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.

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