SELF-SIMILAR
SOLUTIONS
KORTEWEG-DE
I.
M.
VRIES
OF
EQUATIONS
OF
TYPE
Krichever
UDC 517.9
The p r e s e n t p a p e r w a s s t i m u l a t e d by the r e c e n t r e s u l t s [1] on the c o n s t r u c t i o n of s e l f - s i m i l a r s o l u t i o n s
of the m o d i f i e d K o r t e w e g - d e V r i e s e q u a t i o n s and the s i n - g o r d o n e q u a t i o n s . Such s o l u t i o n s c a n be d e s c r i b e d
by o r d i n a r y e q u a t i o n s of P a i n l e v e t y p e [2], f o r w h i c h a c o m m u t a t i o n r e p r e s e n t a t i o n is found [1]. The p r o b l e m
of c o n s t r u c t i o n of s i m u l t a n e o u s e i g e n f u n c t i o n s of a u x i l i a r y o p e r a t o r s r e d u c e s to the R i e m a n n p r o b l e m . A c c o r d ing to [1], the a n a l o g u e s of the d i s p e r s i o n d a t a a r e the t r a n s i t i o n m a t r i c e s at the j u m p s of this f u n c t i o n on
anti--Stoke s."
The p r i n c i p a l d i s t i n c t i o n of the p r o p o s e d s c h e m e is the u s e of Lax p a i r s f o r the i n i t i a l p a r t i a l d i f f e r e n t i a l
e q u a t i o n s . This a l l o w s one to unite in one c o n s t r u c t i o n the c o n s t r u c t i o n of s e l f - s i m i l a r s o l u t i o n s and o t h e r
w e l l - k n o w n c l a s s e s of e x a c t s o l u t i o n s : m u l t i s o l i t o n and r a t i o n a l . The g e n e r a l s o l u t i o n s o b t a i n e d in the r e a l m s
of o u r c o n s t r u c t i o n c o r r e s p o n d to " s o l i t o n s a g a i n s t the b a c k g r o u n d of s e l f - s i m i l a r s o l u t i o n s . " T h e r e is no
d i f f i c u l t y in c a r r y i n g o v e r a l l the c o n s t r u c t i o n s to n o n t r i v i a l a l g e b r a i c c u r v e s ( m u l t i s o l i t o n s o l u t i o n s c o r r e s p o n d in the a l g e b r o g e o m e t r i c c o n s t r u c t i o n s to d e g e n e r a t i o n s of t h e s e c u r v e s to r a t i o n a l ones). This w i l l be
done in the d e t a i l e d p u b l i c a t i o n of the r e s u l t s of this p a p e r .
1. F o r d e f i n i t e n e s s we r e s t r i c t o u r s e l v e s to L a x ' e q u a t i o n s f o r o p e r a t o r s w i t h s c a l a r c o e f f i c i e n t s . W e
note t h a t h e r e , a s a l s o in the c o n s t r u c t i o n of f i n i t e - z o n e s o l u t i o n s of r a n k l > 1 (cf. [3, 4]), the c o n s t r u c t i o n of
s i m u l t a n e o u s e i g e n f u n c t i o n s of s c a l a r d i f f e r e n t i a l o p e r a t o r s u s e s the v e c t o r i a l R i e m a n n p r o b l e m .
L e t ~s be the a n g l e g i v i n g on the X plane the s - t h r a y d e f i n e d by the e q u a t i o n
Rei)~m(exp(~k)--exp(2~i~ n t))]--0,
t~]~, l ~ .
(1)
L e t us a s s o c i a t e w i t h e a c h r a y a c o n s t a n t m a t r i x G s s u c h that Gskk = 1, and the e l e m e n t G kl (k ~ l) c a n be nonz e r o only f o r t h o s e i n d i c e s for w h i c h on the c o r r e s p o n d i n g r a y (1) holds and f o r w h i c h i n a n e i g h b o r h o o d of this
r a y the l e f t s i d e of (1) d e c r e a s e s upon c o u n t e r c l o c k w i s e r o t a t i o n . We note t h a t if G kl ~ 0, t h e n G/sk = 0. The
i n d i c a t e d c o l l e c t i o n of r a y s is i n v a r i a n t w i t h r e s p e c t to r o t a t i o n by 2 ~ / n . L e t ~(s) be the c o r r e s p o n d i n g p e r m u t a t i o n of i n d i c e s . We r e q u i r e t h a t the r e l a t i o n
/o o . . . o
Go(s) = M-XGsM,
~i
31= | 0
t)
0..,0
0
I ... 0 0
\0
(2)
: ;0
be satisfied.
