Mathematics / Matematik
Permanent URI for this collectionhttps://hdl.handle.net/11147/8
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Article Citation - WoS: 9Citation - Scopus: 8Similarity Solutions To Burgers' Equation in Terms of Special Functions of Mathematical Physics(Jagellonian University, 2017) Öziş, Turgut; Aslan, İsmailIn this paper, the Lie group method is used to investigate some closed form solutions of famous Burgers' equation. A detailed and complete symmetry analysis is performed. By similarity transformations, the equation is reduced to ordinary differential equations whose general solutions are written in terms of the error function, Kummer's confluent hypergeometric function Φ(a; b; x) and Bessel functions Jp, showing the strong connection between the best mathematical modelling equations and the special functions of mathematical physics.Article Citation - WoS: 11Citation - Scopus: 11Variations on a Theme of Q-Oscillator(IOP Publishing Ltd., 2015) Pashaev, OktayWe present several ideas in the direction of physical interpretation of q- and f-oscillators as nonlinear oscillators. First we show that an arbitrary one-dimensional integrable system in action-angle variables can be naturally represented as a classical and quantum f-oscillator. As an example, the semi-relativistic oscillator as a descriptive of the Landau levels for relativistic electron in magnetic field is solved as an f-oscillator. By using dispersion relation for q-oscillator we solve the linear q-Schrödinger equation and corresponding nonlinear complex q-Burgers equation. The same dispersion allows us to construct integrable q-NLS model as a deformation of cubic NLS in terms of recursion operator of NLS hierarchy. A peculiar property of the model is to be completely integrable at any order of expansion in deformation parameter around q = 1. As another variation on the theme, we consider hydrodynamic flow in bounded domain. For the flow bounded by two concentric circles we formulate the two circle theorem and construct the solution as the q-periodic flow by non-symmetric q-calculus. Then we generalize this theorem to the flow in the wedge domain bounded by two arcs. This two circular-wedge theorem determines images of the flow by extension of q-calculus to two bases: the real one, corresponding to circular arcs and the complex one, with q as a primitive root of unity. As an application, the vortex motion in annular domain as a nonlinear oscillator in the form of classical and quantum f-oscillator is studied. Extending idea of q-oscillator to two bases with the golden ratio, we describe Fibonacci numbers as a special type of q-numbers with matrix Binet formula. We derive the corresponding golden quantum oscillator, nonlinear coherent states and Fock-Bargman representation. Its spectrum satisfies the triple relations, while the energy levels' relative difference approaches asymptotically to the golden ratio and has no classical limit.Article Citation - WoS: 10Citation - Scopus: 11Some Exact and Explicit Solutions for Nonlinear Schrödinger Equations(Polish Academy of Sciences, 2013) Aslan, İsmailNonlinear models occur in many areas of applied physical sciences. This paper presents the first integral method to carry out the integration of Schrödinger-type equations in terms of traveling wave solutions. Through the established first integrals, exact traveling wave solutions are obtained under some parameter conditions.Conference Object Citation - WoS: 3Citation - Scopus: 3Filamentary Structures of the Cosmic Web and the Nonlinear Schrödinger Type Equation(IOP Publishing Ltd., 2011) Tığrak, Esra; Van De Weygaert, R.; Jones, B. J. T.We show that the filamentary type structures of the cosmic web can be modeled as solitonic waves by solving the reaction diffusion system which is the hydrodynamical analogous of the nonlinear Schrödinger type equation. We find the analytical solution of this system by applying the Hirota direct method which produces the dissipative soliton solutions to formulate the dynamical evolution of the nonlinear structure formation.