Publications (46)
Lie point symmetries of differential--difference equations
D. Levi, P. Winternitz, R. Yamilov
We present an algorithm for determining the Lie point symmetries of differential equations on fixed non transforming lattices, i.e. equations involving both continuous and discrete…
Subgroup type coordinates and the separation of variables in Hamilton-Jacobi and SchrÅdinger equations
E. G. Kalnins, Z. Thomova, P. Winternitz
Separable coordinate systems are introduced in the complex and real four-dimensional flat spaces. We use maximal Abelian subgroups to generate coordinate systems with a maximal num…
Heisenberg Algebra, Umbral Calculus and Orthogonal Polynomials
G. Dattoli, D. Levi, P. Winternitz
Umbral calculus can be viewed as an abstract theory of the Heisenberg commutation relation . In ordinary quantum mechanics is the derivative and $\hat M…
Classical Trajectories for two Ring-Shaped Potentials
M. Kibler, G. -H. Lamot, P. Winternitz
This paper deals with the classical trajectories for two super-integrable systems: a system known in quantum chemistry as the Hartmann system and a system of potential use in quant…
Discrete systems related to some equations of the Painlevé-Gambier classification
S. Lafortune, B. Grammaticos, A. Ramani +1
We derive integrable discrete systems which are contiguity relations of two equations in the Painlevé-Gambier classification depending on some parameter. These studies extend earl…
Solutions to the Optical Cascading Equations
S. Lafortune, P. Winternitz, C. R. Menyuk
Group theoretical methods are used to study the equations describing Ï^{(2)}:Ï^{(2)} cascading. The equations are shown not to be integrable by inverse scattering techniques. On…
Quasiseparation of variables in the Schroedinger equation with a magnetic field
F. Charest, C. Hudon, P. Winternitz
We consider a two-dimensional integrable Hamiltonian system with a vector and scalar potential in quantum mechanics. Contrary to the case of a pure scalar potential, the existence…
General -order superintegrable systems separating in polar coordinates
A. M. Escobar-Ruiz, P. Winternitz, I. Yurdusen
The general description of superintegrable systems with one polynomial integral of order in the momenta is presented for a Hamiltonian system in two-dimensional Euclidean plane…
Superintegrable systems with spin and second-order (pseudo)tensor integrals of motion
I. Yurdusen, O. O. Tuncer, P. Winternitz
We investigate a quantum non-relativistic system describing the interaction of two particles with spin 1/2 and spin 0, respectively. Assuming that the Hamiltonian is rotationally i…
The Conformal Group SU(2,2) and Integrable Systems on a Lorentzian Hyperboloid
M. A. del Olmo, M. A. Rodriguez, P. Winternitz
Eleven different types of "maximally superintegrable" Hamiltonian systems on the real hyperboloid are obtained. All of them correspond to a free…
Generalized Kadomtsev-Petviashvili equation with an infinite dimensional symmetry algebra
F. Gungor, P. Winternitz
A generalized Kadomtsev-Petviashvili equation, describing water waves in oceans of varying depth, density and vorticity is discussed. A priori, it involves 9 arbitrary functions of…
Invariant difference schemes and their application to invariant ordinary differential equations
R. Rebelo, P. Winternitz
We present an exposition of a method of discretizing ordinary differential equations while preserving their Lie point symmetries. This method is very general and can be applied to…
algebra of KP, free fermions and 2-cocycle in the Lie algebra of pseudodifferential operators
A. Yu. Orlov, P. Winternitz
The symmetry algebra of integrable systems is defined. As an example the classical Sophus Lie point symmetries of all higher KP equat…
Symmetries of the Kadomstev-Petviashvili Hierarchy
A. Yu. Orlov, P. Winternitz
The relation between the $\widehat{\Sl}(\infty)$ algebra of flows commuting with the KP hierarchy and the Kac-Moody-Virasoro Lie point symmetries of individual equations is establi…
Weak Transversality and Partially Invariant Solutions
A. M. Grundland, P. Tempesta, P. Winternitz
New exact solutions are obtained for several nonlinear physical equations, namely the Navier-Stokes and Euler systems, an isentropic compressible fluid system and a vector nonlinea…
Symmetry reduction and superintegrable Hamiltonian systems
M. A. Rodriguez, P. Tempesta, P. Winternitz
