Analytical and numerical analysis of the dissipative Kundu-Eckhaus equation

It is well-known that the Kundu-Eckhaus equation (KEE) is a nonlinear equation which belongs to nonlinear Schr¨odinger class and it is commonly used as a model to investigate the dynamics of diverse phenomena in many areas including but are not limited to hydrodynamics, fiber and nonlinear optics, plasmas and finance. However, the effects of dissipation on the dynamics of KEE have not been investigated so far. In this thesis, in order to address this open problem we propose the dissipative Kundu-Eckhaus equation (dKEE) and perform an analytical and numerical analysis of the dKEE. With this motivation, we derive a simple monochromatic wave solution to dKEE. Then, we propose a split step Fourier method (SSFM) for the numerical solution of the dKEE and we test the stability of the SSFM using the analytical solution derived as a benchmark problem. Observing the stability and the accuracy of the scheme, we first investigate the rogue wave dynamics of the dKEE using the SSFM. More specifically, we show that modulation instability (MI) turns the monochromatic wave field into a chaotic one, thus the appearance of rogue waves become obvious. We discuss the properties and characteristics of such rogue waves. Additionally, we depict the amplitude probability distribution functions (PDFs) and discuss the effects of diffusion, Raman and dissipation coefficient as well as the MI parameters on the probability of rogue wave occurrence. Secondly, we investigate the effects of dissipation on the self-localized solitons of the KEE. For this purpose, we propose a Petviashvili method (PM) to obtain the self-localized solitons of the KEE and analyze the effects of dissipation by time stepping of these solitons using the SSFM proposed for dKEE. It is known that, KEE admits stable single, two and N-soliton solutions for the no potential case. It has been recently found that, under the effect of photorefractive and saturable potentials, such solitons of the KEE become unstable. We show that the dissipation parameter can be used to stabilize the single, two and three solitons of the KEE which do not satisfy the necessary Vakhitov-Kolokolov condition for the soliton stability. With this aim, we present the power graphs as functions of soliton eigenvalue and as well as time. Additionally, we depict the soliton shapes for various times to show that they are preserved for time scales long enough for many engineering purposes. We comment on our findings and discuss the applicability and uses of our results. Additionally, we suggest possible directions for the near future research activities