Dynamics Of Perturbations And Generation Of Nonlinear Waves On Continuous Wave Backgrounds

Nonlinear waves such as solitons and rogue waves are excitation structures with unique dynamic properties in nonlinear systems,which widely exist in physical systems such as nonlinear optics,Bose-Einstein condensates,water waves,and plasmas.In recent years,the study of nonlinear waves and their excitations has become one of the most important frontier issues in nonlinear science.Modulation instability is a fundamental property of nonlinear systems.In the beginning,it was only used to qualitatively explain the formation mechanism of waves such as rogue waves and breathers.Later,people used the linear stability analysis method to establish a quantitative correspondence between it and the wave generations,which helped the realization of controllable excitation of various nonlinear waves.Nevertheless,There are still some problems in the current research on perturbation dynamics and nonlinear wave generations on continuous waves: Numerous models describing real physical systems are non-integrable.How to realize the generations of more types of nonlinear waves in non-integrable models? There are a variety of nonlinear waves with large dynamical differences near the critical points of the modulation instability region and the stability region.How to understand the connection between these waves and the important role of critical points? The models describing some physical systems are high-dimensional.What are the unique behaviors of perturbation dynamics on continuous waves in high-dimensional models?In view of these problems,we consider various physical systems such as the nonlinear optical fiber systems with pure fourth-order dispersion,the dispersion-shifted optical fibers,and so on.The dynamic behavior of various forms of perturbations on continuous waves is analyzed and predicted,and the generation schemes of many nonlinear waves on continuous waves are given.These results not only broaden the scope of quantitative analysis of perturbation dynamics,provide a theoretical foundation for the experimental excitation and application of various nonlinear waves,but also deepen our understanding on the intrinsic relationship between modulation instability and wave generations.The main contents and innovations of the paper are as follows:(1)The linear stability analysis is modified to predict the dynamical characters of localized perturbations on continuous waves,including their periodicity and localization in the evolution direction,and their propagating velocity.According to the above character quantities and the initial condition’s form,an approximative solution is constructed to describe the short-distance dynamics of perturbations.(2)Benefited from the asymptotic form of modified linear stability analysis,the quantitative generation condition of one-and multi-peak solitons is presented in the model of dispersion-shifted fibers.Including them,the six kinds of nonlinear waves on continuous waves are generated controllably in numerical simulations.It is confirmed that the third-order dispersion has the effect of enriching the nonlinear wave species.(3)The way to generate high-order rogue waves is presented by the multi-Gaussian perturbations on continuous waves.Compared with other ways,it is easier and more convenient for preparation.In the nonlinear fiber model with anomalous dispersion,the order of rogue waves can be directly controlled by the number of Gaussian perturbations,and the high-order rogue waves can transform between different structures by adjusting initial parameters.In terms of generation mechanism,the generation of high-order rogue waves in this way is the combined results of modulation instability and the interaction between perturbations.(4)The exactly transforming relation between nonlinear waves near the critical points of modulation instability and stability is presented.In the numerical simulations,these waves are generated by a localized perturbation and their transition is realized.Meanwhile,the perturbation’s amplitude is found to have a non-negligible influence on the gain value of modulation instability and the position of critical points,which cannot be analyzed by the linear stability analysis.(5)The modified linear stability analysis is extended into a two-dimensional form for predicting the splitting and reversal rotation of ring-shaped perturbation on homogeneous BoseEinstein condensates.It is demonstrated that the existence of homogeneous background is the direct inducement of reversal-rotation mode.