How I can implement the advection model with PCs

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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

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