One day workshop in honor of Maxim Pavlov's 60th birthday. A relative of the NLS equation revisited

As other hierarchies, the AKNS hierarchy is known to possess a "negative" part (also see Kamchatnov and Pavlov, Phys. Lett. A 301 (2002) 269, arXiv:nlin/0208025). Via the NLS reduction, the first "negative" flow becomes a system for two functions. By elimination of one of them, the system implies a fairly simple PDE with a mixed third order derivative term. Whereas the reduction of this PDE to real dependent variable is completely integrable (i.e., a Lax pair exists), this is most likely not true for complex variable (S. Sakovich, arXiv:2205.09538). In between is the reduction given by the first negative flow of the NLS hierarchy. We present a vectorial binary Darboux transformation for the latter and exploit it to (re-) derive several types of (multi-) soliton solutions of the PDE, including rogue waves. It is derived from a general result of bidifferential calculus. Further properties of the PDE will be discussed. This work is mainly based on arXiv:2202.04512, to appear in Journal of Physics A: Mathematical and Theoretical.