Academic Year 2012

Thursday Nov 22, 12:00

Peakons are non-smooth solutions to a class of nonlinear partial differential equations. They were first discussed by R. Camassa and D. Holm (then both at the Los Alamos National Lab) in the early nineties of the last century. Peakons represent non-smooth nonlinear waves, yet one can think of them as interacting particles. The mathematical theory of peakons, as it turned out, goes back to T. Stietljes. In particular the first complete construction of peakons was done by adapting Stieltjes' method of continued fractions. In this sense the theory of peakons is intimately connected with the theory of orthogonal polynomials and their generalizations. One such generalization suggested by the theory of peakons is a family of Cauchy biorthogonal polynomials which were first used in the study of peakon solutions to the Degasperis-Procesi (DP) equation. In this talk I will retrace the main steps in the story of peakons with an emphasis on the connection with orthogonal and biorthogonal polynomials.

Tuesday, Nov 27th, 12:00 hrs

We show how from the QRT map one can derive quadrirational Yang-Baxter maps. The F-List of quadrirational Yang-Baxter maps will be reconstructed. The notion of octorationality will be introduced and from a specific generalisation of the QRT mapping octorational maps will be obtained.

Friday, Nov 30th, 12:00 hrs

Orthogonal polynomials, Padé Approximants, Generating functions are classical tools in Theoretical Physics and Applied Mathematics. What happens to them when randomness comes into play? Elegant phenomena occur, which may be viewed as the footprints of the noise and in turn allow for new tools for data analysis. In the talk I shall describe various examples taken from the literature or from works I was involved in. The examples shall be accompanied with proofs or pictures.

Thursday, Dec 6th, 12:00 hrs

In this talk I shall give a short review of the old and new results in wave turbulence theory (WTT). The development of WWT can be roughly divided into three periods:
1960-1990: Main result - with effects of finite size not taken into account and under a list of assumptions, e.g. energy cascade is formed by local 3- or 4-wave resonances of linear frequencies, energy distribution over k-scale (i.e. energy spectrum) is described by some power law. Computation method – wave kinetic equation.
1990-2010: experimental and theoretical studies demonstrate the problems: non-local interactions, nonlinear frequencies in an energy cascade, exponential form of energy spectrum, etc. New types of WT are introduced (mesoscopic, discrete, sand-pile, finite-dimensional, laminated, kinetic, …) as attempts to improve the original WTT.
2010-present: an energy cascade is formed not by 3- or 4- wave resonances but by the other mechanism - modulation instability (also called Benjamin-Feir instability). Computation method – increment chain equation method (ICEM). Examples of spectra computation will be given. The ICEM can be applied to any weakly nonlinear dispersive PDEs which can be reduced to the focusing NLS or mNLS.

Friday, Dec 7th, 12:00 hrs

We formulate a discretization of the one-dimensional delta Bose gas (or Lieb-Liniger model). The quantum integrability is shown with the aid of a representation of the affine Hecke algebra and the spectral problem for the discrete model is solved by means of the Bethe Ansatz.

Friday, Dec 7th, 16:00 hrs

Two new integrable twisted hierarchies with D2 symmetries are constructed via the loop algebra factorization method. The splitting type factorization yields the generalized sinh-Gordon equation, this result justifies some far-reaching generalizations of the well-known connection between the sine-Gordon equation, the Backlund transformation, and surfaces with curvature. The non-splitting type factorization yields the Gerdjikov-Mikhailov-Valchev equation which is an anisotropic multicomponent generalization of the classical Heisenberg ferromagnetic equation and is one of the simplest twisted integrable systems. Special analytical features in the associated inverse scattering theory are discussed to solve the Cauchy problem of these twisted flows.

Monday, Dec 10th, 12:00 hrs

I'll discuss 2 classes of recurrences (and corresponding maps) which arise inthe context of cluster mutation. These two families can be represented in determinantal form and using Dodgson condensation we derive some linear equations with periodic coefficients. The invariants of the monodromy matrixare important functions of these coefficients.

