Related papers: A new integrable system on the sphere
We announce a new bi-Hamiltonian integrable two-component system admitting the scalar 3rd-order Burgers equation as a reduction.
Abundant second-order maximally conformally superintegrable Hamiltonian systems are re-examined, revealing their underlying natural Weyl structure and offering a clearer geometric context for the study of St\"ackel transformations (also…
A criterion in terms of differential invariants for a metric on a surface to be Liouville is established. Moreover, in this paper we completely solve in invariant terms the local mobility problem of a 2D metric, considered by Darboux: How…
A generalized version of Bertrand's theorem on spherically symmetric curved spaces is presented. This result is based on the classification of (3+1)-dimensional (Lorentzian) Bertrand spacetimes, that gives rise to two families of…
The notion of integrability is discussed for classical nonautonomous systems with one degree of freedom. The analysis is focused on models which are linearly spanned by finite Lie algebras. By constructing the autonomous extension of the…
We discuss the dynamical quantum systems which turn out to be bi-unitary with respect to the same alternative Hermitian structures in a infinite-dimensional complex Hilbert space. We give a necessary and sufficient condition so that the…
In two-dimensional Euclidean plane, existence of second-order integrals of motion is investigated for integrable Hamiltonian systems involving spin (\emph{e.g.,} those systems describing interaction between two particles with spin 0 and…
This thesis presents our results on Liouville integrable systems of Calogero-Ruijsenaars type: 1. We prove an explicit formula providing canonical spectral coordinates for the rational Calogero-Moser system. 2. We explore action-angle…
We introduce the concept of a standard form for two embedded maximal sphere systems in the doubled handlebody, and we prove an existence and uniqueness result. In particular, we show that pairs of maximal sphere systems in the doubled…
An algebraic definition of Gardner's deformations for completely integrable bi-Hamiltonian evolutionary systems is formulated. The proposed approach extends the class of deformable equations and yields new integrable evolutionary and…
A moving parallel frame method is applied to geometric non-stretching curve flows in the Hermitian symmetric space Sp(n)/U(n) to derive new integrable systems with unitary invariance. These systems consist of a bi-Hamiltonian modified…
A number of examples of Hamiltonian systems that are integrable by classical means are cast within the framework of isospectral flows in loop algebras. These include: the Neumann oscillator, the cubically nonlinear Schr\"odinger systems and…
The coalgebra approach to the construction of classical integrable systems from Poisson coalgebras is reviewed, and the essential role played by symplectic realizations in this framework is emphasized. Many examples of Hamiltonians with…
Some particular examples of classical and quantum systems on the lattice are solved with the help of orthogonal polynomials and its connection to continuous models are explored.
The particular case of the integrable two component (2+1)-dimensional hydrodynamical type systems, which generalises the so-called Hamiltonian subcase, is considered. The associated system in involution is integrated in a parametric form. A…
We investigate a family of integrable Hamiltonian systems on Lie-Poisson spaces $\mathcal{L}_+(5)$ dual to Lie algebras $\mathfrak{so}_{\lambda, \alpha}(5)$ being two-parameter deformations of $\mathfrak{so}(5)$. We integrate corresponding…
The method introduced in (Yehia H M 2006 J. Phys. A: Math. Gen. 39 5807-5824) and (Yehia H M 2012 J. Phys. A: Math. Gen. 45 395209) is extended to construct new families of several-parameter integrable systems, which admit a complementary…
Associated to a convex integral polygon $N$ in the plane are two integrable systems: the cluster integrable system of Goncharov and Kenyon, constructed from the dimer model on bipartite torus graphs, and the Beauville integrable system…
Hamiltonian dynamics describing conservative systems naturally preserves the standard notion of phase-space volume, a result known as the Liouville's theorem which is central to the formulation of classical statistical mechanics. In this…
We suggest that trialgebraic symmetries migth be a sensible starting point for a notion of integrability for two dimensional spin systems. For a simple trialgebraic symmetry we give an explicit condition in terms of matrices which a…