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

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.