Journal of Nonlinear Mathematical Physics

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.

We report the recursion operators for a class of symmetry integrable evolution equations of third order which admit a fourth-order integrating factor. Under some assumptions we obtain the complete list of equations, one of which is a special case of the Schwarzian Korteweg-de Vries equation.

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.

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

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...

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

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...

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...

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...

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...

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...

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...