Question: I'm trying to find the equation of the electric potential of a electric dipole in Cartesian coordinates. 

The equation should take the form of something looks like this:

$$z = x\exp(-x^2-y^2)$$

[![z = x*exp(-x^2-y^2)](https://i.stack.imgur.com/ILZts.png)](https://i.stack.imgur.com/ILZts.png)

Where x and y are Cartesian coordinates and $z$ is the electric potential, and $-\nabla z$ is the electric field (quivers in the plot). 

I know that equation is not true for dipole electric potential. It's just there to show how the plot of the real equation should look like.

Answer: 
It is zero at the equilibrium position: the restoring force is $F = - k x$, where $x = 0$ at the equilibrium position.
The restoring force is zero, it is the inertia of the object that keeps it from stopping at that point.
Once the object crosses the equilibrium point, the restoring force starts acting on it again, but this time in the opposite direction.