Clustering Transformation Of Derivative Nonlinear Schr (o) Dinger Equation And Its Application

The second-type derivative nonlinear Schr¨odinger (DNLSII) equation was introduced as anintegrable model in1979. Very recently, the DNLSII equation has been shown by an experimentto be a model of the evolution of optical pulses involving self-steepening without concomitantself-phase-modulation. In this paper the n-fold Darboux transformation (DT) Tnof the coupledDNLSII equations is constructed in terms of determinants. Comparing with the usual DT of thesoliton equations, this kind of DT is unusual because Tnincludes complicated integrals of seedsolutions in the process of iteration. By a tedious analysis, these integrals are eliminated in Tnexcept the integral of the seed solution. Moreover, this Tnis reduced to the DT of the DNLSIIequation under a reduction condition. As applications of Tn, the explicit expressions of soliton,rational soliton, breather, rogue wave and multi-rogue wave solutions for the DNLSII equationare displayed.