On The Optical Rogue Waves In Nonlinear Fiber Optics

In a strict sense,the terminology of soliton refers to a localized wave-packet which can propagate over a long distance without any distortion of its shape.It results from an exact balance between the dispersion(or diffraction)that tends to expand the wave packet and the nonlinear effect that tends to localize it.Mathematically,the soliton concept is a sophisticated construct based on the integrability of a class of nonlinear differential equations,which can be usually solved via the method of inverse scattering transform.In history,the first observation of solitons was done by a young Scottish engineer named John Scott Russell on the Union Canal near Edinburgh in 1834,which are now known as the Korteweg-de Vries(Kd V)solitons.Since then,or more precisely,in the past one century,the concept of soliton has unfolded and flourished in many disciplines,such as fluid mechanics,optics,plasma physics,Bose-Einstein condensation(BEC),chemistry,and biology.Because of its fundamental scientific interest and potential applications,the soliton has now become one of the most active research topics at the cutting edge of nonlinear science,on both theoretical and experimental sides.In the meantime,the modern nonlinear nomenclature has identified all self-trapped wave-packets as “solitons”,including solitary waves that come from the non-integrable sets..As a special type of solitons,rogue waves refer to the transient wave-packets localized on both space and time.They occur on a finite background(hence termed soliton on a finite background),with their peak amplitude larger than twice the significant wave height of the surrounding waves.A typical example is the Peregrine soliton which was first proposed by Peregrine in 1983,as a rational solution to the focusing nonlinear Schr?dinger(NLS)equation.Originally,the rogue wave terminology was used to describe extreme wave events occurring in the open ocean,who appear out of nowhere while carrying a huge devastating power.It is this unpredictability together with the giant amplitude that makes rogue waves a big threat to the cruising ships and tankers,the oil platforms,and even the water conservancy facilities on the coastline.Afterwards,invoked by the seminal observation of optical rogue waves in a microstructured optical fiber,the field of rogue waves has expanded rapidly,encompassing hydrodynamics,plasma physics,nonlinear optics,acoustics,BEC,and even finance,as has been experienced by soliton.Scientists and researchers of different discipline backgrounds acknowledge that this type of rational soliton on a finite background can describe extreme wave events of different origins,from the monster waves in the ocean to the ultra-high pulse spikes in optics,and even the financial crises in economy.Despite the extensive investigation,however,the fundamental origin of rogue waves as well as the physical mechanism underlying their formation is still in debate.Moreover,some interesting rogue wave dynamics such as anomalous Peregrine solitons,coexisting rogue waves,and Peregrine solitons on a periodic background are not well understood and remain to be seen in a laboratorial environment.In this thesis,we will address these topics in an analytical level,by selecting several representative integrable nonlinear models related to the nonlinear fiber optics.We try to obtain the exact rogue wave solutions of these different models,using the non-recursive Darboux transformation(DT)method.Based on these analytical solutions,a detailed discussion of the intriguing rogue wave dynamics as well as the modulation instability in each specific physical setting is provided.We also perform extensive numerical simulations to confirm the stability of Peregrine solitons formed in self-induced transparency media and those with anomalous peak amplitude resulting from the self-steepening effect,despite the onset of modulation instability.The main results of our work can be summarized as follows.1.We investigated for the first time the anomaly of Peregrine solitons in birefringent nonlin-ear space–time coupling fibers within the framework of the coupled Fokas–Lenells(CFL)equations.The exact explicit rogue wave solutions,including the fundamental Peregrine soliton solutions and their higher-order counterparts,were obtained,with the help of the non-recursive DT method.We revealed that in such a two-component system,owing to the presence of the self-steepening effect,there would emerge an anomalous Peregrine soliton state on one wave component whose peak can grow three times higher than its background level,at the expense of a heavy falling-off on the other wave component.We expect that this finding may shed more light on the fundamental origin of the rogue waves occurring in nature,where an extreme giant peak can appear even from the “noise” back-ground.We also demonstrated other interesting rogue wave dynamics such as coexisting rogue waves and multiple rogue waves on such a vector nonlinear system.2.We then considered the coupled cubic-quintic(CQ)NLS equations which can model the propagation of ultrashort pulses in a long-haul telecommunication fiber.We obtained their Peregrine soliton solutions on either the continuous-wave(cw)or the periodic-wave background,by means of the standard DT technique.We particularly showed that the periodic Peregrine soliton solutions can be expressed as a linear superposition of two fun-damental Peregrine solitons of different cw backgrounds.Because of the self-steepening effect inherent in the CQ-NLS equation,some interesting Peregrine soliton dynamics such as ultrastrong amplitude enhancement and rogue wave coexistence are still present when they are built on a periodic background.We numerically confirm the stability of these analytical solutions against non-integrable perturbations,i.e.,when the coefficient rela-tion that enables the integrability of the vector model is slightly lifted.We also presented the nth-order general rogue wave solutions for such vector system,as well as the analysis of the modulation instability that is responsible for the formation of rogue waves.We believe that our results may help to understand the rogue wave phenomena occurring in telecommunication fiber systems.3.We further investigated the vector rogue wave dynamics occurring in a cryogenic erbium-doped resonant fiber that entails the self-induced transparency effect.We used the NLS and Maxwell-Bloch equations as an elementary model,and obtained their explicit fun-damental rational rogue wave solutions in the context of self-induced transparency for the coupled optical and matter waves.It was exhibited that the optical wave component always features a typical Peregrine-like structure,while the matter waves involve more complicated yet spatiotemporally balanced amplitude distribution.The stability of these rogue waves was then confirmed by numerical simulations,and they were shown to be excited amid the onset of modulation instability.Besides,we investigated as well the super rogue wave dynamics occurring in another resonant nonlinear system named as the classical massive Thirring model.We revealed that in this simple vector system,both rogue wave components,whenever they take the fundamental Peregrine soliton or the su-per rogue wave forms,may possess the same maximum peak-amplitude factor,behaving like those occurring in scalar nonlinear systems.However,due to the coherent coupling,the two super rogue wave components may exhibit drastically different spatiotemporal distributions,despite that they evolve from almost the same background fields.We ex-pect that these results stimulate future experimental studies of rogue waves in optical resonant media.4.We finally discussed the rogue wave dynamics in coupled-mode long-haul telecommuni-cation fibers which can be modelled by the vector Lakshmanan–Porsezian–Daniel(VLPD)equation.Using the nonrecursive DT method,we obtained a special hierarchy of rogue wave solutions of the VLPD equation.In terms of the exact rational solutions,we demon-strated several interesting rogue wave dynamics such as rogue wave doublets,quartets and sextets.The modulation instability responsible for the excitation of rogue waves from an unstable continuous background in such a complex nonlinear system was also discussed.