Rogue waves of the nonlocal Davey–Stewartson I equation

Abstract

Recently, Fokas presented a nonlocal Davey–Stewartson I (DSI) equation (Fokas 2016 Nonlinearity 29 319–24), which is a two-spatial dimensional analogue of the nonlocal nonlinear Schrödinger (NLS) equation (Ablowitz and Musslimani 2013 Phys. Rev. Lett. 110 064105), involving a self-induced parity-time-symmetric potential. For this equation, high-order periodic line waves and line breathers are derived by employing the bilinear method. The long wave limit of these periodic solutions yields two kinds of fundamental rogue waves, namely, kink-shaped and W-shaped line rogue waves. The interaction of fundamental line rogue waves generate higher-order rogue waves, which have several interesting patterns with different curvy profiles. Furthermore, semi-rational solutions are constructed, which are line rogue waves on a background of periodic line waves. Finally, two particular solutions of the nonlocal NLS equation, namely, a first-order rogue wave and a semi-rational solution are obtained as reductions of the corresponding solutions of the nonlocal DSI equation.