Journal of Nonlinear Mathematical Physics

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 a matrix triplet (A, B, C). Here A has only eigenvalues of modulus larger than one. The class of solutions obtained contains the N-soliton and breather solutions as special cases. We also prove that these solutions reduce to known continuous matrix NLS solutions as the discretization step vanishes.

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 integration in quadratures and in closed form, and discuss nonlinear Laplace transformations and symmetric PDE models.

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 efficient and helps to unearth several new types of linearizing transformations besides the known ones in the literature. To make our studies systematic we divide our analysis into two parts. In the first part we confine our investigations to the scalar ODEs and in the second part we focus our attention on a system of two coupled second-order ODEs. In the case of scalar ODEs, we consider second and third-order nonlinear ODEs in detail and discuss the method of deriving maximal number of linearizing transformations irrespective of whether it is local or nonlocal type and illustrate the underlying theory with suitable examples. As a by-product of this investigation we unearth a new type of linearizing transformation in third-order nonlinear ODEs. Finally the study is extended to the case of general scalar ODEs. We then move on to the study of two coupled second-order nonlinear ODEs in the next part and show that the algorithm brings out a wide variety of linearization transformations. The extraction of maximal number of linearizing transformations in every case is illustrated with suitable examples.

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 or hybrid type linearizing transformations like point-Sundman, point-generalized linearizing transformation and Sundman-generalized linearizing transformation in coupled second-order ODEs using the integrals of motion. We propose suitable algorithms to identify all these transformations (with maximal in number) in a straightforward manner. We illustrate the method of deriving each one of the linearizing transformations with a suitable example.

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 of the hyperkähler structure of a space-time with a Taub-NUT metric.

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 Lie groups of dimension three. Furthermore, we show that the left-invariant pseudo-Einstein metric on SL(2) is unique up to a constant. At last, we study the left-invariant pseudo-Riemannian metrics on compact Lie groups and classify the pseudo-Einstein metrics on the low-dimensional compact Lie groups.

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.

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 many modulated amplitude waves with nontrivial phases are shown.

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 — connections on tangent bundle can underlie a common mathematical description of the main equations of gravity and electromagnetism.