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Darboux Transformation of the Second-Type Derivative Nonlinear Schrödinger Equation

Research paper by Yongshuai Zhang, Lijuan Guo, Jingsong He, Zixiang Zhou

Indexed on: 30 Apr '15Published on: 30 Apr '15Published in: Letters in Mathematical Physics



Abstract

The second-type derivative nonlinear Schrödinger (DNLSII) equation was introduced as an integrable model in 1979. Very recently, the DNLSII equation has been shown by an experiment to be a model of the evolution of optical pulses involving self-steepening without concomitant self-phase-modulation. In this paper, the n-fold Darboux transformation (DT) Tn of the coupled DNLSII equations is constructed in terms of determinants. Comparing with the usual DT of the soliton equations, this kind of DT is unusual because Tn includes complicated integrals of seed solutions in the process of iteration. By a tedious analysis, these integrals are eliminated in Tn except the integral of the seed solution. Moreover, this Tn is reduced to the DT of the DNLSII equation 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 equation are displayed.