# Journal of Nonlinear Mathematical Physics

Volume 17, Issue 1, March 2010

**Letter to Editor**

## 1. On Non-Commutative Integrable Burgers Equations

Metin Gürses, Atalay Karasu, Refik Turhan

Pages: 1 - 6

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.

**Letter to Editor**

## 2. On the Geodesic Flow on the Group of Diffeomorphisms of the Circle with a Fractional Sobolev Right-Invariant Metric

Marcus Wunsch

Pages: 7 - 11

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=−12.

**Research Article**

## 3. Approximate Partial Noether Operators of the Schwarzschild Spacetime

Ibrar Hussain, F. M. Mahomed, Asghar Qadir

Pages: 13 - 25

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...

**Research Article**

## 4. Spinors on Kahler–Norden Manifolds

Nedim Değirmenci, Şenay Karapazar

Pages: 27 - 34

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,...

**Research Article**

## 5. Bäcklund-Transformation-Related Recursion Operators: Application to the Self-Dual Yang–Mills Equation

C. J. Papachristou, B. Kent Harrison

Pages: 35 - 49

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...

**Research Article**

## 6. On (*q*, *h*)-Analogue of Fractional Calculus

Jan Čermák, Luděk Nechvátal

Pages: 51 - 68

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...

**Research Article**

## 7. Stanley Decomposition for Coupled Takens–Bogdanov Systems

David Mumo Malonza

Pages: 69 - 85

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....

**Research Article**

## 8. Hypergeometric Solutions to an Ultradiscrete Painlevé Equation

Christopher M. Ormerod

Pages: 87 - 102

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...

**Research Article**

## 9. Isochronous Oscillators

F. Calogero, F. Leyvraz

Pages: 103 - 110

We exhibit the solution of the initial-value problem for the system of 2N + 2 oscillators characterized by the Hamiltonian
H(p^0,pˇ0,p^_,pˇ_,q^0,qˇ0,q^_,qˇ_)=12[p^02−pˇ02+Ω2(q^02−qˇ02)] +q^0−Ωqˇ02b∑n=1N[p^n2−pˇn2+ωn2(q^n2−qˇn2)]+p^0−Ωq^0b∑n=1N[−p^npˇn+ωn2q^nqˇn]
where N is an arbitrary positive...

**Research Article**

## 10. Solvable Systems of Isochronous, Multi-Periodic or Asymptotically Isochronous Nonlinear Oscillators

F. Calogero, F. Leyvraz

Pages: 111 - 120

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.

**Research Article**

## 11. Bernoulli Numbers and Solitons — Revisited

Grzegorz Rządkowski

Pages: 121 - 126

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.