dr. Jonathan Kress (University of New South  Wales, Australia),
Invariant classification of conformally superintegrable systems

Abstrakt:

A Hamiltonian system with a 2n-dimensional phase space is said to be Liouville integrable if it possesses n integrals of the motion that are mutually in involution.  The system is said to be superintegrable if it possesses more than n integrals up to a maximum of 2n-1.  Many well known integrable systems, such as the inverse square central force and harmonic oscillator systems, are in fact superintegrable.

Superintegrable systems with second order constants of the motion have been extensively studied because of their close connection with special functions and separation of variables.  This talk will present a classification of second order non-degenerate superintegrable systems on three-dimensional conformally flat spaces.  Each such system is associated with a configuration of 6 points in the extended-complex plane and invariants of these configurations under the action of the conformal group can be used to classify the systems.  This is a joint work with Joshua Capel.

5.11. 2013 (výjimečně v útery v 16.25 v R1)

Sergei Igonin, Ph.D. (Yaroslavl, Russia),
Integrability of nonlinear (1+1)-dimensional PDEs, Backlund transformations, and infinite-dimensional Lie algebras V.

31. 10. 2013

Sergei Igonin, Ph.D. (Yaroslavl, Russia),
Integrability of nonlinear (1+1)-dimensional PDEs, Backlund transformations, and infinite-dimensional Lie algebras IV.

29. 10. 2013 (výjimečně v útery v 16.25 v R1)

Giovanni Manno, Ph.D. (University of Padova, Italy),
Contact Geometry of Monge-Ampère equations.

Abstrakt:

Monge-Ampère equations (MAEs) with an arbitrary number of independent variables can be interpreted as particular hypersurfaces of a Lagrangian Grassmann bundle. After investigating the relation between MAEs and their characteristics, we discuss how to obtain normal forms for parabolic MAEs with two independent variables.

24. 10. 2013

Sergei Igonin, Ph.D. (Yaroslavl, Russia),
Integrability of nonlinear (1+1)-dimensional PDEs, Backlund transformations, and infinite-dimensional Lie algebras II.

Abstrakt:

I plan to give a series of lectures on some geometric and algebraic methods in the theory of integrable nonlinear PDEs. It is well known that Backlund transformations (BTs) and zero-curvature representations (ZCRs) help to construct interesting explicit solutions for a wide class of nonlinear PDEs. I will begin my lectures with some classical examples, showing how soliton solutions of the Korteweg--de Vries (KdV) equation can be obtained by means of BTs and ZCRs of KdV.

After that, I will describe a general geometric theory for BTs and ZCRs of PDEs. This theory is based on the use of infinite-dimensional Lie algebras, infinite jet bundles, and Krasilshchik--Vinogradov coverings of PDEs.

For any PDE satisfying some non-degeneracy conditions, I will define a family of Lie algebras, which are called the fundamental Lie algebras of this PDE. Fundamental Lie algebras are defined in a coordinate-independent way and are new geometric invariants for PDEs.

Recall that, for every topological space $X$ and every point $a\in X$, one has the fundamental group $\pi_1(X,a)$. The above-mentioned Lie algebras are called fundamental, because their role for PDEs is somewhat similar to the role of fundamental groups for topological spaces.

Fundamental Lie algebras are closely related to ZCRs and BTs. In these lectures, we will concentrate on the case of (1+1)-dimensional PDEs. For such PDEs, it will be shown how fundamental Lie algebras classify all ZCRs up to local gauge equivalence. Also, I will show how to describe fundamental Lie algebras in terms of generators and relations. Using these algebras, one obtains necessary conditions for integrability and necessary conditions for existence of BTs for the considered class of PDEs.

In this construction, jets of arbitrary order are allowed. In the case of low-order jets, fundamental Lie algebras generalize Wahlquist--Estabrook prolongation algebras.

In the structure of fundamental Lie algebras for KdV, Krichever--Novikov, nonlinear Schrodinger, (multicomponent) Landau--Lifshitz type equations, we encounter infinite-dimensional subalgebras of Kac--Moody algebras and infinite-dimensional Lie algebras of certain matrix-valued functions on some algebraic curves. Applications to classification of some PDEs with respect to BTs and methods to construct BTs will be discussed as well.

10. 10. 2013

Maxim Pavlov, Ph.D. (Lebeděvův fyzikální ústav, Moskva, Rusko),
Benney hydrodynamic chain and DKP hierarchy. Their N-component hydrodynamic reductions.
Löwner equations and the Gibbons--Tsarev system. Particular solutions

Abstrakt:

We consider the Gibbons--Tsarev system and discuss possibilities
to construct their particular solutions parameterized by arbitrary constants.

03. 10. 2013

Doc. RNDr. Artur Sergyeyev, Ph.D.,
Coupling constant metamorphosis and integrability

Semináře se obvykle konají ve čtvrtek od 14.45 hod. v budově Matematického ústavu, Na Rybníčku 1 v Opavě, v místnosti R1. Všichni zájemci jsou srdečně zváni.

Schroedinger operators exhibiting spectral transition