The second-type derivative nonlinear Schrodinger (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-steep
ening without concomitant self-phase-modulation. In this paper the $n$-fold Darboux transformation (DT) $T_n$ 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 $T_n$ includes complicated integrals of seed solutions in the process of iteration. By a tedious analysis, these integrals are eliminated in $T_n$ except the integral of the seed solution. Moreover, this $T_n$ is reduced to the DT of the DNLSII equation under a reduction condition. As applications of $T_n$, the explicit expressions of soliton, rational soliton, breather, rogue wave and multi-rogue wave solutions for the DNLSII equation are displayed.
We explore the form of rogue wave solutions in a select set of case examples of nonlinear Schrodinger equations with variable coefficients. We focus on systems with constant dispersion, and present three different models that describe atomic Bose-Ein
stein condensates in different experimentally relevant settings. For these models, we identify exact rogue wave solutions. Our analytical findings are corroborated by direct numerical integration of the original equations, performed by two different schemes. Very good agreement between numerical results and analytical predictions for the emergence of the rogue waves is identified. Additionally, the nontrivial fate of small numerically induced perturbations to the exact rogue wave solutions is also discussed.
We introduce a mechanism for generating higher order rogue waves (HRWs) of the nonlinear Schrodinger(NLS) equation: the progressive fusion and fission of $n$ degenerate breathers associated with a critical eigenvalue $lambda_0$, creates an order $n$
HRW. By adjusting the relative phase of the breathers at the interacting area, it is possible to obtain different types of HRWs. The value $lambda_0$ is a zero point of the eigenfunction of the Lax pair of the NLS equation and it corresponds to the limit of the period of the breather tending to infinity. By employing this mechanism we prove two conjectures regarding the total number of peaks, as well as a decomposition rule in the circular pattern of an order $n$ HRW.
