Solitons, Breathers and Rogue Waves in Nonlinear Media

Abstract

In this thesis, the solutions of the Nonlinear Schrödinger equation (NLSE) and its hierarchy are studied extensively. In nonlinear optics, as the duration of optical pulses get shorter, in highly nonlinear media, their dynamics become more complex, and, as a modelling equation, the basic NLSE fails to explain their behaviour. Using the NLSE and its hierarchy, this thesis explains the ultra-short pulse dynamics in highly nonlinear media. To pursue this purpose, the next higher-order equations beyond the basic NLSE are considered; namely, they are the third order Hirota equation and the fifth order quintic NLSE. Solitons, breathers and rogue wave solutions of these two equations have been derived explicitly. It is revealed that higher order terms offer additional features in the solutions, namely, ‘Soliton Superposition’, ‘Breather Superposition’ and ‘Breather-to-Soliton’ conversion.

How robust are the rogue wave solutions against perturbations? To answer this question, two types of perturbative cases have been considered; one is odd-asymmetric and the other type is even-symmetric. For the odd-asymmetric perturbative case, combined Hirota and Sasa-Satsuma equations are considered, and for the latter case, fourth order dispersion and a quintic nonlinear term combined with the NLSE are considered. Indeed, this thesis shows that rogue waves survive these perturbations for specific ranges of parameter values.

The integrable Ablowitz-Ladik (AL) equation is the discrete counterpart of the NLSE. If the lattice spacing parameter goes to zero, the discrete AL becomes the continuous NLSE. Similar rules apply to their solutions. A list of corresponding solutions of the discrete Ablowitz-Ladik and the NLSE has been derived. Using associate Legendre polynomial functions, sets of solutions have been derived for the coupled Manakov equations, for both focusing and defocusing cases. They mainly explain partially coherent soliton (PCS) dynamics in Kerr-like media. Additionally, corresponding approximate solutions for two coupled NLSE and AL equations have been derived. For the shallow water case, closed form breathers, rational and degenerate solutions of the modified Kortweg-de Vries equation are also presented.