Higher-order rogue wave dynamics for a derivative nonlinear Schrödinger equation

The mixed Chen-Lee-Liu derivative nonlinear Schrödinger (CLL-NLS) equation can be considered as the simplest model to approximate the dynamics of weakly nonlinear and dispersive waves, taking into account the self-steepening effect (SSE). The latter effect arises as a higher-order correction of the nonlinear Schrödinger (NLS) equation, which is known to describe the dynamics of pulses in nonlinear fiber optics, and constitutes a fundamental part of the generalized NLS equation. Similar effects are described within the framework of the modified NLS equation, also referred to as the Dysthe equation, in hydrodynamics. In this work, we derive fundamental and higher-order solutions of the CLL-NLS equation by applying the Darboux transformation. Exact expressions of non-vanishing solitons at boundaries, breathers, and a hierarchy of rogue wave solutions are presented. In addition, we discuss the localization properties of such rogue waves, by characterizing their length and width. In particular, we describe how the localization properties of first-order NLS rogue waves can be modified by taking into account the SSE, presented in the CLL-NLS equation. This is illustrated by use of an analytical and a graphical method. The results may motivate similar analytical studies, extending the family of the reported rogue wave solutions as well as possible experiments in several nonlinear dispersive media, confirming these theoretical results.