Journal of Nonlinear Mathematical Physics

Volume 19, Issue 2, June 2012
Research Article

1. Closed form Solutions to the Integrable Discrete Nonlinear Schrödinger Equation

Francesco Demontis, Cornelis van der Mee
Pages: 136 - 157
In this article we derive explicit solutions of the matrix integrable discrete nonlinear Schrödinger equation by using the inverse scattering transform and the Marchenko method. The Marchenko equation is solved by separation of variables, where the Marchenko kernel is represented in separated form, using...
Research Article

2. Lie Theorem via Rank 2 Distributions (Integration of PDE of Class ω = 1)

Boris Kruglikov
Pages: 158 - 181
In this paper we investigate compatible overdetermined systems of PDEs on the plane with one common characteristic. Lie's theorem states that its integration is equivalent to a system of ODEs, and we give a new proof by relating it to the geometry of rank 2 distributions. We find a criterion for...
Research Article

3. A Systematic Method of Finding Linearizing Transformations for Nonlinear Ordinary Differential Equations I: Scalar Case

V. K. Chandrasekar, M. Senthilvelan, M. Lakshmanan
Pages: 182 - 202
In this paper we formulate a stand alone method to derive maximal number of linearizing transformations for nonlinear ordinary differential equations (ODEs) of any order including coupled ones from a knowledge of fewer number of integrals of motion. The proposed algorithm is simple, straightforward and...
Research Article

4. A Systematic Method of Finding Linearizing Transformations for Nonlinear Ordinary Differential Equations II: Extension to Coupled ODEs

V. K. Chandrasekar, M. Senthilvelan, M. Lakshmanan
Pages: 203 - 225
In this second paper on the method of deriving linearizing transformations for nonlinear ODEs, we extend the method to a set of two coupled second-order nonlinear ODEs. We show that beside the conventional point, Sundman and generalized linearizing transformations one can also find a large class of mixed...
Research Article

5. Hyperkähler Structure of the Taub-NUT Metric

G. Gaeta, M. A. Rodríguez
Pages: 226 - 235
The Taub-NUT four-dimensional space-time can be obtained from Euclidean eight-dimensional one through a momentum map construction; the HKLR theorem [9] guarantees the hyperkähler structure of R8 descends to a hyperkähler structure in the Taub-NUT space. Here we present a detailed and fully explicit construction...
Research Article

6. Left-Invariant Pseudo-Einstein Metrics on Lie Groups

Sheng Chen, Ke Liang
Pages: 236 - 246
In this article, we focus on left-invariant pseudo-Einstein metrics on Lie groups. To begin with, we give some examples of pseudo-Einstein metrics on Lie groups. Also we calculate the Levi-civita connection, and then Ricci tensor associated with left-invariant pseudo-Riemannian metrics on the unimodular...
Research Article

7. Liouvillian and Analytic First Integrals for the Brusselator System

Jaume Llibre, Clàudia Valls
Pages: 247 - 254
We characterize the Liouvillian and analytic first integrals for the polynomial differential systems of the form x′ = a − (b + 1)x + x2y, y′ = bx − x2y, with a, b ∈ ℝ, called the Brusselator differential systems.
Research Article

8. Construction of Modulated Amplitude Waves via Averaging in Collisionally Inhomogeneous Bose–Einstein Condensates

Qihuai Liu, Dingbian Qian
Pages: 255 - 268
We apply the averaging method to analyze spatio-temportal structures in nonlinear Schrödinger equations and thereby study the dynamics of quasi-one-dimensional collisionally inhomogeneous Bose–Einstein condensates with the scattering length varying periodically in space and crossing zero. Infinitely...
Research Article

9. Tidal Tensors in the Description of Gravity and Electromagnetism

Nicoleta Voicu
Pages: 269 - 284
In 2008–2009, L. F. O. Costa and C. A. R. Herdeiro proposed a new gravito-electromagnetic analogy, based on tidal tensors. We show that connections on the tangent bundle of the space-time manifold help in finding an advantageous geometrization of their ideas. Moreover, the combination tidal tensors —...