I will show how combining some 19th century projective geometry with integrable harmonic maps equations leads to a construction of a new instanton, ultimately answering (negatively!) a question of S. T. Yau.
Classical integrable systems exhibit a tower of conserved quatities having local densities built out of traces of powers of the model's Lax matrix. It is argued that this local structure, absent in general models, leads to peculiar thermalisation properties of integrable systems. In particular, their equilibrum properies are expected to be grasped by so-called Generalised Gibbs measures. The...
I will present results of joint work with Maciej Błaszak (UAM, Poznań) and Krzysztof Marciniak (Linköping University). Since the classical works of Novikov et al., there has been a tremendous amount of research devoted to connections between soliton hierarchies and their integrable finite-dimensional reductions, which was mainly focused on stationary flows. Recently, we have revisited this...
A system of real ordinary differential equations $\ddot{y}_k=\sum_{l=1}^n M_{kl}\exp y_l,~k=1,...,n$, where $M$ is a real $n\times n$ constant symmetric matrix, is analysed in detail. The motivation includes several physical applications, which are discussed for the general and some special $n$'s.
A simple symmetry analysis and a discussion of the Lagrangian-Hamiltonian structure is...
The motion of neutral and charged test particles in the Reissner-Nordström spacetime is analysed. Inspired by the recent application of the Biermann-Weierstrass theorem to obtain a general non-radial solution to the geodesic motion in the Schwarzschild spacetime [1], we apply the same method to the Reissner-Nordström spacetime. This approach allows us to describe all types of orbits except for...
Ernst-type equations are elegant reformulations of Einstein's vacuum equations of general relativity when the existence of two commuting Killing vector fields is assumed. Axisymmetric, stationary spacetimes such as rotating black holes and planar gravitational waves are examples of solutions of the Ernst-type equations.
An important mathematical feature of the Ernst-type equations is that...
In 1990-ties I have in collaboration with M.Antonowicz, P.Kulish, S.Orlov developed technique of transferring integrable structures of soliton hierarchies to finite dimensional invariant manifolds defined as stationary/restricted flows. They are systems of ODE´s that, after suitable parametrisation, acquire form of Newton equations with velocity independent forces.
Structures of integrable...
This talk presents a study of nonlinear superpositions of Riemann wave solutions admitted by hyperbolic first-order systems. We focus on the Euler system and non-elastic wave superpositions that cannot be decomposed into pairwise independent interactions of waves. The property of quasi-rectifiability of the families of vector fields imposes certain conditions on the commutators of these vector...
We will consider elliptic Gaudin-type model in an external magnetic field associated with non-skew-symmetric elliptic r-matrix. We will discuss a modified algebraic Bethe ansatz for the considered elliptic r-matrix and obtain the spectra of the corresponding Gaudin-type Hamiltonians in terms of solutions of modified Bethe equations. The applications of the obtained result to the...
We prove that the elliptic photo-gravitational Hill problem is not integrable except in one case, when the gravitational force of the lighter primary acting on the infinitesimal mass is balanced by the radiation-pressure force of this primary. In this exceptional case, the infinitesimal mass moves under the influence of the more massive primary localised at infinity. We show that in this case...
We study the dynamics of a charged rigid body with stationary charge distribution in external constant electric and magnetic fields. The total charge of the body vanishes and the charge distribution is described by symmetric matrix of the 'electrostatic inertia' of the body. The equations of motion are derived and it is shown that they are Hamiltonian with respect to a certain degenerated...
We consider a 2 degree of freedom rational Hamiltonian H. Arnold Liouville integrability requires the existence of an additional first integral. We want to relax the condition of existence of an additional first integral by requiring only the preservation of a codimension 1 foliation transverse to the levels of H. The Malgrange pseudo group is defined in terms of differential invariants, so...
We study Lie-Rinehart algebra structures in a framework provided by duality pairings of modules over unital commutative associative algebra. Thus, we construct new examples of Lie brackets corresponding to a fixed anchor map whose image is a cyclic submodule of the derivation module, and therefore we call them cyclic Lie-Rinehart algebras. Special cases of our construction include Lie...
