# 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

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

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

 ${\partial a\over\partial t}-{\partial b\over\partial x}-{\partial c\over% \partial y}=0$
 ${\partial u\over\partial t}-a=0$
 ${\partial u\over\partial x}-b=0$
 ${\partial u\over\partial y}-c=0$

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

 $du={\partial u\over\partial t}\,dt+{\partial u\over\partial x}\,dx+{\partial u% \over\partial y}\,dy$

from which it follows that

 $du\wedge dx\wedge dy={\partial u\over\partial t}\,dt\wedge dx\wedge dy$
 $du\wedge dy\wedge dt={\partial u\over\partial x}\,dt\wedge dx\wedge dy$
 $du\wedge dt\wedge dx={\partial u\over\partial y}\,dt\wedge dx\wedge dy$

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:

 $da\wedge dx\wedge dy-db\wedge dy\wedge dt-dc\wedge dt\wedge dx$
 $du\wedge dx\wedge dy-a\,dt\wedge dx\wedge dy$
 $du\wedge dy\wedge dt-b\,dt\wedge dx\wedge dy$
 $du\wedge dt\wedge dx-c\,dt\wedge dx\wedge dy$

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 ExampleOfRewritingADifferentialEquationAsAPfaffianSystem 2013-03-22 14:38:49 2013-03-22 14:38:49 rspuzio (6075) rspuzio (6075) 6 rspuzio (6075) Example msc 53B99