Existence of multisite intrinsic localized modes in one-dimensional Debye crystals

The existence of highly localized multisite oscillatory structures (discrete multibreathers) in a nonlinear Klein-Gordon chain which is characterized by an inverse dispersion law is proven and their linear stability is investigated. The results are applied in the description of vertical (transverse, off-plane) dust grain motion in dusty plasma crystals, by taking into account the lattice discreteness and the sheath electric and/or magnetic field nonlinearity. Explicit values from experimental plasma discharge experiments are considered. The possibility for the occurrence of multibreathers associated with vertical charged dust grain motion in strongly coupled dusty plasmas (dust crystals) is thus established. From a fundamental point of view, this study aims at providing a rigorous investigation of the existence of intrinsic localized modes in Debye crystals and/or dusty plasma crystals and, in fact, suggesting those lattices as model systems for the study of fundamental crystal properties.

Figure 1

The dispersion relation of the TDL excitations: frequency square (normalized by ${\omega }_g^2$), as given by (12), vs (reduced) wave number $k{r}_0$. The value of ${\omega }_0∕{\omega }_g$ (∼ coupling strength) increases from top to bottom. Note that upper (less steep, continuous) curve is more likely to occur in a real (weakly coupled) DP crystal.

Figure 2

(a) The on-site potential is depicted against the vertical displacement $x$. (b) The frequency $\omega$ is depicted against the amplitude of the oscillation $x_{\max }$. (c) The frequency $\omega$ is depicted against the action variable $J$. All the figures have been obtained by using the set of values III in Table .

Figure 3

$F\left(x_{20}\right)$ for the 1:1 and 2:3 resonances.

Figure 4

(Color online) The destabilization scenario represented by the eigenvalues of the Floquet matrix for growing values of $\mid ϵ\mid$.

Figure 5

(Color online) Time evolution of a 1:1 two-breather for $ϵ=-0.016$.

Figure 6

(Color online) Time evolution of a 2:3 two-breather for $ϵ=-0.003$.

Figure 7

(a) The on-site potential is depicted against the vertical displacement $x$. (b) The frequency $\omega$ is depicted against the amplitude of the oscillation $x_{\max }$. (c) The frequency $\omega$ is depicted against the action variable $J$. All the figures have been obtained using the set of values IV in Table .

Figure 8

(Color online) Snapshots of an nonsymmetric multibreather for four successive time moments.