Study On The Transmission Characteristics Of Breather Solution In Hierarchy Nonlinear Schr?dinger Equations

Along with the continuous development of modern science and technology and the deepening of research,people pay more and more attention to the nonlinear effects in the system.The nonlinear effect of the system is the basis for uncovering many complicated phenomena.And the nonlinear Schr?dinger equation is one of the core models for exploring the nonlinear effects of the system.The solutions of the nonlinear Schr?dinger equation include soliton solutions,breather solutions,rouge wave solutions and different combinations of these solutions.The nonlinear phenomena reflected by these solutions are real in nature,and they have an important impact on our production and life.As research continues to deepen,researchers have discovered these phenomena in other scientific fields.Therefore,it is necessary to study the dynamic characteristics of these nonlinear phenomena,which will have certain theoretical guiding significance for future scientific research and production life.The research in this paper mainly includes the following parts:(1)The development history of nonlinear Schr?dinger equations,solitons,breathers and rouge waves is briefly described and the theoretical results of researchers in these fields in recent years are listed.In addition,the practical application of the research results of nonlinear phenomena in production and life is also introduced.(2)Based on the standard nonlinear Schr?dinger equation,the first-order and high-order breather solutions of the equation are obtained,and the dynamic characteristics of collision and separation,degenerate state and parallel transmission are studied.In addition,when the characteristic frequency of the breather tends to zero,the rouge wave limit of the breather solution can be obtained.The study found that the amplitude of the rouge waves,the number of secondary peaks,and the order of the central rouge wave after the rouge wave split,the number of side peaks are related to thesolution's order N.(3)Based on the extended nonlinear Schr?dinger equation,the first-order soliton solution of the equation is obtained,and the conversion relationship between the breather and the soliton of the odd-order equation and the odd-even equation is studied.The results show that when the group velocity and phase velocity are not equal,the real and imaginary parts of the soliton have breather characteristics.When the group velocity is equal to the phase velocity,the breather characteristics of the real and imaginary parts of the soliton disappear,the breather transforms into solitons,and the real part and amplitude of the soliton are evenly symmetric,and the imaginary part is oddly symmetric.(4)Based on the nonlinear Schr?dinger equation containing both second-order and fourth-order linear and nonlinear terms,the first-order breather solution is obtained,and the transition conditions of the breather to different types of solitons,breathers and periodic waves are studied.In addition,with the aid of the recurrence of the Darboux transformation,the second-order breather solution of the equation is obtained,the collision characteristics between breather with breather,breather with soliton,breather with periodic wave are studied,and the dynamic characteristics of parallel transmission and degenerate state of the breather were studied.