Fundamental Solitons For The Higher-order Nonlinear Schrödinger Equation With A Pt -symmetric Potential

Tez (Yüksek Lisans) -- İstanbul Teknik Üniversitesi, Fen Bilimleri Enstitüsü, 2016Thesis (M.Sc.) -- İstanbul Technical University, Institute of Science and Technology, 2016Bu tezde, dördüncü mertebeden ve bir dış potansiyel içeren doğrusal olmayan Schrödinger denklemlerinin optik soliton çözümlerinin matematiksel, sayısal ve kararlı olma analizleri bu çalışmada sırasıyla açıklanmıştır.In this thesis, the mathematical, numerical and stability analysis of optical solitons of nonlinear Schrödinger equation with fourth-order dispersion term and PT -symmetric potential is explored, respectivelyYüksek LisansM.Sc

Exact Solutions for Domain Walls in Coupled Complex Ginzburg - Landau Equations

The complex Ginzburg Landau equation (CGLE) is a ubiquitous model for the evolution of slowly varying wave packets in nonlinear dissipative media. A front (shock) is a transient layer between a plane-wave state and a zero background. We report exact solutions for domain walls, i.e., pairs of fronts with opposite polarities, in a system of two coupled CGLEs, which describe transient layers between semi-infinite domains occupied by each component in the absence of the other one. For this purpose, a modified Hirota bilinear operator, first proposed by Bekki and Nozaki, is employed. A novel factorization procedure is applied to reduce the intermediate calculations considerably. The ensuing system of equations for the amplitudes and frequencies is solved by means of computer-assisted algebra. Exact solutions for mutually-locked front pairs of opposite polarities, with one or several free parameters, are thus generated. The signs of the cubic gain/loss, linear amplification/attenuation, and velocity of the coupled-front complex can be adjusted in a variety of configurations. Numerical simulations are performed to study the stability properties of such fronts.Comment: Journal of the Physical Society of Japan, in pres

Parametric resonance of ground states in the nonlinear Schrödinger equation

AbstractWe study the global existence and long-time behavior of solutions of the initial-value problem for the cubic nonlinear Schrödinger equation with an attractive localized potential and a time-dependent nonlinearity coefficient. For small initial data, we show under some nondegeneracy assumptions that the solution approaches the profile of the ground state and decays in time like t-1/4. The decay is due to resonant coupling between the ground state and the radiation field induced by the time-dependent nonlinearity coefficient

Topological control of extreme waves

From optics to hydrodynamics, shock and rogue waves are widespread. Although they appear as distinct phenomena, transitions between extreme waves are allowed. However, these have never been experimentally observed because control strategies are still missing. We introduce the new concept of topological control based on the one-to-one correspondence between the number of wave packet oscillating phases and the genus of toroidal surfaces associated with the nonlinear Schrödinger equation solutions through Riemann theta functions. We demonstrate the concept experimentally by reporting observations of supervised transitions between waves with different genera. Considering the box problem in a focusing photorefractive medium, we tailor the time-dependent nonlinearity and dispersion to explore each region in the state diagram of the nonlinear wave propagation. Our result is the first realization of topological control of nonlinear waves. This new technique casts light on shock and rogue waves generation and can be extended to other nonlinear phenomena

Self-Organization, Coherence and Turbulence in Laser Optics

In the last decades, rapid progress in modern nonlinear science was marked by the development of the concept of dissipative soliton (DS). This concept is highly useful in many different fields of science ranging from field theory, optics, and condensed matter physics to biology, medicine, and even sociology. This chapter aims to present a DS appearance from random fluctuations, development, and growth, the formation of the nontrivial internal structure of mature DS and its breakup, in other words, a full life cycle of DS as a self-organized object. Our extensive numerical simulations of the generalized cubic-quintic nonlinear Ginzburg-Landau equation, which models, in particular, dynamics of mode-locked fiber lasers, demonstrate a close analogy between the properties of DS and the general properties of turbulent and chaotic systems. In particular, we show a disintegration of DS into a noncoherent (or partially coherent) multisoliton complex. Thus, a DS can be interpreted as a complex of nonlinearly coupled coherent “internal modes” that allows developing the kinetic and thermodynamic theory of the nonequilibrious dissipative phenomena. Also, we demonstrate an improvement of DS integrity and, as a result, its disintegration suppression due to noninstantaneous nonlinearity caused by the stimulated Raman scattering. This effect leads to an appearance of a new coherent structure, namely, a dissipative Raman soliton

Spin and magnetization effects in plasmas

We give a short review of a number of different models for treating magnetization effects in plasmas. In particular, the transition between kinetic models and fluid models is discussed. We also give examples of applications of such theories. Some future aspects are discussed.Comment: 18 pages, 1 figure. To appear in Plasma Physics and Controlled Fusion, Special Issue for the 37th ICPP, Santiago, Chil