# Method of characteristics of a system of first order pdes

Consider the system of first order PDEs

$\left\{ \begin{eqnarray} \frac{\partial}{\partial t} v_1 + \frac{\partial}{\partial x_1} p_1 + \eta(x_1) v_1 = 0 \\ \frac{\partial}{\partial t} v_2 + \frac{\partial}{\partial x_2} p_2 + \eta(x_2) v_2 = 0 \\ \frac{\partial}{\partial t} p_1 + \frac{\partial}{\partial x_1} v_1 + \eta(x_1) p_1 = f(t) \\ \frac{\partial}{\partial t} p_2 + \frac{\partial}{\partial x_2} v_2 + \eta(x_2) p_2 = 0 \end{eqnarray} \right.$

with initial condition $v_1 = v_2 = p_1 = p_2 = 0$ at $t=0$, where $v_1,v_2,p_1,p_2 : \mathbb{R}^3 \to \mathbb{R}$ are the unknown functions, $\eta, f: \mathbb{R} \to \mathbb{R}$ are given smooth functions.

I would like to consider this system of PDEs with the method of characteristics. I know the method for a single PDE, but I don't know how to find the characteristics of such a system.

Regards.

I could understand that this system is a hyperbolic one.

Take $U=(v_1,v_2,p_1,p_2)'$, then the system can be rewritten as

$\frac{\partial}{\partial t} U + \left( \begin{array}{cccc} ~0 ~ & 0 ~ & 1 ~ & 0 \\ 0 & 0 & 0 & 0 \\ 1 ~ & 0 ~ & 0 ~ & 0 \\ 0 & 0 & 0 & 0 \end{array} \right) \frac{\partial}{\partial x_1} U + \left( \begin{array}{cccc} ~0 ~ & 0 ~ & 0 ~ & 0 \\ 0 & 0 & 0 & 1 \\ 0 ~ & 0 ~ & 0 ~ & 0 \\ 0 & 1 & 0 & 0 \end{array} \right) \frac{\partial}{\partial x_2} U + \left( \begin{array}{cccc} ~\eta(x_1) ~ & 0 ~ & 0 ~ & 0 \\ 0 & \eta(x_2) & 0 & 0 \\ 0 ~ & 0 ~ & \eta(x_1) ~ & 0 \\ 0 & 0 & 0 & \eta(x_2) \end{array} \right) U =\left( \begin{array}{cccc} 0 \\ 0 \\ f(t) \\ 0 \end{array} \right) .$

Take $B_1 = \left( \begin{array}{cccc} ~0 ~ & 0 ~ & 1 ~ & 0 \\ 0 & 0 & 0 & 0 \\ 1 ~ & 0 ~ & 0 ~ & 0 \\ 0 & 0 & 0 & 0 \end{array} \right)$, $B_2 = \left( \begin{array}{cccc} ~0 ~ & 0 ~ & 0 ~ & 0 \\ 0 & 0 & 0 & 1 \\ 0 ~ & 0 ~ & 0 ~ & 0 \\ 0 & 1 & 0 & 0 \end{array} \right)$, then define a matrix

$B (y_1,y_2)= y_1 B_1 + y_2 B_2 = \left( \begin{array}{cccc} ~0 ~ & 0 ~ & y_1 ~ & 0 \\ 0 & 0 & 0 & y_2 \\ y_1 ~ & 0 ~ & 0 ~ & 0 \\ 0 & y_2 & 0 & 0 \end{array} \right)$.

The eigenvalues of $B$ are $-y_1, y_1, -y_2, y_2$, and the corresponding eigenvectors are $(-1, 0, 1, 0)', (1, 0, 1, 0)', (0, -1, 0, 1)', (0, 1, 0, 1)'$.

Since the eigenvalues of $B$ are distinct and all real and the four eigenvectors are linearly independent, the system of first order PDEs is hyperbolic.

• This question has been answered on http://mathoverflow.net/. I would like to express my gratitude to Robert Bryant for giving the answer and explaining the method of characteristics, and to Igor Khavkine for discussing the method of characteristics. Jan 8, 2015 at 1:56