Research On The Rogue Wave Solutions And The Interaction Solutions Of Several Classes Of Nonlinear Soliton Equations

The exact solutions of soliton equations are an important research contents in soliton theory,which have been widely applied in many fields,such as nonlinear physics,chemistry,optics and so on.The research on exact solutions for soliton equations can not only explain many nonlinear phenomena in various fields more essentially,but also provide a powerful theoretical basis and application tools for engineering applications and other disciplines.There are many types of the exact solutions for nonlinear soliton equation,for instance,rational solutions,soliton solutions,breather solutions,rogue wave solutions,lump solutions and interaction solutions.In this paper,we will study the soliton solutions,breather solutions,rogue wave solutions and interaction solutions of several classes of nonlinear soliton equations by the Darboux transformation methods and the algebraic methods.The main contents are as follows:1.A new generalized nonlinear Schr ¨odinger(GNLS)equation is solved by the Darboux transformation methods.Based on the linear Lax pair,when we set the different seed solutions,the soliton solutions,breather solutions and rogue wave solutions of the GNLS equation are constructed by using the classical Darboux transformation method and the generalized Darboux transformation method,respectively.What's more,the dynamic properties of the first,second and third order exact solutions are illuminated by some interesting figures plotted with the help of Maple.2.The interaction solutions of the(2+1)-dimensional Boiti-Leon-Manna-Pempinelli(BLMP)equation and the(2+1)-dimensional Ito equation are studied by an direct algebraic method.This method is based on the Hirota bilinear equations,combining quadratic functions and exponential functions,and then constructing appropriate test functions,interaction solutions between lump solutions and multiple kinky soliton solutions of the(2+1)-dimensional BLMP equation as well as the interaction solutions between lump solutions and multiple strip soliton solutions of the(2+1)-dimensional Ito equation are investigated,respectively.The dynamical behaviors and the interaction collision process of the obtained solutions are also analysed in detail.It is found that the locations of the interaction collision between lumps and multiple kinky solitons are related to the symbols and values of the parameters in the exponential functions,the collision positions can be at the top,middle or bottom.3.Further generalization of the previous algebraic method is made to study the interactions among the different types of solutions of the(3+1)-dimensional Jimbo-Miwa(JM)equation and the dynamic behavior and collision process of the obtained interaction solutions are discussed in detail.The expressions of the test functions of this method include quadratic functions,hyperbolic functions and trigonometric functions,thus,the new interaction solutions between lump solutions,kinky soliton solutions and periodic solutions are obtained.