Abstract:
Nonlinear Evolution Equations (NLEEs) are a type of mathematical equation, commonly a
partial differential equation (PDE), that describe the evolution of physical phenomena with
nonlinear behaviours across time. These equations are critical in many scientific disciplines,
including mathematical modelling and soliton theory. The solutions of the NLEEs
represent the solitary waves (SW) called solitons. The solitary waves or solitons have a
very unique history.Solitons are observed as a dispersion and nonlinearity balance. They
are common in many different sectors and have intriguing qualities that make them useful
in a variety of applications. The chirped solitons are a recently popular solitary wave
phenomenon. Chirp is used in spread spectrum communications as well as some sonar and
radar devices.
In this thesis, firstly we will obtain some chirped periodic and solitons wave by using Jacobian
elliptic function (JEF) for for higher order NLSE with anomalous dispersion regime.
We also obtain some solitary waves (SW) like dark, bright, kink, hyperbolic, periodic and
other solutions for the governing model. The chirp that corresponds to each of these optical
solitons is also determined. We will also display the graph of our solutions in different dimensions.
Secondly we will study different analytical solutions for Susceptible-Infectious-
Recovered (SIR) epidemic model with specific nonlinear incidence rate and spatial diffusion
like lump waves (LW), rogue waves (RW), periodic wave (PW) and periodic-cross
lump waves (PCLW). This model offers useful information for containment methods by
simulating and understanding the geographical spread of infectious illnesses. It advances
our knowledge of how the dynamics of an epidemic are influenced by the movement of
infected and susceptible people across space.
