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Eigenstates and instabilities of chains with embedded defects.

J D'Ambroise¹, P G Kevrekidis, S Lepri

¹Department of Mathematics, Bard College, Annandale-on-Hudson, New York 12504, USA.

Chaos (Woodbury, N.Y.)

|July 5, 2013

Summary

This study presents a general method for analyzing eigenvalue problems in one-dimensional lattices with defects. The approach allows for semi-analytical computation of spectra for both linear and nonlinear systems, revealing oscillatory instabilities in nonlinear states.

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Area of Science:

Condensed matter physics
Quantum mechanics
Nonlinear dynamics

Background:

Schrödinger lattices are fundamental models in physics.
Defects in lattices can significantly alter their properties.
Analyzing eigenvalue problems is crucial for understanding lattice behavior.

Purpose of the Study:

To develop a general approach for solving eigenvalue problems in one-dimensional lattices with embedded defects.
To investigate the spectral properties of both linear and nonlinear lattice defects.
To analyze the stability of nonlinear waves in the presence of defects.

Main Methods:

A general approach based on matching solutions at the defect site.
Semi-analytical computation of spectra for linear defects.
Computation of linearization spectra for nonlinear defects.

Main Results:

The method provides a handle for semi-analytical spectral computation, reducing to polynomial equations for linear cases.
Nonlinear defects exhibit complex spectral properties.
Calculations reveal oscillatory instabilities in strongly nonlinear states.

Conclusions:

The developed matching approach is effective for analyzing eigenvalue problems in lattices with various types of defects.
The study highlights the prevalence of oscillatory instabilities in nonlinear lattice systems.
This work contributes to the understanding of scattering states and wave stability in complex lattice structures.