Synchronization of dust acoustic waves in a forced Korteweg–de Vries–Burgers model

The synchronization of dust acoustic waves to an external periodic source is studied in the framework of a driven Korteweg–de Vries–Burgers equation that takes into account the appropriate nonlinear and dispersive nature of low-frequency waves in a dusty plasma medium. For a spatiotemporally varying source term, the system is shown to demonstrate harmonic (1:1) and superharmonic (1:2) synchronized states. The existence domains of these states are delineated in the form of Arnold tongue diagrams in the parametric space of the forcing amplitude and forcing frequency and their resemblance to some past experimental results is discussed.

Figure 1

PSD for the times series of KdV (solid line) and KdV-B (dash-dotted line) equations with initial perturbation Eq. (4). Insets (I) and (II) show the phase space plots and time series, respectively, for KdV (solid line) and KdV-B (dash-dotted line) models.

Figure 2

The harmonic (1:1) synchronization in the fKdV-B model with $f_s<f_0$ and $f_s>f_0$. The time series of the fKdV-B model (solid line) and the forcing (dash-dotted line) at driver frequency (a) $f_s=21$ Hz with threshold amplitude $A_s=0.40A_0$ and (b) $f_s=23$ Hz with threshold amplitude $A_s=0.40A_0$. (c) PSD of time series (a). (d) PSD of time series (b). The inset (I) is the phase space plot and the inset (II) is the Lissajous figure which reflects the frequency locking at the driver frequency.

Figure 3

The superharmonic (1:2) synchronization in the fKdV-B model with $f_s<2f_0$ and $f_s>2f_0$. The time series of the fKdV-B model (solid line) and the forcing (dash-dotted line) at driver frequency (a) $f_s=43$ Hz with threshold amplitude $A_s=0.60A_0$ and (b) $f_s=45$ Hz with threshold amplitude $A_s=0.50A_0$. (c) PSD of times series (a). (d) PSD of time series (b). The inset (I) is the phase space plot and the inset (II) is the Lissajous figure which reflects the frequency locking at half of the driver frequency.

Figure 4

The Arnold tongue diagram for harmonic (1:1) and superharmonic (1:2) synchronization states in the fKdV-B model. The amplitude is varied from $A_s=0.10A_0$ to $A_s=0.70A_0$ for 1:1, and $A_s=0.20A_0$ to $A_s=0.70A_0$ for 1:2 synchronization.