Journal of Nonlinear Mathematical Physics

Following Feigin and Fuchs, we compute the first cohomology of the Lie superalgebra 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 the same, but 𝔬𝔰𝔭(1|2)-relative, cohomology. We explicitly give 1-cocycles spanning these cohomology. We classify generic formal 𝔬𝔰𝔭(1|2)-trivial deformations of the -module structure on the superspaces of symbols of differential operators. We prove that any generic formal 𝔬𝔰𝔭(1|2)-trivial deformation of this -module is equivalent to a polynomial one of degree ≤ 4. This work is the simplest superization of a result by Bouarroudj [On 𝔰𝔩(2)-relative cohomology of the Lie algebra of vector fields and differential operators, J. Nonlinear Math. Phys. No. 1 (2007) 112–127]. Further superizations correspond to 𝔬𝔰𝔭(N|2)-relative cohomology of the Lie superalgebras of contact vector fields on 1|N-dimensional superspace.

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.

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: the "complex-Liouville" (or "harmonic") case and the "linear/null" (or "Jordan block") case. By means of explicit examples, we show that, in all cases, a suitable glueing of coordinate patches of the different structures allows us to separate natural systems with indefinite kinetic energy all over 𝕄².

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 equation. The method is based on the search of the Jacobi Last Multipliers for the equations. We exemplify the simplicity and elegance of Jacobi's method by applying it to the same two equations as Ibragimov did. We show that the Lagrangians obtained by Ibragimov are particular cases of some of the many Lagrangians that can be obtained by Jacobi's method.

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.

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 system with q ∈ ℕ\{0, 1} being seen that these changes are also valid for q ∈ ℂ\{0, 1}.

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 estimates than in the mentioned paper.

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 conservation laws and nonlocal transformations for the equations. We demonstrate our approach and derive second degree potential symmetries for the Burgers' hierarchy and the Calogero–Degasperis–Ibragimov–Shabat hierarchy.

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