We construct complete sets of invariant quantities that are integrals of motion for two Hamiltonian systems obtained through a reduction procedure, thus proving that these systems…
Symmetry Classification of Diatomic Molecular Chains
S. Lafortune, S. Tremblay, P. Winternitz
A symmetry classification of possible interactions in a diatomic molecular chain is provided. For nonlinear interactions the group of Lie point transformations, leaving the lattice…
Fourth-order superintegrable systems separating in Polar Coordinates. II. Standard Potentials
A. M. Escobar-Ruiz, J. C. López Vieyra, P. Winternitz +1
Superintegrable Hamiltonian systems in a two-dimensional Euclidean space are considered. We present all real standard potentials that allow separation of variables in polar coordin…
Symmetry preserving discretization of SL(2,R) invariant equations
A. Bourlioux, R. Rebelo, P. Winternitz
Nonlinear ODEs invariant under the group SL(2,R) are solved numerically. We show that solution methods incorporating the Lie point symmetries provide better results than standard m…
Continuous Symmetries of Difference Equations
D. Levi, P. Winternitz
Lie group theory was originally created more than 100 years ago as a tool for solving ordinary and partial differential equations. In this article we review the results of a much m…
Symmetries of Discrete Dynamical Systems Involving Two Species
D. Gomez-Ullate, S. Lafortune, P. Winternitz
The Lie point symmetries of a coupled system of two nonlinear differential-difference equations are investigated. It is shown that in special cases the symmetry group can be infini…
Group foliation and non-invariant solutions of the heavenly equation
L. Martina, M. B. Sheftel, P. Winternitz
The main physical result of this paper are exact analytical solutions of the heavenly equation, of importance in the general theory of relativity. These solutions are not invariant…
New infinite families of th-order superintegrable systems separating in Cartesian coordinates
A. M. Escobar-Ruiz, R. Linares, P. Winternitz
A study is presented of superintegrable quantum systems in two-dimensional Euclidean space allowing the separation of variables in Cartesian coordinates. In addition to the H…
Point Symmetries of Generalized Toda Field Theories
S. Lafortune, P. Winternitz, L. Martina
A class of two-dimensional field theories with exponential interactions is introduced. The interaction depends on two ``coupling'' matrices and is sufficiently general to include a…
Point Symmetries of Generalized Toda Field Theories II Applications of the Symmetries
L. Martina, S. Lafortune, P. Winternitz
The Lie symmetries of a large class of generalized Toda field theories are studied and used to perform symmetry reduction. Reductions lead to generalized Toda lattices on one hand,…
Superintegrable systems with spin induced by coalgebra symmetry
D. Riglioni, O. Gingras, P. Winternitz
A method for deriving superintegrable Hamiltonians with a spin orbital interaction is presented. The method is applied to obtain a new superintegrable system in Euclidean space $\m…
Superintegrable Systems in Darboux spaces
E. G. Kalnins, J. M. Kress, W. Miller +1
Almost all research on superintegrable potentials concerns spaces of constant curvature. In this paper we find by exhaustive calculation, all superintegrable potentials in the four…
First integrals of ordinary difference equations beyond Lagrangian methods
V. Dorodnitsyn, E. Kaptsov, R. Kozlov +1
A new method for finding first integrals of discrete equations is presented. It can be used for discrete equations which do not possess a variational (Lagrangian or Hamiltonian) fo…
First integrals of ordinary difference equations which do not possess a variational formulation
P. Winternitz, V. Dorodnitsyn, E. Kaptsov +1
The paper presents a new method for finding first integrals of ordinary difference equations which do not possess Lagrangians, nor Hamiltonians. As an example we solve a third orde…
Hyperspherical Harmonics, Separation of Variables and the Bethe Ansatz
J. Harnad, P. Winternitz
The relation between solutions to Helmholtz's equation on the sphere and the $[{\gr sl}(2)]^n$ Gaudin spin chain is clarified. The joint eigenfuctions of the Laplacian an…
Integrable and superintegrable systems with spin in three-dimensional Euclidean space
P. Winternitz, I. Yurdusen