Let v st and Ps', 1 _< s' < Nn be two collections of complex numbers,
invariant with respect to rotation by
27r/n, and ~-(s') be the corresponding
permutation of indices. To them we associate constant vectors a s, =
(ash ..... O~s, n) and fis' = (fls'l, .... fis,n ) such that a~-(s') = °~s'M, fiT(S') = flS 'M.
The pairs (vs', as,),
(#s', fis') are called "Tyurin parameters"
as in [3, 4].
To each collection of enumerated
~D(x, t, X) = (¢~, . . . .
'$n) s u c h t h a t
"data of the inverse
problem"
corresponds
a unique
vector-function
1°. The f u n c t i o n ¢ has the f o r m v~x, t, k)~0, w h e r e v(x, t, M is a p i e c e w i s e m e r o m o r p h i c v e c t o r - f u n c t i o n
on the e x t e n d e d h p l a n e , and the m a t r i x ~'0 = exp (kQx + h m Q m t ) , Q k l = exp [ ( 2 u i / n ) k ] 6 k l .
2 °. O u t s i d e ~¢, ¢ is p i e c e w i s e m e r o m o r p h i c ,
w h e r e ¢_+s(r) = ~ (re~'%+-i°).
h a s j u m p s on the r a y s r e i ~ s , r > 0, w h i l e
~+~ (r) = ¢-~ (r)V~,
(3)
M o s c o w S t a t e U n i v e r s i t y . T r a n s l a t e d f r o m F u n k t s i o n a l ' n y i A n a l i z i E g o P r i l o z h e n i y a , Vol. 14, No. 3, pp.
8 3 - 8 4 , J u l y - S e p t e m b e r , 1980. O r i g i n a l a r t i c l e s u b m i t t e d D e c e m b e r 17, 1979o
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©1981 P l e n u m P u b l i s h i n g C o r p o r a t i o n
3 °. The poles of ,~ lie at the points v s ' , and for the r e s i d u e s of its c o m p o n e n t s one c a n u s e the r e l a t i o n
%,z resvs,~/= as'k resvs,~ ,
Moreover,
t < k, 1~ n .
(4)
let
n
>J ~,~*z (x, t, ~,) = 0.
(5)
4 °.
LEMMA 1. T h e r e e x i s t s a u n i q u e v e c t o r - f u n c t i o n ¢ s a t i s f y i n g the e n u m e r a t e d r e q u i r e m e n t s and n o r r e a l i z e d by the c o n d i t i o n v(x, t, h) ~ (1, . . . .
1), h ~ ~.
The p r o b l e m of c o n s t r u c t i n g $ is e q u i v a l e n t w i t h the R i e m a n n p r o b l e m on c o n s t r u c t i n g v(x, t, M. C o n d i t i o n 2 ° is t r a n s f o r m e d h e r e into
V+s (r) =
v-s (r)Gs
(3D
(r),
w h e r e ~ (r) = ~o (rei(Ps)GsTo* (rei %). The r e s t r i c t i o n s on the f o r m of Gs a r e s u f f i c i e n t for the e x i s t e n c e of a s o l u t i o n
of the R i e m a n n p r o b l e m . In f a c t , we c o n s i d e r the p i e c e w i s e c o n s t a n t f u n c t i o n G, e q u a l to Gs i n the s e c t o r s
% -< ~ -< ~s + 5q~, 1 o u t s i d e t h e m . T h e n ¢1 = ¢ G-1 s a t i s f i e s a l l the r e q u i r e m e n t s i m p o s e d on ¢, e x c e p t for one:
it has j u m p s on the r a y s ~s + 6¢. F o r s m a l l 6~ > 0, it follows f r o m the r e s t r i c t i o n s on the f o r m Gs that the
j u m p f u n c t i o n Gs (r) for ~s + 5~ d e c r e a s e s e x p o n e n t i a l l y as r - - ~. The n u m b e r of l i n e a r l y i n d e p e n d e n t s o l u t i o n s of the p r o b l e m (3), (4), (5) is e q u a l to n. C o n d i t i o n (6) s i n g l e s out a v e c t o r - f u n c t i o n ~ w h i c h is u n i q u e up
to p r o p o r t i o n a l i t y .
