# Rational and semi-rational solutions of the nonlocal Davey-Stewartson equations

## TL;DR

This paper derives explicit breather, rational, and semi-rational solutions for nonlocal Davey-Stewartson equations using the bilinear method, revealing complex localized wave phenomena in these nonlocal integrable systems.

## Contribution

It introduces new solution types for nonlocal DS equations, including semi-rational solutions combining lumps, breathers, and line waves, expanding understanding of their wave dynamics.

## Key findings

- Periodic breathers in x-direction for x-nonlocal DS equations
- Line rogue waves in nonlocal DSII that appear and vanish quickly
- Semi-rational solutions show interactions of rogue waves and periodic waves

## Abstract

In this paper, the partially party-time ($PT$) symmetric nonlocal Davey-Stewartson (DS) equations with respect to $x$ is called $x$-nonlocal DS equations, while a fully $PT$ symmetric nonlocal DSII equation is called nonlocal DSII equation. Three kinds of solutions, namely breather, rational and semi-rational solutions for these nonlocal DS equations are derived by employing the bilinear method. For the $x$-nonlocal DS equations, the usual ($2+1$)-dimensional breathers are periodic in $x$ direction and localized in $y$ direction. Nonsingular rational solutions are lumps, and semi-rational solutions are composed of lumps, breathers and periodic line waves. For the nonlocal DSII equation, line breathers are periodic in both $x$ and $y$ directions with parallels in profile, but localized in time. Nonsingular rational solutions are ($2+1$)-dimensional line rogue waves, which arise from a constant background and disappear into the same constant background, and this process only lasts for a short period of time. Semi-rational solutions describe interactions of line rogue waves and periodic line waves.