Two-Soliton Solutions

Kink-Kink Collision

[Maple Plot]

At the initial time the two kinks are widely spaced and move toward each other. It can be seen that kinks pass through each other without any changes. In this example the two twists propagate in the ribbon and the total charge is conserved. The total charge of the kink-kink pair is equal to the sum of charges, [Maple Math] , so that [Maple Math] .

Kink-AntiKink Collision

[Maple Plot]

Colliding kink and antikink also do not suffer any changes. The total charge of the kink-antikink pair is equal to the sum of charges, [Maple Math] , so that [Maple Math] . During the kink-antikink collision, at a certain moment, the two twists disappear.

Breather Solutions

Standing Breather

[Maple Plot]

In this figure the so-called breather solution, which is one of the most beautiful sine-Gordon solutions, is presented. Note that the ends of the ribbon are not twisted one with respect to another. Consequently, the charge of the breather is [Maple Math] which coincides with the charge of the kink-antikink pair. This coincidence is not accidental, because the breather actually is the kink and antikink bound together in an oscillatory system.

Large amplitude Breather

[Maple Plot]

The moving breather looks attractive as well.

Small amplitude Breather

[Maple Plot]

This is also the breather. It resembles the ripple of a water surface.