4 1. DIFFERENTIA L EQUATION S AN D THEI R GEOMETR Y
E X A M P L E 2.2 . Th e Liouville equatio n u
xy
= e u i s Darboux-integrable , wit h
(2.9) I = y, I = u
yy
--u2y, J = x, J = u
xx
--u2x.
Likewise, th e syste m (2.6 ) i s easil y integrate d t o giv e th e well-know n formul a
(2.10) = log l 2 3x\v)92'{x)
{9l(y) + 92{x)Y
The proble m o f classifyin g al l Darboux-integrabl e hyperboli c equation s (2.5 )
is difficult , an d stil l open . A numbe r o f importan t result s o n thi s proble m ca n
be foun d i n [Go] , [Ve] , [BGH1], [AK2] , [JA ] an d [ZS] . I n th e cas e o f /-Gordo n
equations, w e have a classical resul t o f Lie, [Go] , which show s tha t th e tw o example s
given abov e essentiall y exhaus t al l th e possibilities .
T H E O R E M 2.3 . An f-Gordon equation
d2u
2'"»
£*-'«
is Darboux-integrable if and only if
(2.12) / / " - (/') 2 = 0 ,
where f is not identically zero.
It i s eas y t o construc t conservatio n law s fro m Darboux-integrabl e equation s
from th e knowledg e o f th e function s i" , /, J , J whic h ar e invarian t fo r th e character -
istics. Fo r example , th e 1-forms
(2.13) UJ
X
= (u
xx
- -ul)dx, uj
y
= (u
yy
- -u 2
y
)dy,
are close d wheneve r u i s a solutio n o f th e Liouvill e equation . I n Chapte r 9 , w e
will prov e tha t Darboux-integrabl e equation s hav e conservatio n law s o f arbitraril y
high orde r an d degree . W e wil l als o giv e necessar y an d sufficien t condition s fo r
Darboux-integrability a t any orde r fo r a genera l hyperboli c equatio n (2.5 ) i n term s
of th e vanishin g o f certai n recursivel y computabl e loca l invariants .
The propert y o f Darbou x integrabilit y tha t w e hav e jus t describe d applie s onl y
to som e specia l partia l differentia l equation s i n tw o independen t variables . I t i s
therefore natura l t o as k i f there exis t partia l differentia l equation s i n mor e tha n tw o
independent variable s whic h ar e o f interes t an d whic h ar e amenabl e t o a simila r
geometric treatment . Thi s questio n wil l b e considere d i n Chapter s 4 , 5 an d 6 ,
where w e wil l se e tha t ther e i s a geometri c transformatio n an d integratio n theor y
for involutiv e over-determine d system s o f second-order partia l differentia l equation s
of th e for m
(2.14) y
kl
+ a k
kl
{x\ . . . , x n)yk + a!
kl
(x\ . . . , x n)yi + c
kl
{x\ . . . , x n)y = 0 ,
where 1 k ^ I n , an d wher e ther e i s n o summatio n o n repeate d indices . Thes e
over-determined system s gover n th e conserve d densitie s fo r a clas s o f hyperboli c
systems o f conservatio n law s studie d b y D . Serre , [Se] , an d Tsarev , [Tsa] , whic h
we wil l conside r i n Chapte r 6 . Th e geometri c conten t o f thi s transformatio n theor y
is base d o n th e classica l theor y o f lin e congruence s i n th e projectiv e differentia l
geometry o f submanifolds , [Ch2] , [Ch3] . I t wil l b e reviewe d i n Chapte r 4 i n th e
case o f immerse d surfaces , an d i n Chapte r 5 fo r higher-dimensiona l submanifolds .
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