Nonlocal Symmetry And Rogue Wave Of Several Nonlinear Equations

In recent years,with the rapid development of science and technology,nonlinear science has become a new subject,and nonlinear equations play a more and more important role in describing the complex physical phenomena in various fields of science.In this paper,we have studied some problems of nonlinear equations by computer algebra,such as nonlocal symmetry,exact solution,Hirota bilinear method,rogue wave,and party-time-symmetry.The main contents are as follows:In Chapter 1,we briefly introduce the research background and development of soliton theory,and some methods to solve the exact solutions of nonlinear partial differential equations.In Chapter 2,for the(2+1)-dimensional Gardner equation,the truncated Painlev?e method is developed to obtain the nonlocal residual symmetry and B¨acklund transformation.And,the symmetry group transformation can be compute from the extended system.Moreover,(2+1)-dimensional Gardner equation is proved to be consistent Riccati expansion(CRE)solvable.With the help of the Riccati equation and the CRE method,we obtain the soliton-cnoidal wave interaction solution of the equation.In Chapter 3,for the Benjamin Ono equation,the Hirota bilinear method and long wave limit method are applied to obtain the breathers and the rogue wave solutions.Bright and dark rogue waves exist in the Benjamin Ono equation,and their typical dynamics are analyzed and illustrated.The semi-rational solutions possessing rogue waves and solitons are also obtained,and demonstrated by the three dimensional figures.Furthermore,the hybrid of rogue waves and breathers solutions are also found in the Benjamin Ono equation.In Chapter 4,for the(2+1)dimension nonlocal nonlinear Schr¨odinger(NLS)equation with the self-induced parity-time( )-symmetric potential is introduced.General periodic solutions are derived by the bilinear method,these periodic solutions behave as growing and decaying periodic line waves arising from the constant background and decaying back to the constant background again.By taking long wave limits of the obtained periodic solutions,rogue waves are obtained.In Chapter 5,we present the conclusion and prospect of this paper.