Rational and semi-rational solutions to the asymmetric Nizhnik–Novikov–Veselov system

Abstract

In this paper, we use the binary Darboux transformation technique to derive an uniform mathematical expression of all kinds of solutions to the (2 + 1)-dimensional asymmetric Nizhnik–Novikov–Veselov system. For the same seeding solution, a family of eigenfunctions associated with the same eigenvalue is obtained, which is used to construct rational and semi-rational solutions. Interestingly, there exists a category of localized rational solutions that show nontrivial interaction scenarios, namely the pulses undergo a scattering process after the head-on collision. The semi-rational solutions are characterized by two generic evolution scenarios: fission and fusion processes. We also find a subclass of dark rogue waves, namely trains of line solitons that evolve to significant strongly localized transient waves.