The o p e r a t o r s ~x, 3t and the o p e r a t o r of m u l t i p l i c a t i o n by An t r a n s f o r m ~ into a f u n c t i o n s u b j e c t to the
s a m e r e s t r i c t i o n s as ¢, but v(x, t, M, w h i c h f i g u r e s i n c o n d i t i o n 1°, c a n have a pole a t h = 00. I n the s t a n d a r d
w a y , c o m p a r i n g s i n g u l a r p a r t s one gets
T H E O R E M 1. T h e r e e x i s t u n i q u e o p e r a t o r s
Ln=
~0
u i ( z , t) Oix,
Lm=
~-J ~ i ( x ' t ) O/x
i~O
w i t h s c a l a r c o e f f i c i e n t s , s u c h that
L ~ = ~=¢,
(0t -- L,~)¢ = 0.
COROLLARY 1. The o p e r a t o r s L n and L m s a t i s f y the e q u a t i o n
[Ln, Ot - - Lm] = O.
COROLLARY 2. Suppose i n the g i v e n i n v e r s e p r o b l e m the T y u r i n p a r a m e t e r s a r e a b s e n t , i . e . , N = 0.
T h e n the c o n s t r u c t i o n i n d i c a t e d l e a d s to a s e l f - s i m i l a r s o l u t i o n of Lax' equation.
The a s s e r t i o n of the c o r o l l a r y follows f r o m the fact that for N = 0 the data a r e i n v a r i a n t w i t h r e s p e c t to
the s c a l e g r o u p , w h i c h m e a n s ¢(x, t, k) = ¢(/3x, fimt, p - l M . S i m p l e c a l c u l a t i o n shows that the n u m b e r of p a r a m e t e r s i s e q u a l to r e ( n - 1), i . e . , for m u t u a l l y p r i m e n and m it is e q u a l to the d i m e n s i o n of the s p a c e of s e l f similar solutions.
If a l l Gs = 1, t h e n the c o n s t r u c t i o n i n d i c a t e d g i v e s a s o l i t o n s o l u t i o n .
R e m a r k . Change of (5) to a s y s t e m of c o n d i t i o n s a n a l o g o u s to [5] i n c l u d e s i n this s c h e m e a l s o the c o n s t r u c t i o n of r a t i o n a l s o l u t i o n s .
II.
Two-Dimensional
Systems
of
KdeV
Type
L e t JJ(x, y , t, ~) be d e f i n e d by 1°-4 ° w i t h the one change i n 1°:
~F0 (x, y, t, )~) = exp (~Qx ~- ~m,Qm,y _~_~mQmt)"
W i t h o u t l o s s of g e n e r a l i t y one c a n a s s u m e that ms < m. H e r e ¢~x, y, t, ;Q e x i s t s and is unique.
T H E O R E M 2. T h e r e a l s o e x i s t u n i q u e s c a l a r o p e r a t o r s L m and L m l , w h o s e c o e f f i c i e n t s depend on x , y ,
t, s u c h that
235
( O y - - Lm~ ) ~p = (Or - - L~n) ~ = O.
COROLLARY 1. [Oy - Lml , 3 t - Lm] = 0.
COROLLARY 2. F o r N = 0, the solutions of the equations obtained a r e s e l f - s i m i l a r .
F o r m s = 2, m = 3 we get a s e l f - s i m i l a r solution of the K a d o m t s e v - P e t v i a s h v i l i equation.
