Breathers,Rogue Waves And Breather-Rogue Waves On A Periodic Background For The Modified Nonlinear Schr(?)Dinger Equation

Nonlinear partial differential equations are often used to describe nonlinear phenomena in hydrodynamics,nonlinear optics and other fields.It has been found that there are many localized exact solutions for the integrable nonlinear evolution equations,including solitons,breathers,rogue waves and different combinations of these solutions.These exact solutions can explain the phenomena in oceanography,hydrodynamics,nonlinear optics and other fields,and promote the development of solitons and integrable systems to a certain extent.Therefore,the exact solutions of the integrable nonlinear evolution equations have become the hot research objects of many scholars.As an important physical model,the modified nonlinear Schr(?)dinger(MNLS)equation has a wide range of applications in many fields of engineering technology and natural science.For example,it can be used to describe the propagation of subpicosecond or femtosecond optical pulse in a monomodal fiber,and it can also be used to model the propagation of the modulated Alfvén wave in cold plasmas along the magnetic field.Based on the generalized Darboux transformation method,the breathers,rogue waves and breather-rogue waves on a periodic background are presented by constructing the odd-order solution of the MNLS equation.The breathers on the plane wave background are derived by constructing the even-order solution.The nonlinear dynamic behaviors of the breathers,rogue waves and breather-rogue waves are analyzed in detail by adjusting the parameters and drawing two-,three-dimensional images.The main research contents are as follows:(1)The symmetry properties are given based on the Lax pair derived from the Wadati-Konno-Ichikawa system.Then,the n-order Darboux transformation in the determinant form of the MNLS equation is constructed according to the definition and properties of the Darboux transformation.Furthermore,the explicit expression of the generalized Darboux transformation is obtained by means of Taylor expansion and limit techniques.(2)The localized exact solutions of the MNLS equation are explored according to the explicit expression of the potential formula of the generalized Darboux transformation.From the odd-order solution,three families of localized exact solutions are obtained on a periodic background,including the first-,second-and third-order rogue waves,two types of the breathers: Akhmediev breather and spatio-temporal breather,and three types of the breather-rogue waves:(a)the interaction solution between one rogue wave and one breather;(b)the interaction solution between a rogue wave and two breathers;(c)the interaction solution between a triangular second-order rogue wave and a breather.In addition,from the even-order solution,the breathers on the plane wave background are derived including the single breather solution,the interaction solution between double breathers and among three breathers.Finally,the nonlinear dynamics behaviors of these localized nonlinear waves are analyzed via the drawn images.