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2013
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# Pantelis Damianou

P.O. Box 20537, 1678 Nicosia, Cyprus
University of Cyprus

## Publications:

 Damianou P. A. Poisson Brackets after Jacobi and Plücker 2018, vol. 23, no. 6, pp.  720-734 Abstract We construct a symplectic realization and a bi-Hamiltonian formulation of a 3-dimensional system whose solution are the Jacobi elliptic functions. We generalize this system and the related Poisson brackets to higher dimensions. These more general systems are parametrized by lines in projective space. For these rank 2 Poisson brackets the Jacobi identity is satisfied only when the Plücker relations hold. Two of these Poisson brackets are compatible if and only if the corresponding lines in projective space intersect. We present several examples of such systems. Keywords: Poisson structures, Plücker relations Citation: Damianou P. A.,  Poisson Brackets after Jacobi and Plücker, Regular and Chaotic Dynamics, 2018, vol. 23, no. 6, pp. 720-734 DOI:10.1134/S1560354718060072
 Damianou P. A., Sabourin H., Vanhaecke P. Intermediate Toda Systems 2015, vol. 20, no. 3, pp.  277-292 Abstract We construct a large family of Hamiltonian systems which interpolate between the classical Kostant–Toda lattice and the full Kostant–Toda lattice and we discuss their integrability. There is one such system for every nilpotent ideal $\mathcal{I}$ in a Borel subalgebra $\mathfrak{b}_+$ of an arbitrary simple Lie algebra $\mathfrak{g}$. The classical Kostant–Toda lattice corresponds to the case of $\mathcal{I}=[\mathfrak{n}_+, \mathfrak{n}_+]$, where $\mathfrak{n}_+$ is the unipotent ideal of $\mathfrak{b}_+$, while the full Kostant–Toda lattice corresponds to $\mathcal{I}=\{0\}$. We mainly focus on the case $\mathcal{I}=[[\mathfrak{n}_+, \mathfrak{n}_+], \mathfrak{n}_+]$. In this case, using the theory of root systems of simple Lie algebras, we compute the rank of the underlying Poisson manifolds and construct a set of (rational) functions in involution, large enough to ensure Liouville integrability. These functions are restrictions of well-chosen integrals of the full Kostant–Toda lattice, except for the case of the Lie algebras of type $C$ and $D$ where a different function (Casimir) is needed. The latter fact, and other ones listed in the paper, point at the Liouville integrability of all the systems we construct, but also at the nontriviality of obtaining the result in full generality. Keywords: Toda lattices, integrable systems Citation: Damianou P. A., Sabourin H., Vanhaecke P.,  Intermediate Toda Systems, Regular and Chaotic Dynamics, 2015, vol. 20, no. 3, pp. 277-292 DOI:10.1134/S1560354715030053
 Damianou P. A., Vanhaecke P. Foreword 2011, vol. 16, no. 3-4, pp.  185-186 Abstract Citation: Damianou P. A., Vanhaecke P.,  Foreword, Regular and Chaotic Dynamics, 2011, vol. 16, no. 3-4, pp. 185-186 DOI:10.1134/S1560354711030014
 Constandinides K., Damianou P. A. Lotka–Volterra Equations in Three Dimensions Satisfying the Kowalevski–Painlevé Property 2011, vol. 16, no. 3-4, pp.  311-329 Abstract We examine a class of Lotka–Volterra equations in three dimensions which satisfy the Kowalevski–Painlevé property. We restrict our attention to Lotka–Volterra systems defined by a skew symmetric matrix. We obtain a complete classification of such systems. The classification is obtained using Painlevé analysis and more specifically by the use of Kowalevski exponents. The imposition of certain integrality conditions on the Kowalevski exponents gives necessary conditions. We also show that the conditions are sufficient. Keywords: Lotka–Volterra equations, Kowalevski exponents, Painlevé analysis Citation: Constandinides K., Damianou P. A.,  Lotka–Volterra Equations in Three Dimensions Satisfying the Kowalevski–Painlevé Property, Regular and Chaotic Dynamics, 2011, vol. 16, no. 3-4, pp. 311-329 DOI:10.1134/S1560354711030075
 Damianou P. A. Reduction and realization in Toda and Volterra 2008, vol. 13, no. 6, pp.  572-587 Abstract We construct a new symplectic, bi-Hamiltonian realization of the KM-system by reducing the corresponding one for the Toda lattice. The bi-Hamiltonian pair is constructed using a reduction theorem of Fernandes and Vanhaecke. In this paper we also review the important work of Moser on the Toda and KM-systems. Keywords: Toda lattice, bi-Hamiltonian systems Citation: Damianou P. A.,  Reduction and realization in Toda and Volterra, Regular and Chaotic Dynamics, 2008, vol. 13, no. 6, pp. 572-587 DOI:10.1134/S1560354708060075
 Damianou P. A., Papageorgiou V. G. On an Integrable Case of Kozlov–Treshchev Birkhoff Integrable Potentials 2007, vol. 12, no. 2, pp.  160-171 Abstract We establish, using a new approach, the integrability of a particular case in the Kozlov–Treshchev classification of Birkhoff integrable Hamiltonian systems. The technique used is a modification of the so called quadratic Lax pair for $D_n$ Toda lattice combined with a method used by M. Ranada in proving the integrability of the Sklyanin case. Keywords: Toda lattices, Birkhoff integrable systems, integrability, Hamiltonian systems Citation: Damianou P. A., Papageorgiou V. G.,  On an Integrable Case of Kozlov–Treshchev Birkhoff Integrable Potentials, Regular and Chaotic Dynamics, 2007, vol. 12, no. 2, pp. 160-171 DOI:10.1134/S1560354707010029
 Damianou P. A. Multiple Hamiltonian Structures for Toda Systems of Type $A-B-C$ 2000, vol. 5, no. 1, pp.  17-32 Abstract Results on the finite nonperiodic $A_n$ Toda lattice are extended to the Bogoyavlensky–Toda systems of type $B_n$ and $C_n$. The investigated areas include master symmetries, recursion operators, higher Poisson brackets and invariants. A conjecture which relates the degrees of higher Poisson brackets and the exponents of the corresponding Lie group is verified for these systems. Citation: Damianou P. A.,  Multiple Hamiltonian Structures for Toda Systems of Type $A-B-C$, Regular and Chaotic Dynamics, 2000, vol. 5, no. 1, pp. 17-32 DOI:10.1070/RD2000v005n01ABEH000121