Find a formula for the vector hyperbolic problem

Consider the vector hyperbolic problem

$u_t + Au_x = −u$

$A=\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} $

with $u = (u, v)^T$ (transpose to indicate u is a column vector in the equation above).

Find a formula using characteristics for the problem with given initial data $u(x, 0) = u_0(x)$.
Reformulate the vector problem as a second-order scalar problem for $u(x, t)$.


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