Interactions of parametrically driven dark solitons. I. Néel-Néel and Bloch-Bloch interactions.

We study interactions between the dark solitons of the parametrically driven nonlinear Schrödinger equation, Eq. 1 . When the driving strength, h , is below sqrt[gamma(2)+1/9], two well-separated Néel walls may repel or attract. They repel if their initial separation 2z(0) is larger than the distance 2zu between the constituents in the unstable stationary complex of two walls. They attract and annihilate if 2z(0) is smaller than 2zu. Two Néel walls with h lying between sqrt[gamma(2)+1/9] and a threshold driving strength hsn attract for 2z(0)<2zu and evolve into a stable stationary bound state for 2z(0)>2zu. Finally, the Néel walls with h greater than hsn attract and annihilate-irrespective of their initial separation. Two Bloch walls of opposite chiralities attract, while Bloch walls of like chiralities repel-except near the critical driving strength, where the difference between the like-handed and oppositely handed walls becomes negligible. In this limit, similarly handed walls at large separations repel while those placed at shorter distances may start moving in the same direction or transmute into an oppositely handed pair and attract. The collision of two Bloch walls or two nondissipative Néel walls typically produces a quiescent or moving breather.