Solving Coupled PDEs: A Comprehensive Guide

[Hazmitaz]

Hello,

It has taken me a long time to try and figure out what a system of coupled PDEs actually IS-and I still can't get a straight answer.

For example I have a system:

\dot{M}=-L_vM

\dot{N}=-L_v+wN

where here ,L, represents the lie derivative and M, N , v, w, are all elements of the sets of vector fields on a manifold. And I want to write this system as a pair of coupled PDEs with independent variables (x,t)\inℝxℝ.
I hope this question is clear and please answer sooner rather than later if you can.

 

[Hazmitaz]

Any thoughts at all-passing notions even would really help. Even if it's to say I'm making no sense or asking something trivial.

In all honesty I really do have this problem: what are coupled PDEs? I can't get a straight answer anywhere.

Although I am very new to Physics Forums-and apologise for joining just to inundate with questions-I plan, as soon as I'm out of my current deadline to join other threads-is there a protocal I've missed?