Darboux transformation and general soliton solutions for the reverse space–time nonlocal short pulse equation

Abstract

We construct the Darboux transformation for the reverse space–time (RST) nonlocal short pulse equation by considering a nonlocal symmetry reduction of the Ablowitz–Kaup–Newell–Segur spectral problem and a covariant hodograph transformation. Some essential differences between the RST nonlocal models and the usual (local) ones are demonstrated by means of a series of explicit solutions and the solving process. The multi-bright soliton solution on vanishing background, as well as the multi-dark soliton, multi-breather and higher-order rogue wave solutions corresponding to nonvanishing background of the RST nonlocal defocusing short pulse equation are derived through the Darboux transformation. The classification and dynamics of these explicit solutions with loop-, cuspon- and smooth-type are presented. The asymptotic analysis is rigorously performed for the two-bright soliton, two-dark soliton and two-breather solutions. Some novel wave patterns such as the double-loop and mixed loop-cuspon rogue waves which are different from their regular counterparts are shown. Other types of singular solutions along certain space–time lines are also obtained.