Abstract:
A soliton is a unique type of solitary wave that emerges as the solution to a nonlinear partial
differential equation (PDE). A solitary wave is a wave that stays concentrated in a specific
region, retains its shape unchanged and move steadily at a constant speed in one direction
only. John Scott Russell first noticed solitons as solitary waves in water in 1834. In 1895,
the Korteweg-De Varies equation was developed to explain solitons in shallow water. It was
established in 1965 that the nonlinear Schr¨odinger equation (NLSE) can support soliton
solution. In 1970 the use of solitons in fiber optics revolutionized telecommunications.
Since then, solitons have been used in a variety of domains expanding our understanding
of wave dynamics and nonlinear processes.
Bright soliton are the solitons which are above x-axises and dark soliton are those which
are below the x-axises. The objective of this thesis is to be utilize the ordinary differential
equation (ODE) method to obtain dark, bright and moving front soliton in various forms
of nonlinear equations. Our aim is to generate these soliton types within the context of
the equations being stuied. Using the ODE method we successfully obtain solutions for
dark, bright and moving front solitons in a range of NLSE. This enables us to showcase the
presence and behaviour of these solitons within different nonlinear systems.
