I am looking for an analytic solution for the following two equations in the variables $v(x,t)$ and $u(x,t)$:
$$
\begin{cases}
\dfrac{\partial v}{\partial t} = -m\dfrac{\partial u}{\partial t} \\
\dfrac{\partial u}{\partial t} =-n \dfrac{\partial v}{\partial t} -av^5
\end{cases}
$$
The boundary conditions are 
$$
v(0,t)=E\qquad u(l,t)=0
$$
The constants  $m,n,a,E$ and $l$  are  positive and non-zero.
Thanks in advance.<br>

Note: This is a simplified version of my earlier (unanswered) question posted on February 13, 2019.