Chowdury, Md Amdadul Huq
Description
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...[Show more] 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.
Items in Open Research are protected by copyright, with all rights reserved, unless otherwise indicated.