Breather statics and dynamics in Klein-Gordon chains with a bend

In this paper, we examine a nonlinear model with an impurity emulating a bend. We justify the geometric interpretation of the model and connect it with earlier work on models including geometric effects. We focus on both the bifurcation and stability analysis of the modes that emerge as a function of the strength of the bend angle, but we also examine dynamical effects including the scattering of mobile localized modes (discrete breathers) off of such a geometric structure. The potential outcomes of such numerical experiments (including transmission, trapping within the bend as well as reflection) are highlighted and qualitatively explained. Such models are of interest both theoretically in understanding the interplay of breathers with curvature, but also practically in simple models of photonic crystals or of bent chains of DNA.

Figure 1

Schematic presentation of a bend in connection with the discrete equation (2). The parameter $\gamma$ stands for the bend-induced interaction between the next-nearest neighbors (in the vicinity of the bend), whereas $C$ represents the nearest neighbor interaction in the rectilinear chain.

Figure 2

(a) Linear mode spectrum with respect to the bending parameter $\alpha =\gamma ∕C$ for a coupling constant $C=0.20$. (b,c) Profile of the impurity mode for $\alpha >0$ (b) or $\alpha <0$ (c) and $\mid \alpha \mid =2.5$. Note that the localization of the impurity modes is non-negligible only for high values of $\alpha$.

Figure 3

Bifurcation diagram for the solutions with a Morse on-site potential. The energy of the breathers is plotted as a function of the bending parameter $\alpha =\gamma ∕C$. The coupling constant is $C=0.13$ (top) or $C=0.26$ (bottom). The numbers indicate the site where the solutions are centered; an integer number corresponds to a site-centered solution, whereas a half-integer number corresponds to a bond-centered solution. Stable solutions (for an infinite system) are represented by full lines whereas unstable solutions are represented by dashed lines.

Figure 4

Spatial profiles of the breathers centered at $n=0$ (a), $n=0.5$ (b), $n=1$ (c), and $n=1.5$ (d) for $\alpha =0.0005$. The results correspond to a chain of oscillators with an on-site Morse potential with parameters ${\omega }_{\mathrm{b}}=0.8$ and $C=0.26$.

Figure 5

Evolution with respect to the coupling constant of the Floquet multiplier arguments $\theta$ (left) and the modulus of the Floquet multipliers (right) for a one-site (top) and a two-site (bottom) breather with Morse on-site potential with frequency ${\omega }_{\mathrm{b}}=0.8$. The one-site breather is stable for $C∊\left(0,0.1297\right)$ and for $C∊\left(0.2382,0.4\right)$ it recovers its stability except for size-dependent instability bubbles via oscillatory and subharmonic bifurcations. The 2-site breather is stable for $C∊\left(0.1300,0.2381\right)$.

Figure 6

(a) Spatial profiles of the breathers centered at $n=1$ for $\alpha =-0.003$ and the same characteristics to those of Figure 4. (a) shows the spatial profile of unstable solution; (b) represents the Floquet multipliers corresponding to this state. The instabilities that can be appreciated near $\theta =\pi$ are due to the collision of extended eigenmodes and disappear in the case of an infinite lattice.

Figure 7

Bifurcation diagram for the solutions with a ${\varphi }^4$ on-site potential. The energy of the breathers is plotted as a function of the bending parameter $\alpha =\gamma ∕C$. The numbers indicate the site where the solutions are centered; the half-integer numbers actually correspond to two-site breathers, but the notation has been kept in consonance with Fig. 3. Stable solutions are represented by full lines whereas unstable solutions are represented by dashed lines.

Figure 8

Spatial profiles of the breathers centered at $n=0$ (a), $n=0.5$ (b), $n=1$ (c) and $n=1.5$ (d) for $\alpha =0.1$. The results correspond to a chain of oscillators with an on-site ${\varphi }^4$ potential with breather frequency ${\omega }_{\mathrm{b}}=1.2$ and coupling constant $C=0.09$.

Figure 9

(a) Spatial profiles of the breathers centered at $n=1$ for $\alpha =0.5$ and the same characteristics to those of Fig. 8. This solution corresponds to the stable ground state. (b) represents the Floquet eigenvalues of this solution.

Figure 10

Time evolution of an unstable breather slightly perturbed for $\alpha >0$ (a) and $\alpha <0$ (b).

Figure 11

Time evolution of an unstable breather slightly perturbed for $\alpha >0$.

Figure 12

Evolution of the energy center (left) and the moving breather (right) for a Morse on-site potential with parameters ${\omega }_{\mathrm{b}}=0.8$ and $C=0.13$. Figures (a) and (b) correspond to a reflection ($\alpha =0.008$ and $K=0.00174$), (c) and (d) to a refraction ($\alpha =0.008$ and $K=\mathrm{0.001\hspace{0.17em}80}$) and (e) and (f) to a refraction through the site $n=-1$ and a reflection at $n=1$ leading to a trapping ($\alpha =0.008$ and $K=\mathrm{0.003\hspace{0.17em}92}$).

Figure 13

Dependence of the critical energy $U_c$ on $\alpha$ for a Morse on-site potential with parameters ${\omega }_{\mathrm{b}}=0.8$ and $C=0.13$. Note that for sufficiently large $\alpha$, the curve separates in two parts corresponding to $U_{c+}$ and $U_{c-}$ (see also the relevant discussion in the text).

Figure 14

Potential barrier experienced by the moving breather in the chain with a Morse on-site potential, when it reaches the bending point. Parameters of the breather are ${\omega }_{\mathrm{b}}=0.8$, $C=0.13$, and $\gamma =0.08$. The thick line corresponds to the fit to Eq. (8) and the slim line to Eq. (6) for $K=0.0018$. The oscillations of the last curve are due to the nonuniformity of the translational velocity of the breather. Parameters of the fitting to Eq. (8) are $a=\mathrm{0.001\hspace{0.17em}768}$, $b=2.7068$, and $\delta =\mathrm{0.087\hspace{0.17em}54}$.

Figure 15

Evolution of the energy center (left) and the moving breather (right) for ${\varphi }^4$ on-site and interaction potentials with parameters ${\omega }_{\mathrm{b}}=3$ and $C=0.5352$. (a) and (b) correspond to a refraction $\left(\alpha =0.2\right)$, (c) and (d) to a trapping $\left(\alpha =0.5\right)$, and (e) and (f) to a reflection $\left(\alpha =4\right)$.

Figure 16

Evolution of the energy center for a ${\varphi }^4$ on-site and interaction potentials with parameters ${\omega }_{\mathrm{b}}=3$, $C=0.5352$ and a fixed value of $\alpha =0.3$. (a) corresponds to a trapping case $\left(K=\mathrm{0.001\hspace{0.17em}25}\right)$ and (b) to a well regime $\left(K=0.02\right)$.