Kink topology control by high-frequency external forces in nonlinear Klein-Gordon models

A method of averaging is applied to study the dynamics of a kink in the damped double sine-Gordon equation driven by both external (nonparametric) and parametric periodic forces at high frequencies. This theoretical approach leads to the study of a double sine-Gordon equation with an effective potential and an effective additive force. Direct numerical simulations show how the appearance of two connected $\pi$ kinks and of an individual $\pi$ kink can be controlled via the frequency. An anomalous negative mobility phenomenon is also predicted by theory and confirmed by simulations of the original equation.

Figure 1

The effective potential (19) with ${\epsilon }_3=-50$. Upper panel: $\omega =46$. Lower panel: $\omega =180$. Other parameters: $\mu =1,\lambda =2/5,\gamma =1,{\epsilon }_1=200$, and ${\epsilon }_2=195$.

Figure 2

The effective potential (19) with ${\epsilon }_3=-50$. Upper panel: $\omega =67$. Middle panel: ${\omega }_c=69.81$ (critical frequency to observe the two $\pi$ kinks). Lower panel: $\omega =73$. Other parameters: $\mu =1,\lambda =2/5,\gamma =1,{\epsilon }_1=200$, and ${\epsilon }_2=195$.

Figure 3

Upper panel: effective potential for $\omega =160$ (black solid line) and $\omega =45$ (gray solid line). Lower panel $\left(\omega =160\right)$: kink profiles moving to the left at $t^{*}=80,140,200$ obtained from simulations of the double sine-Gordon Eq. (11) (black solid line) and from the simulations of the effective Eqs. (12) and (13) (red dashed line). Other parameters: $\mu =1,\lambda =2/5,\gamma =1,f=0.1,{\epsilon }_1=200,{\epsilon }_2=175$, and ${\epsilon }_3=0$.

Figure 4

Upper and lower panels show kink profiles at $t^{*}=80,140,200$ and its spatial derivative at $t^{*}=80,200$, respectively, moving to the left obtained from simulations of Eq. (11) (black solid line) and from the simulations of the effective Eqs. (12) and (13) (red dashed line). Other parameters: $\lambda =2/5,\gamma =1,f=0.1,{\epsilon }_1=200,{\epsilon }_2=175,{\epsilon }_3=0$, and $\omega =45$.

Figure 5

Upper and lower panels show kink profiles, moving to the left, at $t^{*}=80,140,200$ for $\omega =46$ and $\omega =180$, respectively. Black solid lines: simulations of Eq. (11). Red dashed lines: simulations of the effective Eqs. (12) and (13). Other parameters: $\mu =1,\lambda =2/5,\gamma =1,f=0.1,{\epsilon }_1=200,{\epsilon }_2=195$, and ${\epsilon }_3=-50$.

Figure 6

Kink profiles moving to the left at $t^{*}=50,100,150$ for $\omega =67\lesssim {\omega }_c=69.81$. Black solid lines: simulations of Eq. (11). Red dashed lines: simulations of the effective Eqs. (12) and (13). Parameters of the simulations: $\lambda =2/5,\gamma =1,f=0.1,{\epsilon }_1=200,{\epsilon }_2=195,{\epsilon }_3=-50$.

Figure 7

Kink profiles moving to the left at $t^{*}=50,100$ for $\omega ={\omega }_c=69.81$. Black solid lines: simulations of Eq. (11). Red dashed lines: simulations of the effective Eqs. (12) and (13). Parameters of the simulations: $\lambda =2/5,\gamma =1,f=0.1,{\epsilon }_1=200,{\epsilon }_2=195,{\epsilon }_3=-50$.

Figure 8

Kink profiles moving to the left at $t^{*}=50,100,150$ for $\omega =73\gtrsim {\omega }_c=69.81$. Black solid lines: simulations of the Eq. (11). Red dashed lines: simulations of the effective Eqs. (12) and (13). Parameters of the simulations: $\lambda =2/5,\gamma =1,f=0.1,{\epsilon }_1=200,{\epsilon }_2=195,{\epsilon }_3=-50$.

Figure 9

Upper and lower panels: contour plot of the kink obtained from simulations of Eq. (11), for $\omega =50$ and $\omega =60$, respectively. Parameters of the simulations: $\lambda =2/5,\gamma =1,f=-0.1,{\epsilon }_1=200,{\epsilon }_2=195,{\epsilon }_3=-50$.

Figure 10

Upper and lower panels: kink velocity versus frequency, $\omega$. Black circles and red diamonds: results from simulations of Eq. (11) and Eqs. (12) and (13), respectively. These results are fitted with linear functions represented by a black solid line and a red dashed line, respectively. Parameters of the simulations: $\lambda =2/5,\gamma =1,f=-0.1,{\epsilon }_1=200,{\epsilon }_2=195,{\epsilon }_3=-50$.