Special polynomials associated with rational solutions of the defocusing nonlinear Schrödinger equation and the fourth Painlevé equation

Abstract

Rational solutions and rational-oscillatory solutions of the defocusing nonlinear Schrödinger equation are expressed in terms of special polynomials associated with rational solutions of the fourth Painlevé equation. The roots of these special polynomials have a regular, symmetric structure in the complex plane. The rational solutions verify results of Nakamura and Hirota [J. Phys. Soc. Japan, 54 (1985) 491–499] whilst the rational-oscillatory solutions appear to be new solutions of the defocusing nonlinear Schrödinger equation.