Static solutions of a D-dimensional Modified Nonlinear Schroedinger Equation

Abstract

We study static solutions of a D-dimensional modified nonlinear Schrödinger equation (MNLSE) which was shown to describe, in two dimensions, the self-trapped (spontaneously localized) electron states in a discrete isotropic electron–phonon lattice [1, 2]. We show that this MNLSE, unlike the conventional nonlinear Schrödinger equation, possesses static localized solutions at any dimensionality when the effective nonlinearity parameter is larger than a certain critical value which depends on the dimensionality of the system under study. We investigate various properties of the equation analytically, using scaling transformations, within the variational scheme and numerically, and show that the results of these studies agree qualitatively and quantitatively. In particular, we prove that, for various values of D, when the coupling constant is larger than a certain critical value (which depends on D), this equation has two solutions, a stable (metastable) and an unstable one. We show that the solutions can be well approximated by a Gaussian ansatz and we also show that, in two dimensions, the equation possesses solutions with a nonzero angular momentum.