$\lambda_2(\rho,u)=u+\sqrt{p'(\rho)}=w(1+t,x)$, where $w$ satisfy the Burgers equation that is $w_t+ww_x=0$,and $\Sigma_2(\rho,u)=u-\int^\rho\frac{\sqrt{p'(s)}}{s}ds$ and $\Sigma_2$ is a constant. How to prove that $(\rho,u)$ satisfy the Euler system $$ \begin{array}{c} \rho_t+(\rho u)_x=0\\ (\rho u)_t+(\rho u^2+p(\rho))_x=0 \end{array} $$
The following is my attempt
$w_t=u_t+\frac{p''(\rho)}{\sqrt{p'(\rho)}}\rho_t,w_x=u_x+\frac{p''(\rho)}{\sqrt{p'(\rho)}}\rho_x,u_t-\frac{\sqrt{p'(\rho)}}{\rho}\rho_t=0,u_x-\frac{\sqrt{p'(\rho)}}{\rho}\rho_x=0$
let $f=\frac{u_t}{\rho_t}=\frac{u_x}{\rho_x}=\frac{\sqrt{p'(\rho)}}{\rho}$,because $w_t+ww_x=0$, we have $u_t+(f\rho)_t+(u+f\rho)(u_x+(f\rho)_x)=0$
that is $u_t+(\rho f)_t+(u^2/2)_x+((\rho f)^2/2)_x+(u\rho f)_x=0$