Theory And Application Of Robust Inverse Scattering And KP Hierarchy Reduction

In this thesis,we mainly discuss some special solutions to the nonlinear equation via the robust inverse scattering transformation,Kadomtsev-Petviashvili(KP)theory and Hirota bilinear method,including three aspects,the first one is giving the rogue wave of the generalized nonlinear Schr¨odinger(NLS)with nonzero boundary condition by using the robust inverse scattering method,the second one is getting the rogue wave of NLSBoussinesq equation with KP hierarchy reduction method and the last one is to present the rogue wave to some high dimensional equation,especially the(2+1)-dimensional equation.Chapter 1 is an introduction about the research background and the development of some relative method.In this thesis,we focus on studying three method,Robust inverse scattering transformation method,KP-hierarchy reduction method and Hirota bilinear method.Then we give the main work of this thesis.Chapter 2 obtains the high-order rogue wave to the generalized NLS equation with robust inverse scattering transformation and the corresponding Riemann-Hilbert method.Compared with the traditional inverse scattering method,the key point of robust inverse scattering method is giving two fundamental solutions of Lax pair and changes the jump matrix from the real axis to a big circle.Then we give the corresponding Riemann-Hilbert problem and obtain the breather and high-order rogue wave.Furthermore,according to the new Riemann-Hilbert problem,we also give the asymptotic of the infinity order rogue wave.During this study,we use a scale transformation and give a new dispersion,then give three cases of asymptotic study.Chapter 3 gives the high-order rogue wave to the NLS-Boussinesq equation with the KP-hierarchy reduction method.As we all know,the Lax pair of Boussinesq equation is3×3,and it is a 3-cycle reduction from KP hierarchy.So the reduction of NLS-Boussinesq equation is difficult.In this chapter,we complete this reduction by a theorem successfully.Then give three kinds of rogue wave: bright rogue wave,four-peal rogue wave,dark rogue wave and their corresponding high-order rogue waves.On the other hand,we give two kinds of breathers to Mel'nikov system by choosing different parameters p and q.These two breathers present different locality and periodicity.Furthermore,by long wave limit method,the lump solution and line rogue wave are given,then the interaction phenomena among all of the obtained solutions are described.This novel pattern is a fantastic phenomenon for the Mel'nikov system.Chapter 4 mainly discusses the localized rogue wave to some high-dimensional equations,such as generalized KP equation,(2+1)-dimensional KdV equation,reduced(3+1)-dimensional Jimbo-Miwa equation with Hirota bilinear method.The key point of this method is giving a new ansatz combining a quadratic function and hyperbolic cosine function.Compared to the line rogue wave in Chapter 3,this rogue wave is localized in the(x,y)-plane from all directions,which is aroused by a pair of resonance soliton and lump solution.Chapter 5 is the summary to the whole thesis and gives the further study.