I need to prove that the Hugoniot Locus of a point $\hat{u}$ of the equation $$u_t + f (u) _x = 0,\qquad f\in C^2$$ is the set of $n$ curves
$$\begin{cases}\tilde{u}_p(\xi, \hat{u})=\hat{u}+\xi r_p(\xi)\\
s_p(\xi, \hat{u})=\lambda_p(\xi)\end{cases}$$
where $r_p$ and $\lambda_p$ are the eigenvectors and respectives eigenvalues of the Jacobian of $f$. Here, $u$ and $f(u)$ are vectors of arbitrary size $n$.
I'm not finding this proof detailed in books, but I found something similar in Smoller's book (1), as I reproduce below. It turns out that I can not understand some things:
At that given solution for $K=0$ we do not have $r=0$ at same manner? I also could not understand very well the following calculations (griffed).
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Edit (May 20)
From page 71 of LeVeque (2), another image based on the discussion (see comments below).
Many thanks.
(1) J. Smoller, Shock Waves and Reaction-Diffusion Equations, Springer, 1994. doi:10.1007/978-1-4612-0873-0
(2) R.J. LeVeque, Numerical Methods for Conservation Laws, Birkhäuser, 1992. doi:10.1007/978-3-0348-8629-1
Best Answer
A state $\tilde u_p(\xi, \hat u)$ belongs to the Hugoniot locus of $\hat u$ (a.k.a. Rankine-Hugoniot set of $\hat u$), if $\tilde u_p(\xi, \hat u)$ and $\hat u$ can be connected through a $p$-shock wave. Therefore, the Rankine-Hugoniot condition $$ s_p \left(\tilde u_p - \hat u\right) = {f(\tilde u_p) - f(\hat u)} $$ must hold, where $\tilde u_p = \tilde u_p(\xi, \hat u)$, and $s_p = s_p(\xi, \hat u)$ is the speed of shock. The parametrization is usually chosen such that $\tilde u_p(0, \hat u) = \hat u$. This equation rewrites as \begin{aligned} s_p \left(\tilde u_p - \hat u\right) &= \int_0^1 \frac{\text d}{\text d \sigma} f(\hat u + \sigma (\tilde u_p - \hat u)) \, \text d \sigma \\ &= \bigg(\underbrace{\int_0^1 f'(\hat u + \sigma (\tilde u_p - \hat u))\, \text d \sigma}_{G(\tilde u_p,\hat u)}\bigg) (\tilde u_p - \hat u) \end{aligned} where $G(\tilde u_p,\hat u)$ represents an average of the Jacobian matrix $f'$ of $f$ between $\hat u$ and $\tilde u_p$. Therefore, we have $\big(G(\tilde u_p,\hat u) - s_p I\big) (\tilde u_p - \hat u) = 0$, i.e., $\tilde u_p - \hat u$ is proportional to the right eigenvector $r_p$ corresponding to the eigenvalue $\lambda_p = s_p$ of the matrix $G(\tilde u_p,\hat u)$. So the proposed equation for Hugoniot loci in OP is correct, but in general, $r_p$ and $λ_p$ are not equal to the eigenvectors and eigenvalues of the Jacobian matrix $f'$ evaluated at $\hat u$.