Kink interactions in the $(1+1)$-dimensional ${\phi }^6$ model

We study kink scattering processes in the $(1+1)$-dimensional ${\phi }^6$ model in the framework of the collective coordinate approximation. We find critical values of the initial velocities of the colliding kinks. These critical velocities distinguish different regimes of collisions. The exact equation of motion for the ${\phi }^6$ model is also solved numerically with the same initial conditions. We discuss advantages and disadvantages of the collective coordinate approximation, and also outline its applicability limits. Resonance phenomena and the so-called escape windows are also observed in the kink collisions.

Figure 1

The potential of the ${\phi }^6$ model.

Figure 2

Ansatz (19) at $a=10$.

Figure 3

The dependence $m(a)$ for the configuration (19).

Figure 4

The dependence $V(a)$ for the configuration (19).

Figure 5

The half-distance $a$ between the kinks as a function of time $t$ for the configuration (19) at $a(0)=10$ and different initial velocities: $v_{\mathrm{in}}=0.3$ (solid curve), $v_{\mathrm{in}}=0.6$ (dashed curve), $v_{\mathrm{in}}=0.9$ (dotted curve). Collective coordinate approximation.

Figure 6

The relative difference (20) between the values of $a_{\min }$, obtained by the two methods, as a function of the initial velocity $v_{\mathrm{in}}$.

Figure 7

Ansatz (21) at $b=10$.

Figure 8

The dependence $m(b)$ for the configuration (21).

Figure 9

The dependence $V(b)$ for the configuration (21).

Figure 10

The dependence $b(t)$ for the configuration (21) for $b(0)=10$ and different initial velocities below $v_{\mathrm{cr}}^{\text{(eff)}}$: $v_{\mathrm{in}}=0.1$ (solid curve), $v_{\mathrm{in}}=0.2$ (dashed curve), $v_{\mathrm{in}}=0.3$ (dotted curve). Collective coordinate approximation.

Figure 11

The dependence $b(t)$ for the configuration (21) for $b(0)=10$ and different initial velocities above $v_{\mathrm{cr}}^{\text{(eff)}}$: $v_{\mathrm{in}}=0.4$ (solid curve), $v_{\mathrm{in}}=0.6$ (dashed curve), $v_{\mathrm{in}}=0.9$ (dotted curve). Collective coordinate approximation.

Figure 12

Space-time picture of the evolution of the configuration $(0,-1,0)$ for $b(0)=10$ and $v_{\mathrm{in}}=0.2$. Numerical solution of Eq. (2).

Figure 13

The dependence $\phi (t,0)$ for the configuration (21) for $b(0)=10$ and $v_{\mathrm{in}}=0.2$.

Figure 14

Space-time picture of the evolution of the configuration $(0,-1,0)$ for $b(0)=10$ and $v_{\mathrm{in}}=0.6$. Numerical solution of Eq. (2).

Figure 15

The dependence $\phi (t,0)$ for the configuration (21) for $b(0)=10$ and $v_{\mathrm{in}}=0.6$.

Figure 16

Ansatz (22) at $c=10$.

Figure 17

The dependence $m(c)$ for the configuration (22).

Figure 18

The dependence $V(c)$ for the configuration (22).

Figure 19

The dependence $c(t)$ for the configuration (22) for $c(0)=10$ and different initial velocities: $v_{\mathrm{in}}=0.3$ (solid curve), $v_{\mathrm{in}}=0.6$ (dashed curve), $v_{\mathrm{in}}=0.9$ (dotted curve). Collective coordinate approximation.

Figure 20

The dependence $\phi (t,0)$ for the configuration (22) for $c(0)=10$ and $v_{\mathrm{in}}=0.043$. Bion formation.

Figure 21

The dependence $\phi (t,0)$ for the configuration (22) for $c(0)=10$ and $v_{\mathrm{in}}=0.2$.

Figure 22

The dependence $\phi (t,0)$ for the configuration (22) for $c(0)=10$ and $v_{\mathrm{in}}=0.044$. Escape window.