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arXiv · 1211.5707

Eigenstates and instabilities of chains with embedded defects

J. D'Ambroise·P. G. Kevrekidis·S. Lepri
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Abstract

We consider the eigenvalue problem for one-dimensional linear Schr\"odinger lattices (tight-binding) with an embedded few-sites linear or nonlinear, Hamiltonian or non-conservative defect (an oligomer). Such a problem arises when considering scattering states in the presence of (generally complex) impurities as well as in the stability analysis of nonlinear waves. We describe a general approach based on a matching of solutions of the linear portions of the lattice at the location of the oligomer defect. As specific examples we discuss both linear and nonlinear, Hamiltonian and $\cP \cT$-symmetric dimers and trimers. In the linear case, this approach provides us a handle for semi-analytically computing the spectrum [this amounts to the solution of a polynomial equation]. In the nonlinear case, it enables the computation of the linearization spectrum around the stationary solutions. The calculations showcase the oscillatory instabilities that strongly nonlinear states typically manifest.

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J. D'Ambroise, P. G. Kevrekidis, S. Lepri. 2012-11-24. Eigenstates and instabilities of chains with embedded defects. https://doi.org/10.1063/1.4803523

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