I am looking for an analytic solution for the following two equations in the variables v(x,t)and u(x,t):
∂v/∂x= -m∂u/∂t     and   ∂u/∂x= -n∂v/∂t-av^5
The boundary conditions are 
v(0,t)=E      and   u(l,t)=0
The constants  m,n,a,E and l  are  positive and non-zero.
Thanks in advance.
Note: This is a simplified version of my earlier (unanswered) question posted on February 13, 2019.