LITERATURE
1.
2.
3.
4.
5.
CITED
H. F l a s c h k e and A. Newell, P r e p r i n t , U n i v e r s i t y of A r i z o n a , T u c s o n (1979).
M. Ablowitz, P r e p r i n t , P o t s d a m (1978).
I. M.
K r i c h e v e r , Funkts. Anal. P r i l o z h e n . , 12, No. 3, 20-31 (1978).
I. M. K r i c h e v e r and S. P. Novikov, Funkts. Anal. P r i l o z h e n . , 12, No. 4, 41-52 (1978).
I. M. K r i c h e v e r , Funkts. Anal. P r i l o z h e n . , 12, No. 1, 76-78 (1978).
ASYMPTOTICS
CLOSE
FUNCTION
OF
AN
DIFFERENTIAL
Ya.
TO
ELLIPTIC
THE
BOUNDARY
OF
A
SPECTRAL
SECOND-ORDER
OPERATOR
V.
Kurylev
UDC 534.213
1. Let A be a p o s i t i v e - d e f i n i t e elliptic differential o p e r a t o r in L 2 (M); M be a bounded d o m a i n in R m,
m _> 2. The o p e r a t o r A is defined by a differential e x p r e s s i o n of the f o r m
+
Au = --
£o3 (x) Oxi a x j - ~, j = l
(1)
i
4=1
on functions s a t i s f y i n g s o m e s e l f - a d j o i n t b o u n d a r y condition (Dirichiet or of the third kind). It will be a s s u m e d
that a~•J' (z) = aj0i (z), ai1 (z), a~ (z) ~ C C ¢ (@) , the b o u n d a r y aM is infinitely d i f f e r e n t i a b l e , and the o p e r a t o r A is unim
f o r m l y elliptic, i.e.,
~
%iJ(x)~i~j~CiSi2, C>0, x ~ M .
i, j = l
2. The fundamental r e s u l t of the p r e s e n t note is the
THEOREM. Let A be a differential o p e r a t o r of the f o r m (1), e(x, y, ;~) be the s p e c t r a l function of the
o p e r a t o r A. T h e n for any e > 0 t h e r e exist c o n s t a n t s C~'2 s u c h that
-- \~neq!
J
for ~ = ~ (z, y) ~ c ~ -(*/')-8, ~ -~ ~.
We explain the use of the notation. T = T(X, y) is the g e o d e s i c d i s t a n c e b e t w e e n x and y, Tneg = Tneg(X , y)
is the "negative eikonal" (cf. [4] for m o r e detail). F o r n(x) >- CX-1/4, Tneg is defined as the g e o d e s i c length
of a m i n i m a l t w o - c o m p o n e n t c u r v e joining x and y and having v e r t e x on aM, for n(x) _< C)~-1/4, Tneg = {s2(x, y) +
(n(x) + n(y))2} 1/2, w h e r e s(x, y) is the g e o d e s i c d i s t a n c e along aM between the p r o j e c t i o n s on aM o f x and y.
H e r e n(x) [ r e s p e c t i v e l y , n(y)] is the g e o d e s i c d i s t a n c e in the m e t r i c g e n e r a t e d by A, f r o m x to aM ( r e s p e c tively, f r o m y to aM). p(x) = det 1/2 II a~j II, w h e r e II a~j(x)H = II aioJ(x)[[-~. F i n a l l y , J m / 2 (z) is the B e s s e l function
of index m / 2 . The s i g n + in (2) c o r r e s p o n d s to a b o u n d a r y condition of the third kind, the s i g n - to a D i r i c h l e t
b o u n d a r y condition.
L e n i n g r a d Section of the V. A. Steklov M a t h e m a t i c a l Institute, A c a d e m y of S c i e n c e s of the USSR. T r a n s lated f r o m F u n k t s i o n a l ' n y i Analiz i P r i l o z h e n i y a , Vol. 14, No. 3, pp. 85-86, J u l y - S e p t e m b e r , 1980. Original
a r t i c l e s u b m i t t e d May 8, 1979.
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