From: foltinek@math.utexas.edu (Kevin Foltinek)
Subject: Re: Proof for Lagrange's method of D.E.
Date: 06 Jan 2000 10:54:03 -0600
Newsgroups: sci.math
Summary: Cauchy characteristics, equations of Charpit and Lagrange

In article <845mg0$h2a$1@lure.pipex.net> "Colin Gillespie"
<coling@theseed.net> writes:

> To solve D.E. of the form
> 
> dQ/dt =A*dQ/ds + B * dQ/dz
> 
> where, Q(s,z,t) and A, B are functions (quite nice) of (s,z,Q). To solve
> this
> I have found the auxiliary equations
> 
> dt/1 = ds/A = dz/B
> [snip]
> If anyone has any references that I could look up it would be much
> appreciated.

Look for "Cauchy characteristics".  Most books on P.D.E.'s contain a
discussion of them.

In general, for a P.D.E. of the form
  F(x1,...,xn,u,u_{x1},...,u_{xn}) = 0
the Cauchy characteristics give you a system of O.D.E.'s called the
equations of Charpit and Lagrange.

The main idea is the following.  The initial conditions for the PDE
will be given by specifying the values of u on a (non-characteristic)
(n-1)-dimensional manifold M; the graph of u over M gives an
(n-1)-dimensional manifold N inside of the (n+1)-dimensional
(x,u)-space.  You flow this manifold N along the vector field defined
by the ODEs of Charpit and Lagrange; the resulting n-dimensional
object is (locally) the graph of the solution u(x).

In concrete terms, say that N was parametrized by (s1,...,s{n-1});
that is, (x1,...,xn,u) are functions of (s1,...,s{n-1}).  Solve the
ODE's, giving (x1,...,xn,u) as functions of t, parametrized by the
(n+1) constants (x1_0,...,xn_0,u_0).  These (n+1) constants are then
chosen to be the points on the set N; that is, they are parametrized
by (s1,...,s{n-1}).  Then you have (x1,...,xn,u) as functions of
(t,s1,...,s{n-1}).  Then solve for u as a function of (x1,...,xn), by
eliminating the variables (t,s1,...,s{n-1}).  (Often, but not always,
it is possible to take x1=s1,...,x{n-1}=s{n-1}, at the very
beginning, so you end up just having to eliminate t.)

Of course in general not much of this can be explicitly written down,
and all you have is a statement that the solution is the manifold
given by the flow of a smaller manifold along a vector field.

Kevin.