Eigenstates and instabilities of chains with embedded defects

Abstract

We consider the eigenvalue problem for one-dimensional linear Schrödinger 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.

References in corpus (5)

• Making Sense of Non-Hermitian Hamiltonians
• Asymmetric Wave Propagation Through Nonlinear PT-symmetric Oligomers
• Interlaced linear-nonlinear optical waveguide arrays
• Pinned modes in lossy lattices with local gain and nonlinearity
• Symmetric and asymmetric localized modes in linear lattices with an embedded pair of ${\chi }^{(2)}$-nonlinear sites

Cited by in corpus (4)

• Nonreciprocal wave scattering on nonlinear string-coupled oscillators
• $\mathcal{P}\mathcal{T}$-symmetric coupler with ${\chi }^{(2)}$ nonlinearity
• Dark Soliton Scattering in Symmetric and Asymmetric Double Potential Barriers
• Self trapping transition for a nonlinear impurity within a linear chain