The Determinant Representation Of The Degenerate Darboux Transformation And Its Application

In the paper the determinant representation of the degenerate Darboux transfor-mation and its applications are studied. In the second chapter, for the complex mKdV equation, the derivation of the determinant represenation Tn of order-n Darboux trans-formation is provided with details, and then the determinant represenation T’n of the degenerate Darboux transformation is obtained by using the degenerate limit λiâ†'λ1 of Tn. In the third chapter, an explicit expression of positon solutions for the com-plex mKdV equation is constructed by using degenerate Darboux transformation. This positon solution is decomposed into a summation of a number of solitons by means of the decomposition of the modulus square. The trajectory and the "phase shift" of the position after collision are discussed approxiamtely. There is a great difference comparing with the singular positon solution of the real mKdV equation. In chapter four, the determinant represenation of the order-n rogue wave solution for the higher-order nonlinear Schrodinger equation is constructed by using the degenerate Darboux transformation.Aanalysising the profiles from the second to the fifth order rogue wave solutions of this equation, we find three basic patterns:fundamental pattern, trianglar pattern and circular pattem.These solutions have two parameters a and β which denote the contribution of the higher-order terms (dispersions and nonlinear effects). In order to analysis the localized properties of the first-order rogue wave solution, we provide the definition of the length and width by the contour line method. According to figures of the higher-order rogue wave solutions, we observe the significant compression ef-fect along t direction. In the fifth chapter, we introduce a nonlocal LPD equation and its Lax pair. Then order-n Darboux transformation and the first-order degenerate Dar-boux transformation of this equation is constructed. By using the degenerate Darboux transformation, a rational soliton solution of the nonlocal LPD equation is constructed explicitly, and the classification of the solution and the time evolution of the corre-sponding PT-symmetrical potential function are studied.