Orthogonal Solutions for a Hyperbolic System

We consider the hyperbolic system
$$ \begin{cases} u_t=a\nabla v+f_1(u,н)\\ v_t=a\nabla u+f_2(u,н)\\ u(0,x)=\xi(x)\\ v(0,x)=\eta(x), \end{cases} $$
and we are looking for necessary and sufficient conditions on the forcing terms $f_i$, $i=1,2$, in order that the semigroup solutions, $u$ and $н$, starting from orthogonal data $\xi,\eta\in L^2(\mathbb R^n)$, remain orthogonal on $\mathbb R_+$.