Journal of Nonlinear Mathematical Physics

We construct the recursion operators for the non-commutative Burgers equations using their Lax operators. We investigate the existence of any integrable mixed version of left- and right-handed Burgers equations on higher symmetry grounds.

We show that the geodesic flow on the infinite-dimensional group of diffeomorphisms of the circle, endowed with a fractional Sobolev metric at the identity, is described by the generalized Constantin–Lax–Majda equation with parameter a = -½.

The objective of this paper is twofold: (a) to find a natural example of a perturbed Lagrangian that has different partial Noether operators with symmetries different from those of the underlying Lagrangian. First we regard the Schwarzschild spacetime as a perturbation of the Minkowski spacetime and investigate the approximate partial Noether operators for this perturbed spacetime. It is shown that the Minkowski spacetime has 12 partial Noether operators, 10 of which are different from the 17 Noether symmetries for this spacetime. It is found that for the perturbed Schwarzschild spacetime we recover the exact partial Noether operators as trivial first-order approximate partial Noether operators and there is no non-trivial approximate partial Noether operator as for the Noether case. As a consequence we state a conjecture. (b) Then we prove a conjecture that the approximate symmetries of a perturbed Lagrangian form a subalgebra of the approximate symmetries of the corresponding perturbed Euler–Lagrange equations and illustrate it by our examples. This is in contrast to approximate partial Noether operators.

It is known that the complex spin group Spin(n, ℂ) is the universal covering group of complex orthogonal group SO(n, ℂ). In this work we construct a new kind of spinors on some classes of Kahler–Norden manifolds. The structure group of such a Kahler–Norden manifold is SO(n, ℂ) and has a lifting to Spin(n, ℂ). We prove that the Levi-Civita connection on M is an SO(n, ℂ)-connection. By using the spinor representation of the group Spin(n, ℂ), we define the spinor bundle S on M. Then we define covariant derivative operator ∇ on S and study some properties of ∇. Lastly we define Dirac operator on S.

By using the self-dual Yang–Mills (SDYM) equation as an example, we study a method for relating symmetries and recursion operators of two partial differential equations connected to each other by a non-auto-Bäcklund transformation. We prove the Lie-algebra isomorphism between the symmetries of the SDYM equation and those of the potential SDYM (PSDYM) equation, and we describe the construction of the recursion operators for these two systems. Using certain known aspects of the PSDYM symmetry algebra, we draw conclusions regarding the Lie algebraic structure of the "potential symmetries" of the SDYM equation.

The paper discusses fractional integrals and derivatives appearing in the so-called (q, h)-calculus which is reduced for h = 0 to quantum calculus and for q = h = 1 to difference calculus. We introduce delta as well as nabla version of these notions and present their basic properties. Furthermore, we give comparisons with the known results and discuss possible extensions to more general settings.

We use an algorithm based on the notion of transvectants from classical invariant theory in determining the form of Stanley decomposition of the ring of invariants for the coupled Takens–Bogdanov systems when the Stanley decompositions of the Jordan blocks of the linear part are known at each stage. The set of systems of differential equations that are in normal form with respect to a particular linear part has the structure of a module of equivariants, and is best described by giving a Stanley decomposition of that module.

We show that an ultradiscrete analogue of the third Painlevé equation admits discrete Riccati type solutions. We derive these solutions by considering a framework in which the ultradiscretization process arises as a restriction of a non-archimedean valuation over a field. Using this framework we may relax the conditions one requires to apply the ultradiscretization process. We derive a family of transcendental solutions that appear as the non-archimedean field valuation of hypergeometric functions.

We exhibit the solution of the initial-value problem for the system of 2N + 2 oscillators characterized by the Hamiltonian

A simple technique is identified to manufacture solvable nonlinear dynamical systems, and in particular three classes whose generic solutions are, respectively, isochronous, multi-periodic, or asymptotically isochronous.

In the present paper we propose a new proof of the Grosset–Veselov formula connecting one-soliton solution of the Korteweg–de Vries equation to the Bernoulli numbers. The approach involves Eulerian numbers and Riccati's differential equation.