The nonlinear Schroedinger equation for the delta-comb potential: quasi-classical chaos and bifurcations of periodic stationary solutions

TL;DR

This paper analyzes the nonlinear Schrödinger equation with a delta-comb potential, revealing classical chaos phenomena, bifurcations, and new nonlinear Bloch band features through analytical solutions and stability analysis.

Contribution

It provides an analytical solution for stationary states in a delta-comb potential and explores the emergence of looped bands and chaos in nonlinear quantum systems.

Findings

Analytical solutions using Jacobi elliptic functions for delta-comb potentials

Identification of bifurcations and transition to spatial chaos

Derivation of critical nonlinearity for looped band emergence

Abstract

The nonlinear Schroedinger equation is studied for a periodic sequence of delta-potentials (a delta-comb) or narrow Gaussian potentials. For the delta-comb the time-independent nonlinear Schroedinger equation can be solved analytically in terms of Jacobi elliptic functions and thus provides useful insight into the features of nonlinear stationary states of periodic potentials. Phenomena well-known from classical chaos are found, such as a bifurcation of periodic stationary states and a transition to spatial chaos. The relation of new features of nonlinear Bloch bands, such as looped and period doubled bands, are analyzed in detail. An analytic expression for the critical nonlinearity for the emergence of looped bands is derived. The results for the delta-comb are generalized to a more realistic potential consisting of a periodic sequence of narrow Gaussian peaks and the dynamical stability of periodic solutions in a Gaussian comb is discussed.