Abstract
The generalized derivative nonlinear Schrödinger equation (g-DNLSE) is a very important integrable system in many fields of physics and applied mathematics. In this paper, we deduce a g-DNLSE which can be reduced to several famous equations. Firstly, for the g-DNLSE, we study the modulational instability which may be the reason for the formation of solitons and rogue waves. Secondly, we select the trivial seed solutions to obtain soliton solutions of the g-DNLSE through two Darboux matrices. Thirdly, using the generalized perturbation (1,N−1)-fold Darboux transformation, we derive smooth-positon solutions and rogue wave solutions of the g-DNLSE. In addition, we obtain the breather solutions and interaction solutions by the generalized perturbation (n,N−n)-fold Darboux transformation. It is worth noting that all of these solutions can change from a strong interaction to a weak interaction by choosing the parameters. This may also be one of the reasons why relevant wave structures presenting diversity. The dynamic behavior of rational soliton solution and multi-rogue wave solution is discussed through numerical simulation, and it may provide a theoretical basis for studying the physical mechanism of multi-rogue wave in optics.
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