Solution
= Solution
Set $u=1+\widehat u(t)\cos(s/L)$ and $v=1+\widehat v(t)\cos(s/L)$ and retain linear terms. With $k=1/L$, the <Fourier mode> has second derivative $-k^2\cos(ks)$. Hence
$$
\boxed{\frac d{dt}\begin{pmatrix}\widehat u\\\widehat v\end{pmatrix}
=\begin{pmatrix}1-k^2&-1\\2Q&-Q(1+Pk^2)\end{pmatrix}
\begin{pmatrix}\widehat u\\\widehat v\end{pmatrix}.}
$$
The trace and <determinant> of this <linearization> are
$$
T(k)=1-Q-(1+QP)k^2,
\qquad D(k)=Q[1+(1-P)k^2+Pk^4].
$$