The Darboux transformation and higher-order rogue wave modes for a derivative nonlinear Schr\"odinger equation

We derive the n-th order solution of the mixed Chen-Lee-Liu derivative nonlinear Schrodinger equation (CLL-NLS) by applying the Darboux transformation (DT). Such solutions together with the n-fold DT, represented by Tn, are given in terms of determinant representation, whose entries are expressed by eigenfunctions associated with the initial “seed” solutions. This kind of DT technique is not common, since Tn is related to an overall factor expressed by integrals of previous potentials in the procedure of iteration. As next step, we annihilate these integrals in the overall factor of Tn, except the only one depending on the initial “seed” solution, which can be easily calculated under the reduction condition. Furthermore, the formulae for higher-order rogue wave solutions of the CLL-NLS are obtained according to the Taylor expansion, evaluated at a specific eigenvalue. As possible applications, the expressions and figures of non-vanishing boundary solitons, breathers and a hierarchy of rogue wave solutions are presented. In addition, we discuss the localization characters of rogue wave by defining their length and width. In particular, we show that these localization characters of the first-order rogue wave can be changed by the self-steepening effect in the CLL-NLS by use of an analytical and a graphical method.

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