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

摘要

In this paper, the partially party-time ($PT$) symmetric nonlocalDavey-Stewartson (DS) equations with respect to $x$ is called $x$-nonlocal DSequations, while a fully $PT$ symmetric nonlocal DSII equation is callednonlocal DSII equation. Three kinds of solutions, namely breather, rational andsemi-rational solutions for these nonlocal DS equations are derived byemploying 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-rationalsolutions are composed of lumps, breathers and periodic line waves. For thenonlocal DSII equation, line breathers are periodic in both $x$ and $y$directions with parallels in profile, but localized in time. Nonsingularrational solutions are ($2+1$)-dimensional line rogue waves, which arise from aconstant background and disappear into the same constant background, and thisprocess only lasts for a short period of time. Semi-rational solutions describeinteractions of line rogue waves and periodic line waves.