Fractal and Chaotic Solutions of the Discrete Nonlinear Schrödinger Equation in Classical and Quantum Systems

DOI

10.2991/jnmp.2001.8.1.4How to use a DOI?

Abstract

We discuss stationary solutions of the discrete nonlinear Schrödinger equation (DNSE) with a potential of the 4 type which is generically 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. As a first (tpical) example we consider a single electron which is strongly coupled with phonons on a 1D chain of atoms -- the (Rashba)­Holstein polaron model. In the adiabatic approximation this system is conventionally described by the DNSE. Another relevant example is that of superconducting states in layered superconductors described by the same DNSE. Amongst many other applications the typical example for a classical lattice is a system of coupled nonlinear oscillators. We present the exact energy spetrum of this model in the strong coupling limit and the corresponding wave function. Using this as a starting point we go on to calculate the wave function for moderate coupling and find that the energy eigenvalue of these structures of the wave function is in exquisite agreement with the exact strong coupling result. This procedure allows us to obtain (numerically) exact solutions of the DNSE directly. 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. These states are analogous to the periodic, quasiperiodic and chaotic structures found in classical chaotic dynamics.

Copyright

© 2006, the Authors. Published by Atlantis Press.

Open Access

This is an open access article distributed under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/).

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TY  - JOUR
AU  - H.S. Dhillon
AU  - F.V. Kusmartsev
AU  - K.E. Kürten
PY  - 2001
DA  - 2001/02/01
TI  - Fractal and Chaotic Solutions of the Discrete Nonlinear Schrödinger Equation in Classical and Quantum Systems
JO  - Journal of Nonlinear Mathematical Physics
SP  - 38
EP  - 49
VL  - 8
IS  - 1
SN  - 1776-0852
UR  - https://doi.org/10.2991/jnmp.2001.8.1.4
DO  - 10.2991/jnmp.2001.8.1.4
ID  - Dhillon2001
ER  -