We perform a group-theoretic analysis of the rdDym equation, extending the results previously obtained by Kumar and co-authors [1,2]. The Lie algebra of point symmetries of the rdDym equation is well known and infinite-dimensional. We identify a 13-dimensional subalgebra characterized by the property of being maximal within the class of finite-dimensional subalgebras that induce arbitrary...
We present a framework in three-dimensional Minkowski space $\mathbb{R}^{1,2}$ which unifies the extended Dym, KdV, modified KdV and modified modified KdV equations via parallel, offset and midsurfaces. Each equation governs a class of surfaces, the members of which are foliated by geodesics of certain properties. These classes of surfaces are linked by reciprocal and Miura-type...
We shall explore a method, initiated by Guichard in 1890, which allows to generate sequences of Voss surfaces by quadratures, starting from an arbitrarily chosen pseudospherical surface and a seed solution of the Moutard equation, by means of two simple transformations. In this talk we
1) identify the Guichard transformations with the well-known mutually inverse recursion operators for...
The aim of this talk is to construct a class of explicit nontrivial rational solutions of the dispersionless Hirota system of PDEs. All the solutions in this class are of homogeneity degree 1 and are quotients of homogeneous polynomials. It is well-known that the solutions of the Hirota dispersionless systems describe Veronese webs. By nontriviality of the solutions it is meant that the...
The Kepler problem is a classical superintegrable dynamical system, possessing five functionally independent integrals of motion. We present a new explicit integrator which preserves exactly (up to round-off errors) all these integrals (and, as a consequence, all phase space trajectories). What is more, our numerical scheme also provides the exact explicit time discretization. We...
Classical tops, such as the Euler, Lagrange, and Kovalevskaya tops, are important examples of rigid body systems in classical mechanics which can serve as useful test cases for numerical simulations, since they preserve several quantities such as energy, the magnitude of angular momentum, and additional integrals of motion. Although the Euler top seems to be the simplest case, it already...
The preservation of energy is an important requirement in the long-time numerical simulation of Hamiltonian partial differential equations. This work develops an energy-preserving mixed finite element approximation of the Korteweg–de Vries (KdV) equation based on the discrete gradient methodology.
Starting from the Hamiltonian formulation of the equation, an energy - preserving time...
Given random walk on a graph, the corresponding discrete-time quantum walk can be constructed using the method proposed by Szegedy. On the other hand, given a partition of the set of states of a Markov chain, one can study the corresponding aggregated process. We extend the aggregation technique to the level of quantum Markov chains. We provide conditions under which application of these two...
In this talk we discuss classical and quantum walks on Cayley graphs of finite Coxeter groups and their reductions obtained by suitable aggregations of the state space. We begin with the definition and classification of finite Coxeter groups, together with the associated Cayley graphs determined by Coxeter generating sets. We then show how graph symmetries and suitable partition properties can...
Phase-space methods provide a powerful alternative formulation of quantum mechanics, offering geometric insight into quantum dynamics through quasiprobability distributions and star products. While these techniques are well established for continuous-variable systems, their application to finite-dimensional quantum systems, particularly qubits, remains an active area of research. In this talk,...
The Random Domino Automaton (RDA)—a slowly driven system in the form of a one-dimensional stochastic cellular automaton—was introduced as a stylized simple model of earthquake statistics to provide a basis for the interrelation of Gutenberg–Richter law and Omori law with the waiting time distribution for earthquakes. The Gutenberg–Richter distribution provides a universal relationship between...
We introduce the Lyapunov Integrability Test (LIT), a new numerical framework for exploring integrability in parameter-dependent nonlinear dynamical systems. Inspired by the philosophy underlying the Mandelbrot set, the proposed approach shifts the focus from analyzing individual trajectories to investigating the global organization of parameter space. Instead of identifying regions associated...
In this prentation we construct evolutionary soliton hierarchies from pencils of Novikov algebras of Stäckel type. We start by defining a special class of associative Novikov algebras, which we call Novikov algebras of Stäckel type, as they are associated with classical Stäckel metrics in Viète co-ordinates. We obtain sufficient conditions for pencils of these algebras so that the...
