Interactions of Parametrically Driven Dark Solitons. I: N\'eel-N\'eel and Bloch-Bloch interactions

We study interactions between the dark solitons of the parametrically driven nonlinear Schr\"odinger equation, Eq.(\ref{NLS}). When the driving strength, $h$, is below $\sqrt{\gamma^2 +1/9}$, two well-separated N\'eel walls may repel or attract. They repel if their initial separation $2z(0)$ is larger than the distance $2z_u$ between the constituents in the unstable stationary complex of two walls. They attract and annihilate if $2z(0)$ is smaller than $2z_u$. Two N\'eel walls with $h$ lying between $\sqrt{\gamma^2 + 1/9}$ and a threshold driving strength $h_{sn}$ attract for $2z(0)<2z_u$ and evolve into a stable stationary bound state for $2z(0)>2z_u$. Finally, the N\'eel walls with $h$ greater than $h_{sn}$ 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 N\'eel walls typically produces a quiescent or moving breather.