How I can implement the advection model with PCs

The model is

{u_t}^{+} +{\gamma}{u_x}^{+}= \lambda(v)(u^{-}-u^{+})-\beta{u^+}{v}

{u_t}^{-} -{\gamma}{u_x}^{-}= \lambda(v)(u^{+}-u^{-})-\beta{u^-}{v}

v_t= \beta{u v}-\alpha{v}, , \text{with}\;\;\; u=u^{+}+u^{-}.

Regarding the initial conditions we use

u^+(x, 0)=u_{*}^{+}+a_1 sin(10xk_1),\; u^-(x, 0)=u_{*}^{-}+a_2 sin(10xk_1),\nonumber\\

v(x, 0)=v_{*}+a_3 sin(10xk_1).

k_1 := 2 \Pi/L, L=1, and I want to use periodic conditions