Study On Some Problems Of Localized Waves And Modulation Instability

Based on three types of symbolic computing software platforms,Maple,Mathematica and Matlab,this paper studies the localized waves,interaction solutions and the corresponding dynamical characteristics for several types of nonlinear integrable systems using modulation instability analysis,generalized Darboux transform and Hirota bilinear method.It mainly includes three aspects: modulation instability analysis,higher-order rogue wave solutions and state transitions for the higher-order nonlinear Schr ¨odinger equation;infinitely many conservation laws,modulation instability analysis and interaction solutions among different types of localized waves for the generalized coupled Fokas-Lenells equation;higher-order localized waves and interaction solutions for the higher-dimensional nonlinear systems.The details of the research are as follows.In chapter 1,the relevant background and research status of localized waves,modulation instability analysis,classical Darboux transform method,generalized Darboux transform method and Hirota bilinear method for nonlinear systems are introduced emphatically.The topic selection and main results of this paper are also briefly elaborated.In chapter 2,the modulation instability distribution characteristics of the infiniteorder nonlinear Schr ¨odinger hierarchy are studied,and the intrinsic connection between the division of modulation instability region and higher-order dispersion term is obtained.The higher-order rogue waves of the sixth-order nonlinear Schr ¨odinger equation are constructed by the generalized Darboux transform method.The influence of higher-order dispersion term on the spectrum,wave width and amplitude of rogue wave is analyzed.Furthermore,the conditions for the state transition between rogue wave and W-type soliton,and the corresponding spectrum are given.In particular,the consistency of the results obtained from the spectral analysis and modulation instability analysis is verified.In chapter 3,the modulation instability distribution characteristics of the generalized coupled Fokas-Lenells equation are studied.It is found that the gain function is intrinsically related to the background amplitude,background frequency,disturbance frequency and physical parameters,and the area of modulation instability region decreases with the increase of background amplitude.Then,from the Riccati-type equations satisfied by spectral problems,infinite conservation laws of the equation are constructed.Finally,the higher-order rogue wave and its interaction with bright and dark solitons or breathers are obtained by the generalized Darboux transform.The structures of these localized waves are parameter-controllable.In chapter 4,the localized waves and interaction solutions of the 3+1-dimensional Hirota bilinear equation,3+1-dimensional generalized Jimbo-Miwa equation and 3+1-dimensional nonlinear evolution equation are studied.Firstly,based on Hirota bilinear method combined with long wave limit and parametric complex techniques,the first-order breather,lump and line rogue wave of the 3+1-dimensional Hirota bilinear equation are obtained.The phase shift,propagation direction,shape and energy of the corresponding solutions are parameter-controllable.Then,the higher-order localized waves and interaction solutions of the 3+1-dimensional generalized Jimbo-Miwa equation and a nonlinear evolution equation are investigated,including the second-order breather,second-order line rogue wave,and the pairwise interaction of soliton,breather,lump and rogue wave solutions.Their dynamics are vividly portrayed with images.In chapter 5,the bright and dark higher-order rational solutions and N-wave resonance solutions of the 3+1-dimensional Kudryashov-Sinelshchikov equation are constructed by Hirota bilinear method.By introducing a polynomial function,both bright and dark structures of the rogue wave type and W-type rational solutions are obtained.By analyzing the first-to third-order rational solutions,it is found that the corresponding relation between the order of rational solution and the extremum.Two polynomial functions are introduced to combine with the first polynomial,we can obtain not only the above-mentioned solutions,but also the higher-order rogue waves or W-type solutions,which can be split into corresponding multiple first-order rational solutions through parameter regulation.Under specific constraints and dispersion relations,N-wave resonance solutions of the equation are obtained,including bright-fusion,dark-fusion,bright-fission and dark-fission resonance solutions.In chapter 6,a brief summary of the whole paper is presented,and further perspectives on the subsequent research work are discussed.