Journal of Nonlinear Mathematical Physics

Volume 14, Issue 3, October 2007

1. Variational derivation of the Camassa-Holm shallow water equation

Delia Ionescu-Kruse
Pages: 311 - 320
We describe the physical hypothesis in which an approximate model of water waves is obtained. For an irrotational unidirectional shallow water flow, we derive the Camassa- Holm equation by a variational approach in the Lagrangian formalism.

2. Second-order recursion operators of third-order evolution equations with fourth-order integrating factors

Norbert Euler, Marianna Euler
Pages: 321 - 323
We report the recursion operators for a class of symmetry integrable evolution equa- tions of third order which admit a fourth-order integrating factor. Under some as- sumptions we obtain the complete list of equations, one of which is a special case of the Schwarzian Korteweg-de Vries equation.

3. Tauberian Theorems In Quantum Calculus

Ahmed Fitouhi, Kamel Brahim
Pages: 324 - 340
In this paper we attempt to establish some tauberian theorems in quantum calculus. This constitutes the beginning of the study of the q-analogue of analytic theory of numbers which is the aim of a forthcoming paper.

4. Fractional q-Calculus on a time scale

Ferhan M. Atici, Paul W. Eloe
Pages: 341 - 352
The study of fractional q-calculus in this paper serves as a bridge between the fractional q-calculus in the literature and the fractional q-calculus on a time scale Tt0 = {t : t = t0 q n , n a nonnegative integer } ∪ {0}, where t0 ∈ R and 0 < q < 1. By use of time scale calculus notation, we find...

5. On Separation of Variables for Integrable Equations of Soliton Type

Julia Bernatska, Petro Holod
Pages: 353 - 374
We propose a general scheme for separation of variables in the integrable Hamilto- nian systems on orbits of the loop algebra sl(2, C) × P (λ, λ −1 ). In particular, we illus- trate the scheme by application to modified Korteweg—de Vries (MKdV), sin(sinh)- Gordon, nonlinear Schr ̈odinger, and...

6. A Whittaker-Shannon-Kotelnikov sampling theorem related to the Askey-Wilson functions

Fethi Bouzeffour
Pages: 375 - 388
A Whittaker-Shannon-Kotel’nikov sampling theorem related to the Askey-Wilson functions is proved. Applications to finite continuous Askey-Wilson transform are given.

7. A two-point boundary value problem on a Lorentz manifold arising in A. Poltorak's concept of reference frame

Yuri E. Gliklikh, Peter S. Zykov
Pages: 388 - 397
In A. Poltorak's concept, the reference frame in General Relativity is a certain manifold equipped with a connection. The question under consideration here is whether it is possible to join two events in the space-time by a time-like geodesic if they are joined by a geodesic of the reference frame connection...

8. Euler-Poincaré Formalism of (Two Component) Degasperis-Procesi and Holm-Staley type Systems

Partha Guha
Pages: 398 - 429
In this paper we propose an Euler-Poincaré formalism of the Degasperis and Procesi (DP) equation. This is a second member of a one-parameter family of partial dif- ferential equations, known as b-field equations. This one-parameter family of pdes includes the integrable Camassa-Holm equation as a first...

9. Linearization of one-dimensional nonautonomous jump-diffusion stochastic differential equations

Gazanfer Unal, Ismail Iyigunler, C Masood Khalique
Pages: 430 - 442
Necessary and sufficient conditions for the linearization of one-dimensional nonau- tonomous jump-diffusion stochastic differential equations are given. Stochastic inte- grating factor is introduced to solve the linear jump-diffusion stochastic differential equations. Closed form solutions to certain...

10. Integrability analysis of the Emden-Fowler equation

K S Govinder, P G L Leach
Pages: 443 - 461
The Emden-Fowler equation of index n is studied utilising the techniques of Lie and Painlev ́e analysis. For general n information about the integrability of this equation is obtained. The link between these two types of analyses is explored. The special cases of n = −3, 2 are also examined. As a...

11. Partial integrability of the anharmonic oscillator

Robert Conte
Pages: 462 - 473
We consider the anharmonic oscillator with an arbitrary-degree anharmonicity, a damping term and a forcing term, all coefficients being time-dependent: u ′′ + g1 (x)u ′ + g2 (x)u + g3 (x)u n + g4 (x) = 0, n real. Its physical applications range from the atomic Thomas-Fermi model to Emden gas dynamics...

12. The Riemann-Hilbert Formalism For Certain Linear and Nonlinear Integrable PDEs

Dimitrios A. Pinotsis
Pages: 474 - 493
We show that by deforming the Riemann-Hilbert (RH) formalism associated with certain linear PDEs and using the so-called dressing method, it is possible to derive in an algorithmic way nonlinear integrable versions of these equations. In the usual Dressing Method, one first postulates a matrix RH problem...