human
Give me a the Python code you requested given the following:
I am working on the file "Nonlinear_elasticity.ipynb".
The first task is:
In this chapter we investigate a nonlinear model of elastic strain in heterogeneous materials.  This system is equivalent to the $p$-system of gas dynamics, although the stress-strain relation we will use here is very different from the pressure-density relation typically used in gas dynamics.  The equations we consider are:  
\begin{align}
\epsilon_t(x,t) - u_x(x,t) & = 0 \
(\rho(x)u(x,t))_t - \sigma(\epsilon(x,t),x)_x & = 0.
\end{align}  
Here $\epsilon$ is the strain, $u$ is the velocity, $\rho$ is the material density, $\sigma$ is the stress,
and ${\mathcal M}=\rho u$ is the momentum. 
The first equation is a kinematic relation, while the second represents Newton's second law.  This is a nonlinear 
conservation law with spatially varying flux function, in which  
\begin{align}
q & = \begin{bmatrix} \epsilon \ \rho u \end{bmatrix}, & f(q,x) & = \begin{bmatrix} -{\mathcal M}/\rho(x) \ -\sigma(\epsilon,x) \end{bmatrix}.
\end{align}  
If the stress-strain relationship is linear -- i.e. if $\sigma(\epsilon,x)=K(x)\epsilon$ -- then this system is equivalent to the acoustics equations that we have
studied previously.  Here we consider instead a quadratic stress response:
\begin{align}
\sigma(\epsilon,x) = K_1(x) \epsilon + K_2(x) \epsilon^2.
\end{align}
We assume that the spatially-varying functions $\rho, K_1, K_2$ are piecewise constant, taking values
$(\rho_l, K_{1l}, K_{2l})$ for $x<0$ and values $(\rho_r, K_{1r}, K_{2r})$ for $x>0$.  This system has been investigated numerically in <cite data-cite="leveque2002"><a href="riemann.html#leveque2002">(LeVeque, 2002)</a></cite>, <cite data-cite="leveque2003"><a href="riemann.html#leveque2003">(LeVeque & Yong, 2003)</a></cite>, and <cite data-cite="2012_ketchesonleveque_periodic"><a href="riemann.html#2012_ketchesonleveque_periodic">(Ketcheson & LeVeque, 2012)</a></cite>.
Note that if we take $\rho=1$, $\sigma=-p$, and $\epsilon=v$, this system is equivalent to the p-system of Lagrangian gas dynamics
\begin{align}
v_t - u_x & = 0 \
u_t - p(v)_x & =...
Can you write Python code for it?
assistant
#Here is the Python code you requested.

# %load exact_solvers/nonlinear_elasticity.py

def dsigma(eps, K1, K2):
    "Derivative of stress w.r.t. strain."
    return K1 + 2*K2*eps

def lambda1(q, xi, aux):
    eps = q[0]
    rho, K1, K2 = aux
    return -np.sqrt(dsigma(eps, K1, K2)/rho)

def lambda2(q, xi, aux):
    return -lambda1(q,xi,aux)

def make_plot_function(q_l, q_r, aux_l, aux_r):
    states, speeds, reval, wave_types = \
        nonlinear_elasticity.exact_riemann_solution(q_l,q_r,aux_l,aux_r)
        
    def plot_function(t,which_char):
        ax = riemann_tools.plot_riemann(states,speeds,reval,wave_types,
                                        t=t,t_pointer=0,
                                        extra_axes=True,
                                        variable_names=['Strain','Momentum'])
        if which_char == 1:
            riemann_tools.plot_characteristics(reval,lambda1,(aux_l,aux_r),ax[0])
        elif which_char == 2:
            riemann_tools.plot_characteristics(reval,lambda2,(aux_l,aux_r),ax[0])
        nonlinear_elasticity.phase_plane_plot(q_l, q_r, aux_l, aux_r, ax[3])
        plt.show()
    return plot_function        
        
def plot_riemann_nonlinear_elasticity(rho_l,rho_r,v_l,v_r):
    plot_function = make_plot_function(rho_l,rho_r,v_l,v_r)
    interact(plot_function, t=widgets.FloatSlider(value=0.,min=0,max=1.,step=0.1),
             which_char=widgets.Dropdown(options=[None,1,2],
                                         description='Show characteristics'));

aux_l = np.array((1., 5., 1.))
aux_r = np.array((1., 2., 1.))
q_l = np.array([2.1, 0.])
q_r = np.array([0.0, 0.])

plot_riemann_nonlinear_elasticity(q_l, q_r, aux_l, aux_r)