Study of Multiple Solitons and Nonlinear Waves

Abstract

Soliton theory is a significant topic in mathematical physics and applied mathematics with its rapid development since the 1960s. Solitons, also known as SWs, are found in the solutions of many kinds of nonlinear evolution equation (NLEE) or nonlinear partial dif- ferential equations (NLPDEs). They have several unique properties and can be used to explaina number of significant physical phenomena. A soliton is a distinct type of SW that possesses the unique characteristic of maintaining its shape and velocity unchanged across collisions. The balance between the nonlinear effects and dispersion leads to a soliton so- lution. Solitons are used widely in both pure and applied mathematics, particularly in areas such as differential equations, algebraic and differential geometry, Lie groups, and Lie al- gebras. The SIR (Susceptible-Infected-Removed) model is a simple mathematical model of epi- demic outbreaks. The present thesis surveys new analytical results about the SIR model. The SIR model is one of the most basic models for describing the temporal dynamics of an infectious disease in a population. It compartmentalizes population into one of three categories: those who are Susceptible to the disease, those who are currently Infectious, and those who have Recovered (with immunity). At its most basic level, the SIR model is a set of ODEs that describes the number (or proportion) of people in each compartment over time. This thesis will analyze the SIR model and provide some solutions through the use of exponential, hyperbolic, and trigonometric functions. Our goal in this work is to find lump soliton solution (LS), rogue wave solution (RW), lump with one kink (LSK), peri- odic waves solution (PW), interaction solution between lump, periodic and one kink as well as interaction solution between lump, periodic and two kink soliton wave for different types of NLPDEs. Additionally, M type interaction with periodic, M type rational soliton wave solution, periodic cross kink wave solution, breather wave, homoclinic breather wave, Kuznetsov Ma breathers, W shaped soliton, mixed type solutions will be obtained