Characteristics

Let us consider the equation:  

$${\bf L}u-b({\bf r}){\partial^{2}u \over \partial t^{2}}=0$$ (18)

where:

$${\bf L}=\sum_{i,k=1}^3 a_{ik}({\bf r}){\partial ^{2}\over \partial x_{i}\partial x_{k}} + \ldots$$

$(x_{1}=x, x_{2}=y, x_{3}=z, {\bf r}=(x,y,z))$. The three dots represent terms of lower order.

Let $\psi(x,y,z,t)=0$ be a surface of arbitrary shape and let the values of $u,{\partial u\over \partial x_i} (i=1,2,3)$ and ${\partial u\over \partial t}$ be given on the surface. Is this data sufficient for the reconstruction of the unique solution of the equation (18)? Does the Cauchy problem for equation (18) with the initial data on $\psi$ have a unique solution? The answer will be positive if it is possible to reconstruct all second derivatives on the surface $\psi$.

Let us introduce a system of functions $\omega_{i}(x,y,z,t) (i=1,2,3,4)$,with $\omega_{1} \equiv \psi (x,y,z,t)$ and with the change of variables:

$$x_{s}^{\prime} = \omega_{s}({\bf r},t),{\:}s=1,2,3,4.$$

On the surface $\omega_{1} \equiv x_{1}^{\prime}=0$ all values $x_{2}^{\prime},x_{3}^{\prime}, x_{4}^{\prime}$ do vary. That means the given data allows one to calculate all mixed derivatives ${\partial^{2}u\over \partial x_{i}^{\prime} \partial x_{j}^{\prime}} (i,j=1,2,3,4)$and second derivatives ${\partial^{2}u\over \partial {x_{i}^{\prime}}^{2}} (i=2,3,4)$.Is it possible to calculate ${\partial^{2}u\over \partial {x_{1}^{\prime}}^{2}}$?

Let us express the derivatives with respect to the old variables through the derivatives with respect to the new variables, omitting terms that are not interesting for us, such as:

$$u_{x_{i}}=u_{x_{1}^{\prime}} {\partial\omega_{1}\over \parti... ...tial x_{i}} {\partial \omega_{1} \over \partial x_{k}} + \ldots$$

and so on.

The equation (18) can be rewritten in the form:  

$$a_{11}^{\prime} u_{x_{1}^{\prime}x_{1}^{\prime}}+ \ldots = 0$$ (19)

where:

$$a_{11}^{\prime}= \sum_{i,k=1}^{3} a_{ik} {\partial\omega_{1}... ...}}-b{\left({\partial\omega_{1} \over \partial t}\right) } ^{2}.$$

All terms denoted by dots are known on the surface $\omega_{1}=0$,so the only condition that supplies a possibility to determine $u_{x_{1}^{\prime} x_{1}^{\prime}}$ is $a_{11}^{\prime} \neq 0$.If $a_{11}^{\prime}=0$, then the Cauchy problem is unsolvable. In this case, surface $\psi=0$ is a special one. And we call it a characteristic (or characteristic surface).

So  

$$\sum_{i,k=1}^{3} a_{ik} {\partial\psi \over \partial x_{i}}{... ...k}} - b{\left( {\partial\psi \over \partial t}\right) }^{2} = 0$$ (20)

is a characteristic equation for the equation (18). But what does this have to do with discontinuities?