Partial Diff Eq classification

[tsunamiBTP]

I have rather nasty vector partial diff eq which I think has been addressed before; however, when I vistied a few websites to classify the equation the notation of these websites is not necessarily what I am accustomed to. I have attached the equation here & was wondering if anyone could point me in the right direction so I can see how others approached solving the problem.

 

I suspect this has applications in electromagnetism, but it may be applicable elsewhere, MAYBE gas dynamics?

 

I have a solution but would like to compare mine with others.

 

Appreciate any feedback

eq of choice.pdf

[Xittenn]

Somebodies model for the expansion and collapse of the universe?