# 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 |
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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 |