Chaotic solutions of the nonlinear Schrodinger equation in classical and quantum systems

TL;DR

This paper explores chaotic and irregular solutions of the nonlinear Schrödinger equation across quantum and classical systems, revealing complex spatial structures and providing numerical methods for exact solutions.

Contribution

It introduces a reformulation of the discrete NSE as a 2D map and develops a numerical procedure for exact solutions, enabling analysis of chaotic quantum and classical lattice systems.

Findings

Chaotic spatial structures can arise in quantum states.

The reformulation as a 2D map allows classical chaos analysis methods.

Numerical solutions can be obtained with arbitrary accuracy.

Abstract

We discuss stationary solutions of the nonlinear Schrodinger equation (NSE) applicable to several quantum spin, electron and classical lattice systems. We show that there may arise chaotic spatial structures in the form of incommensurate or irregular quantum states and trajectories in space. As a first (typical) example we consider a single electron which is strongly coupled with phonons on a 1D chain of atoms. In the adiabatic approximation the system is conventionally described by a discrete set of NSEs. Another apt example is that of superconducting states in layered superconductors described by the same NSE. Amongst many other applications the typical example for a classical lattice is a system of coupled nonlinear oscillators. We reformulate this discrete NSE to the form of a 2D mapping. By this we may investigate a quantum problem by methods conventionally applied to classical chaotic dynamics. We find three types of solutions: periodic, quasiperiodic and chaotic. We then develop a procedure which allows us to obtain numerical solutions of the NSE directly. This procedure may be used to any arbitrary accuracy and so these solutions are exact to the degree of precision specified. Both methods give a consistent result. When applied to our typical example we find that the wave function of an electron on a deformable lattice (and other quantum or classical discrete systems) may exhibit incommensurate and irregular structures.