Research On Darboux Transformation,soliton Solutions And Rogue Wave Solutions Of Coupled MKdV Equations

Darboux transformation is a powerful and effective method for solving exact math-ematical physics equations.This paper studies the mKdV equations in two coupling forms,and mainly works in two aspects:Darboux transformation and exact solutions of two-component complex mKdV equations,coupled complex mKdV equations are con-structed;the dynamic behaviors of exact solutions are analyzed,including structures and interactions of solutions,etc.The paper is organized as followsIn the first chapter,we introduce the soliton theory and the research status of the mKdV equationIn the second chapter,firstly,we give the Lax pair and construct the binary Darboux transformation of two-component complex mKdV equations.Then,we present the gen-eral N-soliton solutions.Further,we investigate the dynamic behaviors of bright-bright solitons,bright-dark solitons,bright-W-shaped solitons,breathers,periodic solutionsIn the third chapter,we study the coupled complex mKdV equations associated with the 3 × 3 spectral problem.Based on modulation instability,we show that two types of N-th order rational rogue wave solutions.The rich dynamics of the rational and semi-rational rogue wave solutions are revealed.