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ExteriorDifferentialSystems[CauchyCharacteristics] - find the Cauchy characteristic vector fields for a Pfaffian system

Calling Sequences

     CauchyCharacteristics(Theta)

Parameters

   Theta   - a list of 1-forms on a manifold M

 

Description

• 

 For a general exterior differential system ℐ, a Cauchy characteristic is a vector field X such ιXω ∈ ℐ for all ω ∈ ℐ.  This condition need only be checked on a set of algebraic generators for ℐ. Thus, for a Pfaffian system defined by 1-forms Θ =θ1,θ2,..., θp, a vector field X is a Cauchy characteristic if  ιXθi = 0 and ιXd θi ∈span  θ1,θ2,... ,θp for each i =1, 2,..., p.  This latter requirement is equivalent to:

  ιXd θi∧θ1∧θ2∧ ⋅⋅⋅ ∧ θp =0.   

• 

The command CauchyCharacteristics(Theta) returns a list of Cauchy characteristics for the Pfaffian system generated by Theta at a generic point of the manifold M.  

• 

Let Cauℐ be the set of all Cauchy characteristics for a general differential system ℐ.  Cauℐ is an integrable distribution in the sense that if X ,Y ∈ Cauℐ, then the Lie bracket X, Y ∈Cauℐ. If Cauℐ has constant rank and u1, u2, ...,um are the first integrals of Cauℐ (that is,  Xuk = 0 for all X ∈ Cauℐ), then there are a set of forms involving only the variables u1, u2, ...,um which generate ℐ.

See Also

DifferentialGeometry

ExteriorDifferentialSystems

Examples

> 

with(DifferentialGeometry): with(ExteriorDifferentialSystems):

 

Example 1

The Pfaffian system defined by any scalar partial differential equation in one dependent variable always admits a Cauchy characteristic vector. We use the PDE ux +zuy + xuz = u  to illustrate this fact.

 

> 

DGsetup([x, y, z, u, p2, p3], M1);

frame name: M1

(1)

 

The Pfaffian system for scalar PDE consists of a single 1 form.

> 

Theta := evalDG([du - (u- z*p2 + x*p3)*dx - p2*dy - p3*dz]);

Θ:=p2⁢z−p3⁢x−u⁢dx−p2⁢dy−p3⁢dz+du

(2)
M1 > 

CauchyCharacteristics(Theta);

D_xz+D_y−x⁢D_zz+u⁢D_uz+p2⁢D_p2z−p2−p3⁢D_p3z

(3)

 

Example 2

The Pfaffian system for any involutive system of 2 second order PDE in two independent variables and one dependent variable always admits a Cauchy characteristic. We illustrate this fact with the system uxx =13 uyy3 ,  uxy  = 12uyy2.

 

M1 > 

DGsetup([x, y, u, p ,q, t], M2);

frame name: M2

(4)

 

The Pfaffian system for (any) number of second order PDE in two independent variables and one dependent variable is always defined by three 1-forms.

M2 > 

Theta := evalDG([du - p*dx - q*dy, dp -1/3*t^3 *dx - 1/2*t^2*dy, dq - 1/2*t^2*dx - t*dy]);

Θ:=−dx⁢p−dy⁢q+du,−13⁢t3⁢dx−12⁢t2⁢dy+dp,−12⁢t2⁢dx−t⁢dy+dq

(5)
M2 > 

CauchyCharacteristics(Theta);

−6⁢D_xt3+6⁢D_yt2−6⁢−q⁢t+p⁢D_ut3+D_p+3⁢D_qt

(6)

 

Example 3

Here is an example of a rank 2 Pfaffian system, defined on a 7-dimensional manifold, which admits two Cauchy characteristics. This example arises in the study of parabolic PDE in two independent variables and one dependent variable with vanishing Goursat invariant. Our example depends upon an arbitrary function ψ λ.

 

M2 > 

DGsetup([x, y, z, p, q, lambda, t], M3);

frame name: M3

(7)

 

Define three 1-forms.

M3 > 

o0 := evalDG(dz - p*dx - q*dy);

o0:=−dx⁢p−dy⁢q+dz

(8)
M3 > 

o1 := evalDG(dp -(lambda^2*t + 2*lambda*diff(psi(lambda) , lambda) - 2*psi(lambda))*dx - (-lambda*t - diff(psi(lambda) , lambda))*dy);

o1:=−λ2⁢t+2⁢λ⁢ⅆⅆλ⁢ψ⁡λ−2⁢ψ⁡λ⁢dx+λ⁢t+ⅆⅆλ⁢ψ⁡λ⁢dy+dp

(9)
M2 > 

o2 := evalDG(dq - (-lambda*t - diff(psi(lambda) , lambda))*dx - t*dy);

o2:=λ⁢t+ⅆⅆλ⁢ψ⁡λ⁢dx−t⁢dy+dq

(10)

 

Our differential system is given in terms of these forms as follows.

M2 > 

Omega := evalDG([o, o1 + lambda*o2]);

Ω:=o,−λ⁢ⅆⅆλ⁢ψ⁡λ−2⁢ψ⁡λ⁢dx+ⅆⅆλ⁢ψ⁡λ⁢dy+dp+λ⁢dq

(11)
M2 > 

Cau := CauchyCharacteristics(Theta);

Cau:=−6⁢D_xt3+6⁢D_yt2−6⁢−q⁢t+p⁢D_ut3+D_p+3⁢D_qt

(12)