Research On Localized-Wave Solutions For The Discrete Nonlinear Equations

As one of the most popular topics in the field of nonlinear science,rogue wave was first used by oceanographer to describe the strange and large amplitude waves that suddenly appeared in the ocean.Breather solutions of certain nonlinear equations are presently well accepted as potential prototypes for the notorious rogue waves in the ocean and other fields of physics.This dissertation is devoted to studying the dynamic behaviors of breather and rogue wave solutions for the discrete nonlinear Schr?dinger(NLS)equation.In the first part,first,by constructing the Darboux transformation of the discrete NLS equation,this paper obtain the first-order breather solution and give the parameter conditions for generating Kuznetsov-Ma breathers,Akhmediev breathers,breathers with a number of bunches and spatio-temporal breathers;Second,this paper analyzes the dynamics of different types of breather-breather and breather-rogue wave interactions;Finally,the dynamics of the degenerate second-order breather solutions are studied by constructing the generalized Darboux transformation of the discrete NLS equation.In the second part,the generation mechanism of rogue waves are studied,first,the one breather solution on a new non-vanishing background of the discrete NLS equation is derived.Through the explicit expressions of one breather’s group and phase velocities,the parameter conditions for the jumps appearing in the velocities are given,which would be one type of generation mechanism of rogue waves.Finally,in order to verify such conclusion,the rogue wave solution is constructed through the generalized Darboux transformation under the conditions of the velocity jumps,which confirms that the generation of rogue waves is related to the discontinuity of velocity.