Dissipative solitons : novel developments
Creators: name: Devine, Natasha Natalia
Publisher: name: The Australian National University
Year: 2010
Subjects: subject: QC174.26.W28 D48 2010
subject: Solitons
subject: Energy dissipation.

description: The notion of dissipative solitons is a useful concept that allows us to describe, in general terms, a variety of phenomena in physics, chemistry, biology and medicine. Some specific features of these formations are common for all of them, independent of the problem that we are solving and the model we are using. Many problems in optics involving gain and loss can be formulated in terms of a single "master equation" that is, in one or another form, of the complex cubic-quintic Ginzburg-Landau equation (CGLE) type. In this thesis, I perform a theoretical study of dissipative solitons and dissipative soliton complexes in the frame of the cubic-quintic complex Ginzburg-Landau equation. The model, based on the CGLE, includes cubic and quintic nonlinearities of both dispersive and dissipative types. The master equation and a nonlinear dynamical system governed by it will be discussed in the Introductory chapter. The cubic-quintic complex Ginzburg-Landau equation has a variety of soliton solutions whose analytic forms are unknown. In order to find those solutions, we used numerical simulations and various approximate techniques. In Chapter 2, Lagrangian techniques are applied to dissipative systems described by the complex Ginzburg-Landau equation (CGLE). In particular, using Lagrangian equations, I re-derive known exact solutions of the CGLE. I also apply the technique to finding approximate solutions for pulsating solitons. The latter are unique formations specific to dissipative systems that can be found mostly using numerical simulations. Lagrangian method allows us to obtain analytical expressions for the basic parameters of these solutions. Chapter 3 shows, numerically, that coupled soliton pairs in nonlinear dissipative systems modeled by the cubic-quintic complex Ginzburg-Landau equation can exist in various forms. They can be stationary, or they can pulsate periodically, quasi-periodically or chaotically, as is the case for single solitons. In particular, new types of vibrating and shaking soliton pairs have been found. Each type is stable in the sense that a given bound state exists in the same form indefinitely. New solutions appear at special values of the equation parameters, thus bifurcating from stationary pairs. Mixed soliton pairs, formed by two different types of single solitons, have been shown. Regions of existence of the new types of pair solutions and corresponding bifurcation diagrams are presented. For equations that cannot be solved exactly, the trial function approach to modelling soliton solutions represents a useful approximate technique. It has to be supplemented with the Lagrangian technique or the method of moments to obtain a finite dimensional dynamical system which can be analyzed more easily than the original partial differential equation; in Chapter 4, using the method of moments for dissipative optical solitons, I show that there are two disjoint sets of fixed points. These correspond to the stationary solitons of cubic-quintic CGLE with concave and convex phase profiles. Numerical simulations confirmed the predictions of the method of moments for the existence of two types of solitons which we call solitons and antisolitons. Their characteristics are distinctly different. In Chapter 5, using the Lagrangian formalism, with a trial function for dissipative optical 2D soliton beams, it is demonstrated that there are two disjoint sets of stationary soliton solutions of the complex cubic-quintic Ginzburg-Landau equation, with concave and convex phase profiles. These correspond to continuously self-focusing and continuously self-defocusing types of 2D solitons. Their characteristics are distinctly different, as the energy for their existence can be generated either at the center or in the outer layers of the soliton beam. These predictions are corroborated with direct numerical simulations of the Ginzburg-Landau equation. Regions of existence in the parameter space of these two types of solutions are found and they are in reasonable agreement with the predictions of the Lagrangian approach. Direct numerical simulations allow us to find more complicated localized solutions around these regions. These solutions lack cylindrical symmetry and/or pulsate in time. Examples of the complex behavior of these beams are presented in Chapter 6. Here, I also present transformations of continuously self-focusing and self-defocusing dissipative solitons are analyzed. Dissipative media admit the existence of two types of stationary self-organized beams: continuously self-focused and continuously self-defocused. Each beam is stable inside of a certain region of its existence. Beyond these two regions, beams lose their stability, and new dynamical behaviors appear. Several types of instabilities related to each beam configuration are shown, and examples of beam dynamics in the areas adjacent to the two regions are presented. We have shown that, in one case beams lose the radial symmetry while in the other one the radial symmetry is conserved during complicated beam transformations. In Chapter 7, dissipative ring solitons with vorticity in the frame of the(2+1)-dimensional cubic-quintic complex Ginzburg-Landau equation are discussed and it is shown that radially symmetric ring structures with any vorticity m can be stable in a finite range of parameters. Beyond the region of stability, the solitons lose the radial symmetry but may remain stable, keeping the same value of the topological charge. Bifurcations into solitons with n-fold bending symmetry, with n independent of m are presented. Solitons without circular symmetry can also display (m+1)-fold modulation behaviour. A sequence of bifurcations can transform the ring soliton into a pulsating or chaotic state which keeps the same value of the topological charge as the original ring. Finally, in Chapter 8, the effect of various perturbations on the fundamental rational solution of the nonlinear Schr{u00F6}dinger equation (NLSE) are studied. This solution describes generic nonlinear wave phenomena in the deep ocean, including the notorious rogue waves. It also describes light pulses in optical fibres. It was found that the solution can survive at least three types of perturbations that are often used in the physics of nonlinear waves. It is also shown that the rational solution remains rational and localized in each direction, thus representing a modified rogue wave. Although the equations that we are using in this chapter are conservative, the solutions are located on a finite background exchanging energy with the background. Thus, the localised part of the solution does represent a dissipative soliton.