# Journal of Nonlinear Mathematical Physics

Volume 14, Issue 1, February 2007

**Research Article**

## 1. Particle trajectories in linear periodic capillary and capillary-gravity deep-water waves

David Henry

Pages: 1 - 7

We show that within the framework of linear theory the particle paths in a periodic gravity-capillary or pure capillary deep-water wave are not closed.

**Research Article**

## 2. An invariant p-adic q-integral associated with q-Euler numbers and polynomials

Ismail Naci Cangül, Veli Kurt, Yilmaz Simsek, Hong Kyung Pak, Seog-Hoon Rim

Pages: 8 - 14

The purpose of this paper is to consider q-Euler numbers and polynomials which are q-extensions of ordinary Euler numbers and polynomials by the computations of the p-adic q-integrals due to T. Kim, cf. [1, 3, 6, 12], and to derive the "complete sums for q-Euler polynomials" which are evaluated by using...

**Research Article**

## 3. q-Euler numbers and polynomials associated with p-adic q-integrals

Taekyun Kim

Pages: 15 - 27

The main purpose of this paper is to present a systemic study of some families of multiple q-Euler numbers and polynomials. In particular, by using the q-Volkenborn integration on Zp, we construct p-adic q-Euler numbers and polynomials of higher order. We also define new generating functions of multiple...

**Research Article**

## 4. The Initial-Boundary Value Poblem for the Korteweg-de Vries Equation on the Positive Quarter-Plane

Pham Loi Vu

Pages: 28 - 43

The paper deals with a problem of developing an inverse-scattering transform for solving the initial-boundary value problem (IBVP) for the Korteweg-de Vries equation on the positive quarter-plane: pt - 6ppx + pxxx = 0, x 0, t 0, (a) with the given initial and boundary conditions: p(x, 0) = p(x), p(x)...

**Research Article**

## 5. New approach to the complete sum of products of the twisted (h, q)-Bernoulli numbers and polynomials

Yilmaz Simsek, Veli Kurt, Daeyeoul Kim

Pages: 44 - 56

In this paper, by using q-Volkenborn integral[10], the first author[25] constructed new generating functions of the new twisted (h, q)-Bernoulli polynomials and numbers. We define higher-order twisted (h, q)-Bernoulli polynomials and numbers. Using these numbers and polynomials, we obtain new approach...

**Research Article**

## 6. Solitons in Yakushevich-like models of DNA dynamics with improved intrapair potential

Giuseppe Gaeta

Pages: 57 - 81

The Yakushevich model provides a very simple pictures of DNA torsion dynamics, yet yields remarkably correct predictions on certain physical characteristics of the dynamics. In the standard Yakushevich model, the interaction between bases of a pair is modelled by a harmonic potential, which becomes anharmonic...

**Research Article**

## 7. A note on the relationship between rational and trigonometric solutions of the WDVV equations

Andrew Riley, Ian A.B. Strachan

Pages: 82 - 94

Legendre transformations provide a natural symmetry on the space of solutions to the WDVV equations, and more specifically, between different Frobenius manifolds. In this paper a twisted Legendre transformation is constructed between solutions which define the corresponding dual Frobenius manifolds....

**Research Article**

## 8. A geometric interpretation of the complex tensor Riccati equation for Gaussian beams

M.F. Dahl

Pages: 95 - 111

We study the complex Riccati tensor equation DcG + GCG - R = 0 on a geodesic c on a Riemannian 3-manifold. This non-linear equation appears in the study of Gaussian beams. Gaussian beams are asymptotic solutions to hyperbolic equations that at each time instant are concentrated around one point in space....

**Research Article**

## 9. On sl(2)-relative cohomology of the Lie algebra of vector fields and differential operators

Sofiane Bouarroudj

Pages: 112 - 127

Let Vect(R) be the Lie algebra of smooth vector fields on R. The space of symbols Pol(T R) admits a non-trivial deformation (given by differential operators on weighted densities) as a Vect(R)-module that becomes trivial once the action is restricted to sl(2) Vect(R). The deformations of Pol(T R), which...

**Research Article**

## 10. Solitons in a double pendulums chain model, and DNA roto-torsional dynamics

Mariano Cadoni, Roberto De Leo, Giuseppe Gaeta

Pages: 128 - 146

It was first suggested by Englander et al to model the nonlinear dynamics of DNA relevant to the transcription process in terms of a chain of coupled pendulums. In a related paper [4] we argued for the advantages of an extension of this approach based on considering a chain of double pendulums with certain...

**Research Article**

## 11. On application of Liouville type equations to constructing Bäcklund transformations

Dmitry Demskoi

Pages: 147 - 156

It is shown how pseudoconstants of the Liouville-type equations can be exploited as a tool for construction of the Bäcklund transformations. Several new examples of such transformations are found. In particular we obtained the Bäcklund transformations for a pair of three-component analogs of the dispersive...