The only ingredient from cluster mutations, needed in this talk, is the skewsymmetric matrix which defines the quiver. From this we define a (pre-)symplectic structure, which is invariant under the action of the map. This defines an invariant, non-degenerate Poisson bracket, either on the wholespace or on the reduced symplectic leaves. When reduced to the algebra offunctions defined by the above periodic coefficients, the Poisson bracket hasone Casimir function, which is the invariant of the monodromy matrix.In this framework we discuss the complete integrability of the maps.

Monday, Dec 10th, 16:00 hrs

The quasi-linear parabolic equationut = (un)xx + f(x)usuxwith n ̸= 0 is a mathematical model for many physical problems that corresponds to nonlinear diffusion with convection. The second term on the right-hand side of this equation is of convective nature. In the theory of unsatured porous medium, the convective part may represent the effect of gravity. The importance of the effect of space-dependent parts on the overall dynamics of this equation is well known. For instance, in porous medium this may account for intrinsic factors like medium contamination with another material or in plasma. This may express the impact that solid impurities arising from the walls have on the enhancement of the radiation channel. In [10] then on classical generators for the porous medium equation with convection were derived. The idea of a conservation law, or more particularly, of a conserved quantity, has its origin in mechanics and physics. Since a large number of physical theories, are usually expressed as systems of nonlinear differential equations, it follows that conservation laws are useful in both general theory and the analysis of concrete systems. Thus the search for conservation laws is important for their physical interpretation. The concepts of self-ad joint and quasi self-adjoint equations were introduced by NHI bragimov in [7]. In [6] a general theorem on conservation laws for arbitrary differential equation which do not require the existence of Lagrangians has been proved. This new theorem is based on the concept of ad joint equations for nonlinear equations. In [7], [8],[3] and [9] the concept of self-ad joint equation has been extended. In some previous papers by using these concepts, the general theorem on conservation laws proved in [6] and the classical symmetries of the porous medium equation we have derived for some porous medium equation conservation laws. In this work by using the property of nonlinear self-ad jointness we construct some conservation laws of the porous medium equation with convection associated to the non classical generators.

Tuesday, Dec 11th, 12:00 hrs (to be confirmed)

We present six open problems on the master Painleve' equation. They deal with:
the matrix Lax pair of Jimbo and Miwa which does not exist when all monodromy exponents are equal to one,
a factorization problem in the three-wave resonant interaction,
an unknown stochastic process integrable with P6,
an unknown Maxwell-Bloch type system,
the Gauss-Codazzi equations of surfaces,
a two-degree of freedom superintegrable Hamiltonian in polar coordinates.
Only one of them is nearly solved.

Tuesday, Dec 11th, 16:00 hrs

Full manifold of the Complex Bloch-Floquet Eigenfunctions is investigated for the zero level of the 2D non-relativistic Pauli Operator describing the motion of charged particle in the periodic magnetic field with zero flux through the elementary cell, and zero electric field. It is completely calculated for the broad class of Algebro-Geometric Operators found in this work. Let us remind that for the case of nonzero flux the Ground State Problem was solved by Aharonov-Casher (1979) for the rapidly decreasing fields, and by Dubrovin-Novikov (1980) for the periodic fields. No Algebro-Geometric Operators where known in the case of nonzero flux. The complex extension of the manifold of ``Magnetic'' Bloch-Floquet eigenfunctions has very bad properties at infinity. We found many good nonsingular ''Algebro-Geometric'' periodic fields (with zero flux through the elementary cell of the lattice) associated with genus zero Complex Riemann Surface. For higher genuses we found periodic operators with very interesting magnetic fields and Bohm-Aharonov Phenomenon. The algebro-geometric case of genus zero leads also to the ''Soliton-Like'' nonsingular magnetic fields with magnetic flux through the disc of radius R asymptotically proportional to the radius R (i.e. total magnetic flux is slowly divergent at R goes to infinity). Especially interesting variety of ground states in the Hilbert Space $\cL_2(\bR^2)$ is found for this case.

Thursday, Dec 13th, 12:00 hrs

A brief review on the connection of integrable lattice equations with Yang-Baxter maps will be presented, followed by a procedure that one can associate integrable lattice models to these maps. Also we present a method to derive discrete Painleve equations and correspondences from Yang-Baxter maps.