The Darboux transformation of the Kundu-Eckhaus equation

We construct an analytical and explicit representation of the Darboux transformation (DT) for the Kundu-Eckhaus (KE) equation. Such solution and n-fold DT Tn are given in terms of determinants whose entries are expressed by the initial eigenfunctions and 'seed' solutions. Furthermore, the formulae for the higher order rogue wave (RW) solutions of the KE equation are also obtained by using the Taylor expansion with the use of degenerate eigenvalues lambda(2k-1) -> lambda(1) =-1/2a + beta c(2) + ic, k = 1, 2, 3,..., all these parameters will be defined latter. These solutions have a parameter beta, which denotes the strength of the non-Kerr (quintic) nonlinear and the self-frequency shift effects. We apply the contour line method to obtain analytical formulae of the length and width for the first-order RW solution of the KE equation, and then use it to study the impact of the beta on the RW solution. We observe two interesting results on localization characters of beta, such that if beta is increasing from a/2: (i) the length of the RW solution is increasing as well, but the width is decreasing; (ii) there exist a significant rotation of the RW along the clockwise direction. We also observe the oppositely varying trend if beta is increasing to a/2. We define an area of the RW solution and find that this area associated with c = 1 is invariant when a and beta are changing.