Journal of Nonlinear Mathematical Physics

Volume 11, Issue 1, February 2004
Research Article

1. On the Quantization of Yet Another Two Nonlinear Harmonic Oscillators

Francesco Calogero
Pages: 1 - 6
In two previous papers the quantization was discussed of three one-degree-of-freedom Hamiltonians featuring a constant c, the value of which does not influence at all the corresponding classical dynamics (which is characterized by isochronous solutions, all of them periodic with period 2: "nonlinear...
Research Article

2. µ-Holomorphic Projective Connections and Conformal Covariance

Mohamed Kachkachi
Pages: 7 - 12
At the quantum level of a bidimensional conformal model, the conformal symmtry is broken by the diffeomorphism anomaly and the conformal covariance is not maintained. Here we interpret geometrically this conformal covariance as an exact holomorphy condition on a two-dimensional Riemann surface on which...
Research Article

3. Reduction of Order for Systems of Ordinary Differential Equations

C. Wafo Soh, F.M. Mahomed
Pages: 13 - 20
The classical reduction of order for scalar ordinary differential equations (ODEs) fails for a system of ODEs. We prove a constructive result for the reduction of order for a system of ODEs that admits a solvable Lie algebra of point symmetries. Applications are given for the case of a system of two...
Research Article

4. The Heun Equation and the Calogero-Moser-Sutherland System III: The Finite-Gap Property and the Monodromy

Kouichi Takemura
Pages: 21 - 46
A new approach to the finite-gap property for the Heun equation is constructed. The relationship between the finite-dimensional invariant space and the spectral curve is clarified. The monodromies are calculated and are expressed as hyperelliptic integrals. Applications to the spectral problem for the...
Research Article

5. On Conditionally Invariant Solutions of Magnetohydrodynamic Equations. Multiple Waves.

A.M. Grundland, P. Picard
Pages: 47 - 74
We present a version of the conditional symmetry method in order to obtain multiple wave solutions expressed in terms of Riemann invariants. We construct an abelian distribution of vector fields which are symmetries of the original system of PDEs subjected to certain first order differential constraints....
Research Article

6. Integrability Conditions for n and t Dependent Dynamical Lattice Equations

R. Yamilov, D. Levi
Pages: 75 - 101
Conditions necessary for the existence of local higher order generalized symmetries and conservation laws are derived for a class of dynamical lattice equations with explicit dependence on the spatial discrete variable and on time. We explain how to use the obtained conditions for checking a given equation....
Research Article

7. Solution of the Goldfish N-Body Problem in the Plane with (Only) Nearest-Neighbor Coupling Constants All Equal to Minus One Half

Francesco Calogero
Pages: 102 - 112
The (Hamiltonian, rotation- and translation-invariant) "goldfish" N-body problem in the plane is characterized by the Newtonian equations of motion ¨zn - i zn = 2 N m=1,m=n an,m zn zm (zn - zm) -1 , written here in their complex version, entailing the identification of the real "physical" plane with...
Research Article

8. Replicator Dynamics and Mathematical Description of Multi-Agent Interaction in Complex Systems

Vasyl V. Gafiychuk, Anatoliy K. Prykarpatsky
Pages: 113 - 122
We consider the general properties of the replicator dynamical system from the stanpoint of its evolution and stability. Vector field analysis as well as spectral properties of such system has been studied. A Lyaponuv function for the investigation of the evolution of the system has been proposed. The...
Research Article

9. Competing Species: Integrability and Stability

P.G.L. Leach, J. Miritzis
Pages: 123 - 133
We examine the classical model of two competing species for integrability in terms of analytic functions by means of the Painlevé analysis. We find that the governing equations are integrable for certain values of the essential parameters of the system. We find that, for all integrable cases with the...
Research Article

10. A Two-Phase Free Boundary Problem for the Nonlinear Heat Equation

S. De Lillo, M.C. Salvatori
Pages: 134 - 140
A two-phase free boundary problem associated with nonlinear heat conduction is cosidered. The problem is mapped into two one-phase moving boundary problems for the linear heat equation, connected through a constraint on the relative motion of their moving boundaries. Existence and uniqueness of the solution...