example of rewriting a differential equation as a Pfaffian system

To show how one may reformulate a differential equation as Pfaff’s problem for a set of differential forms, consider the wave equation

$$\frac{{\partial }^{2}u}{\partial t^2}=\frac{{\partial }^{2}u}{\partial x^2}+\frac{{\partial }^{2}u}{\partial y^2}$$

The first step is to rewrite the equation as a system of first-order equations

$$\frac{\partial a}{\partial t}-\frac{\partial b}{\partial x}-\frac{\partial c}{\partial y}=0$$

$$\frac{\partial u}{\partial t}-a=0$$

$$\frac{\partial u}{\partial x}-b=0$$

$$\frac{\partial u}{\partial y}-c=0$$

To translate these equations into the language of differential forms, we shall use the fact that

$$du=\frac{\partial u}{\partial t}dt+\frac{\partial u}{\partial x}dx+\frac{\partial u}{\partial y}dy$$

from which it follows that

$$du\wedge dx\wedge dy=\frac{\partial u}{\partial t}dt\wedge dx\wedge dy$$

$$du\wedge dy\wedge dt=\frac{\partial u}{\partial x}dt\wedge dx\wedge dy$$

$$du\wedge dt\wedge dx=\frac{\partial u}{\partial y}dt\wedge dx\wedge dy$$

We can do likewise with $a$ or $b$ or $c$ in the place of $u$; there is no point in repeating the formulas for each of these variables.

Multiplying the differential equations through by the form $dt\wedge dx\wedge dy$ and using the above identities to eliminate partial derivatives, we obtain the following system of differential forms:

$$da\wedge dx\wedge dy-db\wedge dy\wedge dt-dc\wedge dt\wedge dx$$

$$du\wedge dx\wedge dy-adt\wedge dx\wedge dy$$

$$du\wedge dy\wedge dt-bdt\wedge dx\wedge dy$$

$$du\wedge dt\wedge dx-cdt\wedge dx\wedge dy$$

From the way these forms were constructed, it is clear that a three dimensional surface in the seven dimensional space with coordinates $x,y,t,a,b,c,u$ which solves Pfaff’s problem and can be parameterized by $x,y,t$ corresponds to the graph of a solution to the system of differential equations, and hence to a solution of the wave equation.

Note: These considerations are purely local. The global topology of the seven-dimensional space will depend on the domain on which the original wave equation was formulated and on the boundary conditions.

Title	example of rewriting a differential equation as a Pfaffian system
Canonical name	ExampleOfRewritingADifferentialEquationAsAPfaffianSystem
Date of creation	2013-03-22 14:38:49
Last modified on	2013-03-22 14:38:49
Owner	rspuzio (6075)
Last modified by	rspuzio (6075)
Numerical id	6
Author	rspuzio (6075)
Entry type	Example
Classification	msc 53B99