Generation Mechanism And Excitation Conditions Of Fundamental Nonlinear Waves On A Plane Wave Background

There are many complex nonlinear waves excitations in nonlinear systems.These complex excitations are usually formed by nonlinear superposition of a variety of fundamental nonlinear excitations.Therefore,it is crucial to study the generation mechanism and excitation conditions of fundamental nonlinear excitation in nonlinear systems for the experimental realization of nonlinear waves,detection of their dynamic characteristics and applications,and it is helpful to understand the complex excitation characteristics of nonlinear systems.Based on the experimental and theoretical research results,in this thesis,we study the generation mechanism and excitation conditions of fundamental nonlinear waves(Rogue waves,Akhmediev breathers,Kuznetsov-Ma breathers,Tajiri-Watanabe breathers,anti-dark solitons,W-shaped solitons,periodic waves,W-shaped soliton trains,and multi-peak solitons)on plane wave background by Darboux transformation,linear stability analysis,integrating-factor method and step-by-step Fourier method in a class of nonlinear Schr¨odinger models which describe optical pulse propagation in nonlinear optical fibers.The corresponding relationship between these fundamental nonlinear excitations and modulation instability is established.Moreover,we reveal the important role of perturbation energy and relative phase in nonlinear excitation.In particular,we find a set of physical parameters(background frequency,perturbation frequency,perturbation energy,and relative phase)that can be used to determine the excitation conditions of fundamental nonlinear wave,and excitation conditions and phase diagram of fundamental nonlinear waves are obtained by these physical parameters.These results provide a theoretical basis for the experimental realization,controllable excitation and applications of fundamental nonlinear waves.The details are as follows:1.Generation mechanism of nonlinear waves and their phase diagram in background frequency and disturbance frequency spaceIn standard nonlinear Schr¨odinger system,the relationship between the fundamental nonlinear waves and the modulation stability is analyzed by Darboux transformation and linear stability analysis methods.And we explain the dynamics characteristics of the fundamental nonlinear waves on plane wave background by modulation instability.Particularly,the phase diagram of nonlinear excitations is established based on background frequency and perturbation frequency which determine the modulation instability characteristics.These results are helpful to understand the dynamic characteristics and generation mechanisms of fundamental nonlinear waves on a plane wave background.2.Role of perturbation energy in determining the excitation conditions of nonlinear wavesWe analyze the correspondence between the fundamental nonlinear waves and the modulation instability in the fourth-order nonlinear Schr¨odinger system.It is found that the anti-dark soliton and the nonrational W-shaped soliton exist in the modulation instability regime and the Kuznetsov-Ma breather could exists in the modulation stability regime,this result is contrary to the prediction of linear stability analysis.In order to understand this phenomenon,the parameter perturbation energy is introduced.It is found that the solitons excitation in the modulation instability regime is the result of the balance of perturbation energy and modulation instability gain.In modulation stable regime,the modulation instability gain is zero,and the perturbation energy of the Kuznetsov-Ma breather is greater than zero,gain and perturbation energy cannot be balanced,so Kuznetsov-Ma breather can exist in the modulation stability regime.These results reveal the important role that perturbation energy plays in nonlinear excitation.In particular,most of the nonlinear waves that coexist in the background frequency and perturbation frequency space can be distinguished by perturbation energy.3.Role of relative phase in determining the excitation conditions of nonlinear wavesAfter the introduction of the perturbation energy,the anti-dark soliton and the non-rational W-shaped soliton,the periodic wave and the W-shaped soliton train can still coexist in the three parameter spaces of background frequency,perturbation frequency and perturbation energy.In order to distinguish these nonlinear waves,we reconstruct the analytical solution of the fourthorder nonlinear Schr¨odinger system and introduce the relative phase parameters.It is found that the relative phase can distinguish between anti-dark and non-rational W-shaped solitons and periodic waves and W-shaped soliton train.Furthermore,the influence of relative phase on the excitation characteristics of different nonlinear waves is discussed.It is found that the relative phase affects the dynamic characteristics of solitons and periodic waves structures without affecting the excitation characteristics of rogue waves and breathers.These results reveal the important role of the relative phase in nonlinear wave excitations on plane waves background.Then,based on the background frequency,perturbation frequency,perturbation energy and relative phase,the excitation condition of the fundamental nonlinear wave on the plane wave background can be completely determined.4.Excitation conditions and phase diagrams of fundamental nonlinear waves on a plane wave backgroundExcitation conditions of fundamental nonlinear waves on plane wave background are obtained based on the background frequency,perturbation frequency,perturbation energy and relative phase which can completely determined excitation conditions of nonlinear waves.Furthermore,based on the theoretical analysis results,we numerically simulated the evolution of non-ideal initial states satisfying excitation conditions of different nonlinear wave.The results show that the nonlinear waves evolving from non-ideal initial states satisfying different conditions are consistent with the theoretical analysis.This result confirms that our theoretical analysis is reliable.In addition,in order to clearly show the relationship between the excitation conditions and background frequency,perturbation frequency,perturbation energy,and relative phase.We present the phase diagram of fundamental nonlinear wave excitations in the space of these parameters.According to our theoretical analysis results,the corresponding nonlinear wave structures can be obtained experimentally by inputting simple form initial state that satisfies the corresponding excitation conditions,this result facilitates the experimental excitation of nonlinear waves.In addition,based on the relationship between the characteristics of the nonlinear waves and the set of physical parameters,such as the relationship between the periodicity of the distribution direction and the perturbation frequency,the periodicity in propagation direction and perturbation energy,the peak and width of solitons and periodic waves and the relative phase,excitation characteristics of nonlinear waves can be controlled by adjusting the corresponding physical parameters,these results also provide a theoretical basis for applications of nonlinear waves.In particular,these results are independent of specific physical systems,so these results are not only applicable to nonlinear wave excitation in nonlinear optics,but also to nonlinear wave excitation in other systems,such as Bose-Einstein condensation,ferrimagnet,and plasma.