Examples

Solve Eq. (2.18) on the interval $[-L,\,L]$ using the following parameters:

Space interval	$L$ =20
Space discretization step	$\triangle x=0.1$
Time discretization step	$\triangle t=0.05$
Amount of time steps	$T=1800$
Velocity of the kink	$c=0.2$

Initial conditions are

a)

Kink solution:

$$f(x) = 4\arctan\biggl(\exp\bigl(\frac{x}{\sqrt{1-c^2}}\bigr)\biggr),$$

$$g(x) = -2\frac{c}{\sqrt{1-c^2}} (\frac{x}{\sqrt{1-c^2}}).$$

Figure 2.11: Space-time plot of the kink, moving with the velocity $c=0.2$

b)

Antikink solution:

$$f(x) = 4\arctan\biggl(\exp\bigl(-\frac{x}{\sqrt{1-c^2}}\bigr)\biggr),$$

$$g(x) = -2\frac{c}{\sqrt{1-c^2}} (\frac{x}{\sqrt{1-c^2}}).$$

Figure 2.12: Space-time plot of the antikink, moving with the velocity $c=0.2$

c)

Kink-kink colision:

$$f(x) = 4\arctan\biggl(\exp\bigl(\frac{x+L/2}{\sqrt{1-c^2}}\bigr)\biggr)+4\arctan\biggl(\exp\bigl(\frac{x-L/2}{\sqrt{1-c^2}}\bigr)\biggr),$$

$$g(x) = -2\frac{c}{\sqrt{1-c^2}} (\frac{x+L/2}{\sqrt{1-c^2}})+2\frac{c}{\sqrt{1-c^2}} (\frac{x-L/2}{\sqrt{1-c^2}}).$$

Figure 2.13: Space-time representation of kink-kink collision

d)

Kink-antikink colision:

$$f(x) = 4\arctan\biggl(\exp\bigl(\frac{x+L/2}{\sqrt{1-c^2}}\bigr)\biggr)+4\arctan\biggl(\exp\bigl(-\frac{x-L/2}{\sqrt{1-c^2}}\bigr)\biggr),$$

$$g(x) = -2\frac{c}{\sqrt{1-c^2}} (\frac{x+L/2}{\sqrt{1-c^2}})-2\frac{c}{\sqrt{1-c^2}} (\frac{x-L/2}{\sqrt{1-c^2}}).$$

Figure 2.14: Space-time representation of kink-antikink collision

e)

Breather solution:

$$f(x) = 0,$$

$$g(x) = 4\sqrt{1-c^2} (x\sqrt{1-c^2})$$

Figure 2.15: The breather solution, oscillating with the frequency $\omega =0.2$