A systematic search for superintegrable quantum Hamiltonians describing the interaction between two particles with spin 0 and 1/2, is performed. We restrict to integrals of motion…
Third-order superintegrable systems separable in parabolic coordinates
I. Popper, S. Post, P. Winternitz
In this paper, we investigate superintegrable systems which separate in parabolic coordinates and admit a third-order integral of motion. We give the corresponding determining equa…
Polynomial Poisson Algebras for Classical Superintegrable Systems with a Third Order Integral of Motion
I. Marquette, P. Winternitz
We present polynomial Poisson algebras for the 8 classical potentials in two-dimensional Euclidian space that separate in cartesian coordinates and allow a third order integral of…
Solutions of (2+1)-dimensional spin systems
Z. Thomova, P. Winternitz, W. J. Zakrzewski
We use the methods of group theory to reduce the equations of motion of two spin systems in (2+1) dimensions to sets of coupled ordinary differential equations. We present solution…
Superintegrable systems with spin in two- and three-dimensional Euclidean spaces
P. Winternitz, I. Yurdusen
The concept of superintegrability in quantum mechanics is extended to the case of a particle with spin s=1/2 interacting with one of spin s=0. Non-trivial superintegrable systems w…
Integrable and superintegrable systems with spin
P. Winternitz, I. Yurdusen
A system of two particles with spin s=0 and s=1/2 respectively, moving in a plane is considered. It is shown that such a system with a nontrivial spin-orbit interaction can allow a…
Loop Algebra Symmetries and Commuting Flows for the Kadomtsev-Petviashvili Hierarchy
A. Yu. Orlov, P. Winternitz
The relation between the $\widehat\gl(\infty)$ symmetry of the Kadomtsev-Petviashvili hierarchy and the Kac-Moody-Virasoro Lie point symmetries of the individual equations is estab…
Fourth order superintegrable systems separating in Polar Coordinates. I. Exotic Potentials
Adrian M. Escobar-Ruiz, J. C. López Vieyra, P. Winternitz
We present all real quantum mechanical potentials in a two-dimensional Euclidean space that have the following properties: 1. They allow separation of variables of the Schrödinger…
Spherical Functions for the Quantum Group su_q(2)
P. Winternitz, G. Rideau
The representation theory of the quantum group su is used to introduce -analogues of the Wigner rotation matrices, spherical functions, and Legendre polynomials. The meth…
An infinite family of superintegrable deformations of the Coulomb potential
S. Post, P. Winternitz
We introduce a new family of Hamiltonians with a deformed Kepler- Coulomb potential dependent on an indexing parameter k. We show that this family is superintegrable for all ration…
Superintegrability in a two-dimensional space of non-constant curvature
E. G. Kalnins, J. M. Kress, P. Winternitz
A Hamiltonian with two degrees of freedom is said to be superintegrable if it admits three functionally independent integrals of the motion. This property has been extensively stud…
General Nth order integrals of the motion
S. Post, P. Winternitz
The general form of an integral of motion that is a polynomial of order N in the momenta is presented for a Hamiltonian system in two-dimensional Euclidean space. The classical and…
A class of solvable Lie algebras and their Casimir Invariants
L. Snobl, P. Winternitz
A nilpotent Lie algebra n_{n,1} with an (n-1) dimensional Abelian ideal is studied. All indecomposable solvable Lie algebras with n_{n,1} as their nilradical are obtained. Their di…
Lie Point Symmetries and Commuting Flows for Equations on Lattices
D. Levi, P. Winternitz
Different symmetry formalisms for difference equations on lattices are reviewed and applied to perform symmetry reduction for both linear and nonlinear partial difference equations…
On the solutions of the model in dimensions
A. M. Grundland, P. Winternitz, W. J. Zakrzewski
We use the methods of group theory to reduce the equations of motion of the model in (2+1) dimensions to sets of two coupled ordinary differential equations. We decouple a…
Classical and Quantum Integrable Systems in $\wt{\gr{gl}}(2)^{+*}$ and Separation of Variables
John Harnad, P. Winternitz
Classical integrable Hamiltonian systems generated by elements of the Poisson commuting ring of spectral invariants on rational coadjoint orbits of the loop algebra $\wt{\gr{gl}}^{…