Collective coordinate descriptions of a kink in a driven-damped $φ^4$ model

Abstract

Extending a recent effective theory formulation for the dynamics of kinks in the sine-Gordon model [1], we propose an analogous effective description of $φ^4$ kinks. Three different reduced models based on the kink position, width and internal mode amplitude are introduced and compared systematically with the numerical solution of the equation with space- and time-dependent perturbations. In all cases considered, the model based on the kink position and width agrees the best with the full numerical solution. As long as the external driving frequency of the perturbation remains moderate, it captures with remarkable accuracy the intricate dynamical processes taking place in the system.

Figures (18)

• Figure 1: The trajectory of a kink that initially rests ($v=0$) at point $x_0=-12$. In this simulation, $\varepsilon_2=0$, which means that $\lambda=1$. The heterogeneity in the system is described only by the function $\mathcal{F}$ through the value of the parameter $\varepsilon_1=0.1$. (a) Comparison of the trajectory obtained from the field solution (black solid line), the first effective model (red dashed line), and the third approximate model (green dashed line). The color bar on the right side of panel (a) refers to the values of the function $\mathcal{F}$. (b) Comparison of the variable $\gamma$ obtained from the field solution (black dotted line) and obtained from the first effective model (red dotted line). (c) Variable $b$ obtained from the field solution (black dotted line) and the third approximate model (green dotted line).
• Figure 2: Movement of a kink initially at rest ($v=0$) and located at $x_0=-12$. In this case, $\varepsilon_2=0$, so the system’s inhomogeneity arises solely from the function $\mathcal{F}$, with the parameter set to $\varepsilon_1=0.1$. (a) The kink trajectory computed from the field solution (black solid curve) is compared with that predicted by the second effective model (purple dashed curve). The color bar on the right side of the panel indicates the values of the function $\mathcal{F}$. (b) The evolution of the variable $b$ obtained from the field solution (black dotted curve) is compared with the corresponding result from the second effective model (purple dotted curve).
• Figure 3: Shifting the kink between two maxima of the function $\mathcal{F}$. As in the previous figure, $\varepsilon_1=0.1$ and $\varepsilon_2=0$. The initial position of the kink is $x_0=-18$. Initially, to enable the kink to slide, a minimum speed of $v=10^{-4}$ was assumed. (a) The continuous black line shows the trajectory obtained from the field solution, the red dashed line shows the trajectory obtained from the first effective model, and the green dashed line was obtained from the third approximate model. The color bar to the right of the panel refers to the values of the function $\mathcal{F}$. (b) Comparison of the variable $\gamma$ obtained from the field solution (black dotted line) and the first approximate model (red dotted line). (c) The variable $b$ obtained from the third effective model (green dotted line) and the field solution (black dotted line).
• Figure 4: Displacement of the kink between two maxima of the function $\mathcal{F}$ for the initial conditions \ref{['phi_wp-x']}. As in the previous figure, $\varepsilon_1=0.1$ and $\varepsilon_2=0$. The kink is initially located at $x_0=-18$. To allow it to move, a small initial velocity of $v=10^{-4}$ was introduced. (a) The solid black curve represents the trajectory obtained from the full field solution, while the purple dashed curve corresponds to the trajectory predicted by the second effective model. The color bar on the right indicates the values of the function $\mathcal{F}$. (b) Comparison of the variable $b$ as computed from the field solution (black dotted curve) and from the second approximate model (purple dotted curve).
• Figure 5: Trajectory when $\varepsilon_1=0.1$ and $\varepsilon_2=0.05$. In this case, both $\mathcal{F}$ and $\lambda$ have a non-trivial shape. As before, the color bar to the right of the first panel refers to the value of the effective potential. The initial position of the kink is $x_0(0)=-16.5$, and the initial velocity is $v=0$. The parameter $k$ is equal to $\frac{\pi}{3}$. (a) Comparison of trajectories from three computations: the full field model, i.e., the "ground truth" (shown by solid line), the first effective model (shown by dashed red line), and the third approximate model (shown by dashed green line). (b) The black dotted line shows the time dependence of the variable $\gamma$ in the field model, while the red dotted line shows the same variable obtained in the first effective model. (c) Again, the black dotted line shows the time course of variable $b$ obtained from the field model, while the green dotted line shows the result obtained from the third effective model.