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Eigenstates and instabilities of chains with embedded defects
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10.1063/1.4803523
/content/aip/journal/chaos/23/2/10.1063/1.4803523
http://aip.metastore.ingenta.com/content/aip/journal/chaos/23/2/10.1063/1.4803523
View: Figures

Figures

Image of FIG. 1.
FIG. 1.

The left set of panels corresponds to the linear -symmetric dimer with  = 2 and while the right set of panels to the case of the trimer with  = 3 and . Each set contains a (top) plot of linear stability eigenfrequencies as a function of increasing γ as computed an a lattice of length 200. The eigenvalue agreement between numeric (circles) and exact (x's) computations is shown for (bottom left) and (bottom right).

Image of FIG. 2.
FIG. 2.

Profiles of modulus of unstable eigenvectors are shown for the linear dimer (left) and the linear trimer (right). These eigenvectors correspond to the eigenvalues with negative imaginary part for , which are seen in the bottom right panels of Fig. 1 . Again, numerical computations (circles) are shown to agree with the exact computation (x's) and the lattice length is 200.

Image of FIG. 3.
FIG. 3.

The left set of plots corresponds to the linear Hamiltonian dimer with  = 2 and while the right set of plots to the case of the linear trimer with  = 3 and . Each set contains a (top) plot of linear stability eigenvalues and a (bottom) plot of the eigenvector for the defect (point spectrum) mode. All plots show agreement between the numerical (circles) and semi-analytical (x's) results. Here, the lattice length is 40.

Image of FIG. 4.
FIG. 4.

The left set of panels corresponds to the -symmetric dimer with  = 2, , and while the right set of panels to the case of the trimer with  = 3, , and . Each set contains a (top) plot of linear stability eigenfrequencies of the stationary solution corresponding to  = 0.7 and as computed on a lattice of length 20. Plots of the eigenvalue ν indicating agreement between numeric (circles) and exact (x's) results are shown for the dimer with (bottom left) and (bottom right), and for the trimer with (bottom left) and (bottom middle) and (bottom right). For the trimer, when γ is small the dominant pair of (unstable) eigenfrequencies of negative imaginary part increases in magnitude as γ increases until . At this point, the complex pair recedes and a single dominant purely imaginary emerges at . This single eigenfrequency increases in magnitude until when it begins to decrease. At , a new dominant complex pair increases in magnitude and continues to increase as γ increases.

Image of FIG. 5.
FIG. 5.

Profiles of the moduli of unstable eigenvectors are shown for the nonlinear dimer. The left four plots show eigenvectors for and the right four plots for . These eigenvectors correspond to eigenvalues with negative imaginary part which are seen in Fig. 4 . Again, numerical computations are shown to agree with the exact (x's) results and the length of the lattice is 20. Notice the localization in both components of the eigenvector in the case of the -symmetry breaking case of .

Image of FIG. 6.
FIG. 6.

Profiles of the moduli of unstable eigenvectors are shown for the nonlinear trimer with lattice length 20. The left four plots show eigenvectors for and the right six plots for . These eigenvectors correspond to eigenvalues with negative imaginary part which are seen in Fig. 4 . The agreement is similar to those of the earlier figures.

Image of FIG. 7.
FIG. 7.

The left set of panels correspond to the -symmetric dimer with  = 2, , and while the right set of panels to the case of the trimer with  = 3, , and . Each set contains contour plots of extremal stability eigenfrequencies for extended solutions as in Eq. (28) on a lattice of length 20 with (top left), (top right), (bottom left), and (bottom right).

Image of FIG. 8.
FIG. 8.

The left set of panels correspond to the Hamiltonian dimer with  = 2, , and while the right set of panels to the case of the trimer with  = 3, , and . Each set contains contour plots of extremal stability eigenfrequencies for extended solutions as in Eq. (28) on a lattice of length 20 with (top left), (top right), (bottom left), and (bottom right).

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/content/aip/journal/chaos/23/2/10.1063/1.4803523
2013-05-08
2014-04-24
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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Eigenstates and instabilities of chains with embedded defects
http://aip.metastore.ingenta.com/content/aip/journal/chaos/23/2/10.1063/1.4803523
10.1063/1.4803523
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