Two (2+1)-dimensional integrable nonlocal nonlinear Schrödinger equations: Breather, rational and semi-rational solutions

Recently, an integrable system of coupled (2+1)-dimensional nonlinear Schrödinger (NLS) equations was introduced by Fokas (Eq. (18) in Nonlinearity 29, 319324 (2016)). Following this pattern, two integrable equations [Eqs. (2) and (3)] with specific parity-time symmetry are introduced here, under different reduction conditions. For Eq. (2), two kinds of periodic solutions are obtained analytically by means of the Hirota's bilinear method. In the long-wave limit, the two periodic solutions go over into rogue waves (RWs) and semi-rational solutions, respectively. The RWs have a line shape, while the semi-rational states represent RWs built on top of the background of periodic line waves. Similarly, semi-rational solutions consisting of a line RW and line breather are derived. For Eq. (3), three kinds of analytical solutions,viz., breathers, lumps and semi-rational solutions, representing lumps, periodic line waves and breathers are obtained, using the Hirota method. Their dynamics are analyzed and demonstrated by means of three-dimensional plots. It is also worthy to note that Eq. (2) can reduce to a (1+1)-dimensional “reverse-space” nonlocal NLS equation by means of a certain transformation, Lastly, main differences between solutions of Eqs. (2) and (3) are summarized.