Two-Solitons Solutions

By applying the Backlund transformation to the trivial solution and one-soliton solution, the two-soliton solutions can be obtained.

Examples of two-soliton solutions and two-soliton collisions:

Kink-Kink Collision

Before the collision, Kinks move toward each other with equal velocities. It seems that in the collision they repel each other, however, they actually pass through each other without change in their velocities.

Kink-AntiKink Collision

Before the collision, Kink and Antikink move toward each other with equal velocities. After the collision they move away with the same velocities but in the neighboring level.

Breather

Another two-soliton solution is the Breather. Breather is an oscillatory localized excitation, which is the Kink and Antikink bound together. Kink and Antikink have no enough energy to overcome their mutual attraction and that is why they make the oscillatory system.

In the following the examples of Breather solutions are given.

Standing Breather

Large amplitude Breather

Small amplitude Breather

Small amplitude Breather looks exotically but its envelope has the form of large amplitude Breather.