Journal of Nonlinear Mathematical Physics

Volume 16, Issue 4, December 2009
Research Article

1. Cohomology of the Lie Superalgebra of Contact Vector Fields on 𝕂1|1 and Deformations of the Superspace of Symbols

Imed Basdouri, Mabrouk Ben Ammar, Nizar Ben Fraj, Maha Boujelbene, Kaouthar Kamoun
Pages: 373 - 409
Following Feigin and Fuchs, we compute the first cohomology of the Lie superalgebra 𝒦(1) of contact vector fields on the (1, 1)-dimensional real or complex superspace with coefficients in the superspace of linear differential operators acting on the superspaces of weighted densities. We also compute...
Research Article

2. Transformation of Auto-Bäcklund Type For Hyperbolic Generalization of Burgers Equation

Ekaterina V. Kutafina
Pages: 411 - 420
We consider the hyperbolic generalization of Burgers equation with polynomial source term. The transformation of auto-Bäcklund type was found. Application of the results is shown in the examples, where kink and bi-kink solutions are obtained from the pair of two stationary ones.
Research Article

3. Nonstandard Separability on the Minkowski Plane

Giuseppe Pucacco, Kjell Rosquist
Pages: 421 - 430
We present examples of nonstandard separation of the natural Hamilton–Jacobi equation on the Minkowski plane 𝕄2. By “nonstandard” we refer to the cases in which the form of the metric, when expressed in separating coordinates, does not have the usual Liouville structure. There are two possibilities:...
Research Article

4. An Old Method of Jacobi to Find Lagrangians

M. C. Nucci, P. G. L. Leach
Pages: 431 - 441
In a recent paper by Ibragimov a method was presented in order to find Lagrangians of certain second-order ordinary differential equations admitting a two-dimensional Lie symmetry algebra. We present a method devised by Jacobi which enables one to derive (many) Lagrangians of any second-order differential...
Research Article

5. Euler–Lagrange Equations for Functionals Defined On Fréchet Manifolds

José Antonio Vallejo
Pages: 443 - 454
We prove a version of the variational Euler–Lagrange equations valid for functionals defined on Fréchet manifolds, such as the spaces of sections of differentiable vector bundles appearing in various physical theories.
Research Article

6. Orbital Linearization in the Quadratic Lotka–Volterra Systems Around Singular Points Via Lie Symmetries

Jaume Giné, Susanna Maza
Pages: 455 - 464
In this paper, we consider linearizability and orbital linearizability properties of the Lotka–Volterra system in the neighborhood of a singular point with eigenvalues 1 and -q. In this paper we give the explicit smooth near-identity change of variables that linearizes or orbital linearizes such Lotka–Volterra...
Research Article

7. Averaging in Weakly Coupled Discrete Dynamical Systems

Niklas Brännström
Pages: 465 - 487
In [Y. Kifer, Averaging in difference equations driven by dynamical systems, Asterisque287 (2003) 103–123] a general averaging principle for slow-fast discrete dynamical systems was presented. In this paper we extend this method to weakly coupled slow-fast systems. For this setting we obtain sharper...
Research Article

8. On Nonlocal Symmetries, Nonlocal Conservation Laws and Nonlocal Transformations of Evolution Equations: Two Linearisable Hierarchies

Norbert Euler, Marianna Euler
Pages: 489 - 504
We discuss nonlocal symmetries and nonlocal conservation laws that follow from the systematic potentialisation of evolution equations. Those are the Lie point symmetries of the auxiliary systems, also known as potential symmetries. We define higher-degree potential symmetries which then lead to nonlocal...
Research Article

9. Global Analytic First Integrals for the Simplified Multistrain/Two-Stream Model for Tuberculosis and Dengue Fever

Jaume Llibre, Clàudia Valls
Pages: 505 - 516
We provide the complete classification of all global analytic first integrals of the simplified multistrain/two-stream model for tuberculosis and dengue fever that can be written as x˙=x(β1−b−γ1−β1x−(β1−ν)y),     y˙=y(β2−b−γ2−(β2−ν)x−β2y), with β1, β2, b, γ1, γ2, ν ∈ ℝ.