Rogue wave solutions and the Darboux transformation of the coupled Hirota equations and the Fokas-Lenells equations

Nonlinear partial differential equations (NLPDE) are ubiquitous in the applied science. However, NLPDEs are generally not solvable analytically. Using the Lax pair as a litmus test for integrability, we explore various equations which are known to analytically solvable. The Darboux transformation uses Lie groups to transform a NLPDE via gauge transformations. Typically, the solutions generated by this method tend to be solitons. One of the fields where NLPDEs are applied is oceanic waves. Rogue waves, oceanic waves, that suddenly appear from nowhere and disappear just as suddenly, are thought to be modeled by a the Peregrine soliton. The Peregrine soliton is caused by high levels of localization which create a shift from typical shape into a well of energy localized in both time and space. In this thesis, we develop the methods to derive soliton solutions for the Fokas-Lenells equation, and the coupled Hirota equations via the Darboux transformation. After the soliton solutions are found for the these two families of equations, we show how the points of criticality can cause focusing and create rogue waves. Once the rogue waves are revealed, we discuss how to alter their shape by their manipulating parameters. In addition the dynamic features of these rogue waves are graphically discussed.