How do we plot a domain wall between two strings?

Question

Intuitively, an angular vector field $\theta(x)$ such as the following would describe a domain wall between two vortices/strings of opposite winding number.

I would like to draw this on a computer using say Mathematica or Python.

1. How do I do this? What vector field describes a configuration depicted above?
2. If it's not too much to ask, how do I model a time-evolving field which is constant at time $t=0$ and then as time evolves, the two vortices (fluxes) get created and separated to give the above configuration?

Thank you very much!