2.1.5 Solution along characteristic curves

For a quasi-linear PDE

$$\frac{\partial \rho}{\partial t} + v_{} \! \left( \rho, x, t \right) \frac{\partial \rho}{\partial x} = 0$$ (67)

a curve $c(t)$ with

$$\frac{\mathrm{d} c}{\mathrm{d} t} = v_{}$$ (68)

or sometimes the corresponding map $\mathbb{R} \rightarrow \mathbb{R}^2\!\!: t \rightarrow \left( c \left( t \right) , t \right)$ is called characteristic curve or characteristic. For the linear advection equation (63) these curves have the general form

$$c(t) = v_{} t + x_0 \enspace .$$ (69)

Along $c(t)$ we get for a solution $\tilde{\rho}$ of Eq. (63)

$$\frac{\mathrm{d} }{\mathrm{d} t} \tilde{\rho} \left( c \lef... ...{} \frac{\partial \tilde{\rho}}{\partial x} = 0 \enspace .$$ (70)

The solution $\tilde{\rho}$ is constant along the characteristic $c(t)$.