is soemthing like $$\mathcal{L}(u,v)=<u| F>$$ and $$u=(u_1,u_2)\quad v=(v_1,v_2)\quad F=(F_1,F_2)$$ and
$$F_1=u_{xx}-u^2v-u_t \quad F_2=-v_{xx}-v^2u-v_t$$ and
$$\frac{\delta \mathcal{L}}{\delta u}=(\frac{\delta\mathcal{L}}{\delta u_1},\frac{\delta\mathcal{L}}{\delta u_2}) =F^{\ast}(u)\cdot v=0...
I know the method is involving adjoint equation, lagrange functional and conserwation laws but i dont know how to do it, please help! I know something like this: that we must split our function into two F=(F_1,F_2), also u=(u_1,u_2) and v=(v_1,v_2) and we must calculate adjoint equation F* and...