Partial differential equations, symmetries, invariants, conservations,

[newold22]

Homework Statement

find the invariant of system of real shrodinger-like system of pd equations with independent variables x,t and dependent u, v

Relevant Equations

u_t=u_{xx} - u^2 v
v_t =-v_{xx}-v^2 u

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 bcs of the vector function F the F'* is a matrix
F*'=\begin{bmatrix}
F_1_u_1'* & F_2_u_1'*\\
F_1_u_2'* & F_2_u_2'
\end{bmatrix}
and the quation is F'*v=0
and the conservation law theorem is I(v)=\int_0^1 d\lambda\int <v(\lambda u_1,\lambda u_2 | u_1,u_2)> dx under condition v=grad I(v) and there is second condition for v : v_u= v_u*

 

[BvU]

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[newold22]

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
$$\frac{\delta \mathcal{L}}{\delta u}=(\frac{\delta\mathcal{L}}{\delta u_1},\frac{\delta\mathcal{L}}{\delta u_2}) =F^{\ast}(u)\cdot v=0, where
$$
and
$$
F^{\ast}_u=
$$
is a matrix 2x2 with entries
$$
F^{\ast}_{1_{u_1}}, \quad F^{\ast}_{2_{u_1}}, \quad F^{\ast}_{1_{u_2}}, \quad F^{\ast}_{2_{u_2}}
$$
and for conservation law
$$
\int_0^1 d\lambda \int <v(\lambda u_1,\lambda u_2)| (u_1,u_2)> dx = I(u)
$$
when
$$
v_u=v_u^{\ast} \Rightarrow v=grad I(u)
$$