New insights into Classical Spin Systems and Vortex Hydrodynamics through Solitons and Breathers

Mukhopadhyay, Aritra Kumar (2014) New insights into Classical Spin Systems and Vortex Hydrodynamics through Solitons and Breathers. Masters thesis, Indian Institute of Science Education & Research Kolkata.

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Abstract

Solitons or solitary waves are localized non-dissipative solutions of nonlinear equations having �nite non-zero energy. Historically, they were �rst observed in 1834 by a young engineer named John Scott Russell, who was passing by a canal, when a barge suddenly stopped and he noted that the mass of water in front of its blunt prow "... rolled forward with great velocity, assuming the form of a large solitary elevation, a rounded, smooth and well de�ned heap of water, which continued its course along the channel without change of form or diminution of speed" (Russell, 1844). Thereafter solitons have been discovered in numerous systems ranging from hydrodynamics to optical �bres and �nds numerous applications in both physics and mathematics. On the other hand, a breather is a nonlinear wave in which energy concentrates in a localized and oscillatory fashion. Solitons and breathers are both obtained as solutions of non-linear equations like Nonlinear Schrodinger Equation, Sine Gordon Equation, KdV equation etc. The objective of this thesis is to study the characteristics of some of the important solitons and breathers and �nd new manifestations of these in di�erent physical systems. The thesis is organized as follows. In Chapter 1, I provide an introduction to the simplest type of soliton solutions i.e. the kink solitons. I also discuss some of the interesting consequences of these kinks in condensed matter systems. In Chapter 2, I discuss about the various soliton and breather solutions of the cubic Non-Linear Schrodinger Equation. A detailed discussion on the characteristics of the Peregrine breather is also presented. Chapter 3 is devoted to an introduction on the dynamics of moving curves and surfaces. In Chapter 4, these geometrical properties of moving curves and the characteristics of Peregrine breather has been used to uncover new excitation modes in 1D classical spin systems governed by Heisenberg interaction. Finally in Chapter 5, I look into some of the other aspects of the Peregrine breather and draw an equivalence between this system and the hydrodynamics of vortices.

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