Trying to find the shock solution for a system of conservation laws: $$ u_{t}+ \frac{1}{2}(u^2+v^2)_{x}=0$$ $$v_{t}+(uv)_{x}=0$$ for the left state $\mathbf{u}_{L}=\mathbf{u}_{0}$ and right state $\mathbf{u}_{R}=\mathbf{u}_{2}=\mathbf{u}_{0}+\epsilon\delta\mathbf{u}$ with one intermediate state $\mathbf{u}_{1}$. Using notation $\mathbf{u}_{0}=(u_0,v_0)$ and $\delta\mathbf{u}=(\delta u,\delta v)$.
I am really unsure how to go about this problem and would appreciate any help. My initial workings are:
$$s(u_L-u_R)=\frac{1}{2}(u_L^2+v_L^2)-\frac{1}{2}(u_R^2+v_R^2)\tag{1}$$ $$s(v_L-v_R)=u_Lv_L-u_Rv_R\tag{2}$$ Rearranging (1) into form: $$s(u_L-u_R)=\frac{1}{2}[(u_L+v_L)^2-(u_R+v_R)^2]-u_Lv_L+u_Rv_R$$ $$s(u_L-u_R)=\frac{1}{2}[(u_L+v_L)^2-(u_R+v_R)^2]-s(v_L-v_R)$$ I don't really know where to go from here
This is a good start towards computing a weak solution that includes one single shock wave of speed $s$. When computing the all-shock solution, we need to consider two shock waves of speed $s_1 < s_2$ and introduce an intermediate state ${\bf u}_1$. The first shock wave of speed $s_1$ connects the (left) state ${\bf u}_0$ with the (right) intermediate state ${\bf u}_1$. The second wave of speed $s_2$ connects the (left) intermediate state ${\bf u}_1$ with the (right) state ${\bf u}_2$, see picture below.
Finally, the Rankine-Hugoniot condition yields the system $$ \begin{aligned} s_1 (u_1 - u_0) &= \tfrac12 (u_1^2 + v_1^2 - u_0^2 - v_0^2) \\ s_1 (v_1 - v_0) &= u_1 v_1 - u_0 v_0 \\ s_2 (u_2 - u_1) &= \tfrac12 (u_2^2 + v_2^2 - u_1^2 - v_1^2) \\ s_2 (v_2 - v_1) &= u_2 v_2 - u_1 v_1 \end{aligned} $$ to be solved for the intermediate state $(u_1, v_1)$ and the shock speeds $s_1$, $s_2$ by assuming that ${\bf u}_L = {\bf u}_0$ and ${\bf u}_R = {\bf u}_2$ are known. From ${\bf u}_2 = {\bf u}_0 + \epsilon\, \delta {\bf u}$ we deduce $$ \begin{aligned} s_1 (u_1 - u_0) &= \tfrac12 (u_1^2 + v_1^2 - u_0^2 - v_0^2) \\ s_1 (v_1 - v_0) &= u_1 v_1 - u_0 v_0 \\ s_2 (u_0 - u_1) + \epsilon s_2 \delta u &\simeq \tfrac12 (u_0^2 + v_0^2 - u_1^2 - v_1^2) + \epsilon (u_0 \delta u + v_0 \delta v) \\ s_2 (v_0 - v_1) + \epsilon s_2 \delta v &\simeq u_0 v_0 - u_1 v_1 + \epsilon (v_0 \delta u + u_0 \delta v) \end{aligned} $$ where linearization with respect to the small parameter $\epsilon$ was performed. You might be able to take it from here.