Reductions of nonlocal nonlinear Schr\"odinger equations to Painlev\'e type functions

TL;DR

This paper reduces nonlocal nonlinear Schr"odinger equations to Painlevé type functions, revealing that nonlocal variants lead to known local equations through ODE reductions, with solutions expressed via elliptic functions and Painlevé transcendents.

Contribution

It demonstrates how nonlocal NLS variants can be reduced to classical Painlevé equations, unifying local and nonlocal integrable systems through ODE reductions.

Findings

Stationary solutions expressed with elliptic functions

Similarity solutions related to Painlevé IV transcendent

Nonlocal Painlevé equations reduce to known local equations

Abstract

In this paper, we take ODE reductions of the general nonlinear Schr\"odinger equation (NLS) AKNS system, and reduce them to Painlev\'e type equations. Specifically, the stationary solution is solved in terms of elliptic functions, and the similarity solution is solved in terms of the Painlev\'e IV transcendent. Since a number of newly proposed integrable 'nonlocal' NLS variants (the PT-symmetric nonlocal NLS, the reverse time NLS, and the reverse space-time NLS) are derivable as specific cases of this system, a consequence is that the nonlocal Painlev\'e type ODEs obtained from these nonlocal variants all reduce to previously known local equations.