Hugoniot Locus given by parametric curves

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

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 $$\hat u$$.
• Great, many thanks for all the help you are giving me. Is possible to ensure there is some point $\tilde{u}$ in the segment $[\hat{u},\tilde{u}_p]$ on which $r_p$ and $\lambda_p$ are eigenvectors and eigenvalues of $f'(\tilde{u})$, isn't? – Na'omi May 16 at 13:23
• I am thinking: Do you have acess to page 71 of LeVeque (Numerical Methods of Conservation Laws)? The author assumes that the curves exist, so he proves that in fact the eigenvector and eigenvalues are of $f'(\hat{u})$, what do you think? I am thinking something like this: 1) your proof; 2) argument that I've cited to prove that this eigenvec. and eigenval. are of $f(\tilde{u})$ to some $\tilde{u}$. Once the system is hyperbolic, there are these $n$ curves. 3) We use the argument from LeVeque to prove that the eigenvector and eigenvalues are of $f'(\hat{u})$ in fact... – Na'omi May 16 at 22:37
• At fifth line of your post I guess is $\tilde u_p(0, \hat u) = \hat u$. I could not edit. Many thanks. – Na'omi May 16 at 22:39
• Now I finally understand, Harry49, and I think the question is correct as you, I was confused: setting $f'(\xi):=G(\tilde{u}_p(\xi,\hat{u}),\hat{u})$, I guess we may consider at the proposed equations $r_p(\xi),\lambda_p(\xi)$ as the eigenvectors and eigenvalues of $f'(\xi)$... Many thanks for all the help. – Na'omi May 22 at 14:09