The nth-Darboux Transformation and Explicit Solutions of the PT-Symmetry Second-Type Derivative Nonlinear Schrödinger Equation

Abstract

AbstractNon-local problems have become one of the research hotspots in recent years since Ablowitz–Musslimani constructed an integrable non-local nonlinear $$\hbox {Schr}\ddot{\mathrm{o}}\hbox {dinger}$$ Schr o ¨ dinger (NLS) equation in 2013. In this paper, we first derive the PT-symmetry second derivative nonlinear $$\hbox {Schr}\ddot{\mathrm{o}}\hbox {dinger}$$ Schr o ¨ dinger (PT-DNLSII) equation. Then we present the nth-Darboux transformation (DT) of the PT-DNLSII equation. As applications, starting from the zero seed solution and non-zero periodic seed solution, the explicit expressions of the multi-soliton solutions, bright soliton solution, dark soliton solution and breather soliton solution of the PT-DNLSII equation are worked out.