2.HYPERBOLIC EQUATIONS INTEGRABLE BY THE METHOD OF DARBOUX 3
2. Hyperboli c equation s integrabl e b y th e metho d o f Darbou x
We shal l b e rathe r imprecis e fo r no w i n ou r descriptio n o f th e propert y o f
Darboux integrabilit y fo r second-orde r hyperboli c partia l differentia l equations . A
geometric set-u p i n term s o f exterio r differentia l system s fo r th e stud y o f genera l
second-order hyperboli c partia l differentia l equation s i n tw o independen t variable s
and a precis e definitio n o f Darbou x integrabilit y wil l b e give n i n Chapte r 9 . W e
consider th e basi c exampl e o f the /-Gordo n equatio n
d2u
2 1
» £&-'"
whose characteristic vector fields
1
ar e give n by , [Gl] , [GK] ,
d d d B d
(2.2) D
x
= +u x-— + uxx- \-f- Vu xxx—- h ...,
OX OU OU
x
OUy OU
xx
O d d
d d
(2-3) D v=*:+uvar
+ fiir +
uwizrdu
+
D*f
dy du du
x
du
y xx
where (x, y, u, ux, u
y
,uxx,uyy,...) ar e local coordinates o n the manifol d associate d
to th e infinit e prolongatio n 2 of the /-Gordo n equatio n (2.1).
An /-Gordo n equatio n i s sai d t o b e Darboux-integrable at order two i f ther e
exist functionall y independen t C°° function s 1,1, J, J o f (x,y,u,u x,uy,uxx,uyy)
which satisf y
(2.4) D
X
(I) = D
X
{I) = 0 , D
y
(J) = D
y
(J) = 0.
The notio n o f Darbou x integrabilit y i s define d similarl y a t an y orde r k 2 , an d
also fo r genera l hyperboli c equation s
, . _
/
du du d
2u
d
2u
d
2u.
(2.5) F(x,y,u, , , ^-^ , ^ - ^ - , —^) = 0.
ox oy ox
1
oxoy oy
l
We shal l se e i n Chapte r 9 that i f (2.5 ) i s Darboux-integrable, the n fo r an y pai r o f
monotone function s /i , /2 G C°°(M;R), th e syste m o f partial differentia l equation s
given b y
(2.6) F = 0, I = MI), J = f 2(J),
is completely integrabl e i n th e sens e th e Frobeniu s theorem . Th e Cauch y proble m
for Darboux-integrable equation s (2.5 ) is thus solvable by ordinary differential equa -
tions, i n analogy with the cas e of first-order partia l differentia l equations , which we
reviewed i n Sectio n 1.
EXAMPLE
2.1. Th e wave equation u
xy
= 0 is Darboux-integrable, wit h
(2.7) I = y, I = u y, J = x, J = u x.
Letting / i = g[, $2 = g f
2
, the syste m (2.6 ) gives at onc e
(2.8) u = g
2
{x) + g
1
{y).
These vecto r field s ar e no t Cauch y characteristic .
!The infinit e prolongatio n o f a differentia l equatio n wil l b e define d i n Chapte r 9 .
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