Multi-rogue Waves And Rational Solutions Of The Coupled Nonlinear Schr(?)dinger Equations

Rogue wave was first introduced by Draper who is a famous oceanographer in1970s and it’s a research hotspot of ocean researchers and mathematicians recently. In this dissertation, We mainly research the rational solutions and rogue waves of the coupled nonlinear Schrodinger (CNLS) equations as follow iut+uxx+2(｜u｜2+｜v｜)u=0, ivt+vxx+2(｜u｜2+｜v｜2)v=0.The CNLS equations are an important model reflecting deep water waves, which describes the law when two nonlinear waves propagate in a nonlinear diffusion medium. The existence of rational solutions to the CNLS equations may explains the reason why rogue waves appear in oceans.The modified Darboux transformation method is applied to the CNLS equations. By using the iterative algorithm of the Darboux transformation and undetermined coefficients mothod, the multi-rogue wave solutions of CNLS equations are generated from the plane wave solution. The hierarchies of first-, second-, third-and forth-order rational solutions with free parameters are explicitly presented. Some basic properties of multi-rogue waves and their collision structures are studied on the basis of the solutions obtained. Besides, some important properties of rogue waves obtained in the second chapter are deeply investigated, include the recursion about orders of rational solutions, the decomposition of rogue waves and the conversion of rational solutions. What’s more, the relation of rational solutions between N-CNLS equations and CNLS equations is explained, and we also give the exact first-, second-, third-and forth-order rational solutions of the N-CNLS equations.Our results might provide useful information for investigating the dynamics of multi-rogue waves in the deep oceans, nonlinear optical fibers and other fields.