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

TL;DR

This paper introduces two new integrable (2+1)-dimensional nonlocal nonlinear Schrödinger equations with parity-time symmetry, deriving various solutions including breathers, rogue waves, lumps, and semi-rational states, and analyzing their dynamics.

Contribution

The paper presents two novel integrable (2+1)D nonlocal NLS equations with specific symmetries and derives multiple analytical solutions using Hirota's method, expanding the understanding of their dynamics.

Findings

Derived periodic, rogue, and semi-rational solutions for eq. 2.

Obtained breathers, lumps, and semi-rational solutions for eq. 3.

Analyzed solution dynamics with three-dimensional plots.

Abstract

Recently, an integrable system of coupled (2+1)-dimensional nonlinear Schrodinger (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,\textit{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 \textquotedblleft reverse-space" nonlocal NLS equation by means of a certain transformation, Lastly, main differences between solutions of eqs.2 and 3 are summarized.