The Rogue Wave Solutions For Two Kinds Of Nonlinear Schr(?)dinger Type Equation

This work analyze two kinds of nonlinear Schr¨odinger type equation, in other words, non-linear Schro¨dinger equation and Gerdjikov-Ivanov equation. Gerdjikov-Ivanov equation whichis a higher order nonlinear equation is regarded as a natural extension for nonlinear Schr¨odingerequation.With regarded to nonlinear Schro¨dinger equation, in the special condition, two compo-nents of Manakov system are proportionable. we give determinant expressions for Darbouxtransformation of the Manakov system. As special case, we generate a family of exact andnon-symmetric rogue wave solutions of the nonlinear Schr¨odinger equation up to third-order,localized in both space and time. They are just symmetrical with respect to the t-axis and notsymmetrical to the x-axis. Furthermore, the amplitude of the non-symmetric solution is smallerthan the corresponding symmetrical one. Besides, the main peak is not located at (0,0) on the(x, t) plane.With regarded to Gerdjikov-Ivanov equation, We construct higher order rogue wave solu-tions for the Gerdjikov-Ivanov equation explicitly in term of determinant expression. Dynamicsof both soliton and non-soliton solutions are discussed. A family of solutions with distinctstructures are presented, which are new to the Gerdjikov-Ivanov equation.