Rogue Wave Of The AKNS System And Its Applications In Fluid Mechanics

Rogue wave is the wave that suddenly appears in the ocean even under good marine condi-tion,which exists momently and has tremendous amplitude.People usually say that "this wave appearing from nowhere and disappearing without a trace",which means that it can hardly be traced,not to mention that to be predictable.Obviously,rogue wave is a great threat of the ships and facilities in the ocean and has made several serious marine accidents in the past.As such,it is an important research topic in the area of construction of ships,ocean waves and ocean engineering.In 2007,some physicists announced that they have observed the rogue wave in an optical system.Their research vastly pushed forward the study of the rogue waves in other fields including optics,plasma and fluid,just to name a few.In addition,the research of rogue wave should begin with hydrodynamics equations.Thus equations of the fluid me-chanics become naturally one kind of important topics in the research.Besides,the nonlinear Schrodinger equation(NLS)is the first equation which has been found the possession of the rogue wave.In last decade,the rogue waves of the NLS have been observed in different experiments of optics and fluid mechanics.These important results urge us,in this thesis,use the localized spatio-temporal solutions of the Ablowitz-Kaup-Newell-Segur(AKNS)system as the model to research the rogue waves in fluid mechanics and to provide their carrier waves correspondingly.As such,we firstly investigate the first nontrivial flow,which is the nonlinear Schrodinger equation,from AKNS to get the conditions,parameters,timing of the occurrence of the wave to provide the theoretical basis for the prediction of rogue wave in general.For this consideration,a b-positon solution,namely the degenerate higher-order breathers,is introduced to the NLS equation,and a determinant representation of the order-n b-positon is also given.Then,we provide the double degenerative process of the breather to the rogue wave and suggest that the higher-order rogue wave can be measured through the measurement of profiles of the b-positon at the interactional area.Furthermore,we strictly proved that the height of an order-n rogue wave of the NLS equation under the fundamental pattern is h =(2n + 1)c,where c denotes the height of the asymptotical plane of the rogue wave.Many physical phenomena require the establishment of a wave motion model with two or more components to illustrate different modes,frequencies and polarization.Also,only multi-component systems can theoretically explain the energy exchange between multi-fields.As such,we use NLS equation(AKNS's second-order flow)with two or three components and modified Korteweg-de Vries equation(mKdV,which is AKNS's third-order flow)as examples to analyze multi-component AKNS system.In order to study the generating conditions and parameters of multi-component NLS and mKdV rogue wave systematically,the Darboux transformation of the multi-component NLS and mKdV equations have been improved from these two aspects:1)introduction a phase translation matrix T0;2)the decomposition rule of the expansion for the eigenfunctions at ?0,where AO is a critical point transferring a breather to a rational(or semi-rational)solution.Basing on the improvement of the Darboux transformation,the determinant of the rational and semi-rational solution of the order-n NLS equations with two components are given,where the rational solution is the solution of rogue wave.Then,some new patterns of rogue waves are illustrated with numerical results.In this thesis,the theory and method of constructing and solving two components are extended to the three components and third-order flow,and some new models are obtained.By the first improved part of the Darboux transformation,when using the Darboux trans-formation to construct the multi-component NLS and mKdV equations of rogue wave,this paper presented the algebraic conditions in the constructions of the rogue waves systematically.These algebraic conditions directly or indirectly determine the physical parameters of the rogue wave generation and control in different physical systems,so it plays an important role in the physical experiment to realize rogue waves.According to the second improved part of the Darboux transformation,we find the law of the order of the polynomials in the denominators for the rational solutions of the multi-component NLS and mKdV equations.This law can sum up the patten of the degree of order-n polynomial with multiple component,which also contains the case with the single component.