Coupled high order PDE

Hi, Can I have some help to solve this PDEs system. I am having a lot of troubles with this.
This are the equations:
!Captura1
If you wanna see more about, you can watch this document. https://docs.google.com/document/d/1ix_8FZYo_AkyqzrmUip3VybtI8tlFUrzcmqOP6z94Xc/edit?usp=sharing

I am sorry, but the system is order 4, in 1d with no variational form,
you have to work a solve.

There is a way to transform this system in to a weak form?