The classification of a system of first-order PDEs is based on whether there are directions along which the PDEs reduce to ODEs. To be more precise, assume that we are given a system of equations in unknowns which we write in matrix form as
where , , and is an matrix.
Now we ask whether there is a family of curves along which the PDEs reduce to a system of ODEs, that is, in which the directional derivative of each occurs in the same direction. We consider a row vector to be determined later. Then
We want the above system to have the form of a linear combination of total derivatives of the in the same direction , that is, we want our system to have the form
for some . Consequently, we require
This means that is an eigenvalue of and is a corresponding left eigenvector. Note that as well as can depend on , , and . So, if is an eigenpair, then
and the system of PDEs is reduced to a single ODE along the family of curves, called characteristics, defined by . The eigenvalue is called the characteristics direction. Clearly, because there are unknowns, it would appear that ODEs are required; but if has distinct real eigenvalues, there are ODEs, each holding along a characteristics direction defined by an eigenvalue. In this case we say that the system is hyperbolic.
Definition. The quasilinear system
is hyperbolic if has real eigenvalues and linearly independent left eigenvectors. Once these eigenvectors are distinct, the system is called stricly hyperbolic.
The system is called elliptic if has no real eigenvalues, and it is parabolic if has real eigenvalues but fewer then independent left eigenvectors.
No exhaustive classification is made in the case that has both real and complex eigenvalues. Note that once matrix has distinct, real eigenvalues it has independent left eigenvectors, because distinct eigenvalues have independent eigenvectors.
More general systems of the form
can be considered as well. We refer the reader to a book entitled “An introduction to nonlinear partial differential equations” due to J.D. Logan.
We are now in a position to see why a single first-order quasilinear PDE is hyperbolic. The coefficient matrix for the equation
is just the real scalar function which has the single eigenvalue and its corresponding eigenvector , a constant function. In this direction, once the PDE reduces to the ODE . We refer the reader to the following topic, called characteristic curves, where we consider when the equation has constant coefficients and variable coefficients.
We place here three more examples.
Example 1 (The shallow-water equations). The following system
is trictly hyperbolic.
Example 2. The following system
is elliptic if and is hyperbolic if .
Example 3 (The diffusion equations). The following equation
may be written as the first-order system
and thus is parabolic.