Darboux Transformations And Exact Solutions For Two Nonlinear Evolution Models

In this thesis,we investigate two nonlinear evolution models including the Hirota Self-Induced Transparency(Hirota-SIT)model and the higher-order coupled nonlinear Schr(?)dinger(NLS)model.By using the Darboux transformation(DT)method and sym-bolic computation software Mathematica,soliton,breather,higher-order rogue waves solutions and breather-to-soliton conversions are investigated in detail.First of all,the basic knowledge of three kinds of nonlinear localized waves,which are solitons,breathers and rogue waves,is introduced briefly.The main idea of DT method and the most primitive DT in soliton theory are expounded.At the same time,we show the process of using DT method to find the exact solutions by an example.Secondly,a class of multi-coupled Hirota Self-Induced Transparency model is s-tudied.Via the Lax pair and the determinant form of N-fold DT,the soliton solutions on the zero background are generated,and the effects of higher-order terms on the propagation characteristics of breathers on the plane wave background are discussed.By combining Taylor expansion method with DT method,the higher-order rogue waves of the model are constructed directly,including three typical structures:fundamental structure,triangular structure and circular structure.Based on the breather solutions,the conditions of breather-to-soliton and the interactions among various nonlinear waves are obtained.At the same time,figures of the solutions are plotted and transmission dynamics are analyzed.The Riccati equation is derived by virtue of the Lax pair and the infinite conservation laws of the model are obtained according to the compatibility condition.Finally,the higher-order coupled nonlinear Schr(?)dinger model is investigated.By combining the DT method with the Taylor expansion method,the higher-order rogue waves of the model on the plane-wave background are discussed.Furthermore,by controlling the basic parameters₅),many interesting structures of rogue waves are ob-tained.Under the condition of satisfying the breather-to-soliton relations,the evolution characteristics of anti-dark solitons,multi-peak solitons,periodic waves and W-shaped solitons are analyzed.