Journal of Mathematical Sciences, VoL 73, No. 4, 1995
PASSAGE
TO THE
LIMIT
P2 --*/'1
A. A. Kapaev and A. V. Kitaev
UDC 517.9
A method is given allowing o n e t o consider the well known formal passage to the limit P2 ~ 1;>1 as the double asymptotics of solutions of the equation 1>2 in a special "transition" domain characterized by the ratio a2 / z 3, where c, is the parameter of P2 and z is its argument. It is shown that the sequence of iterated B~cklund transformations of generic solutions of 1='2 is described by a generic solution of 1'1. The iterations of B~icklund transformations of solutions of P~, both rational and separatriz for z ~ - c o , are studied. Bibliography: 12 titles.
The present paper is concerned with the formal passage to the limit (as ~ ~ 0) [1], 722(X2) :
e - 5 "~- ~Ul(Xl),
X2 = --6e -1~ + e2Xl,
C~2 = --4e -15 -- e--3fl,
(1)
transforming the equation P2 : u~ = 2(u2) 3 -1- x2u 2 - v~2 into the equation e l : u "1 = 6(ul) 2 + zl +/3. The aim of the paper is to illustrate the general m e t h o d of [2] with the simplest example; namely, to translate the limit (1) from the formal language of equations into the language of specific solutions, so that (1) could be treated as the double asymptotics (with respect to the argument x2 and the coefficient 82 under a certain specific relation between them) of the solutions of P2To state our result it is necessary to recall the main facts of the theory of the first and second Painlev~ equations (P1 and P2), cf. [3-5]. First of all, the solutions of P2 and P1 are one-to-one parametrized by the m o n o d r o m y data, the latter constituting manifolds M2 and M1 respectively. These are complex manifolds of dimension three and two, respectively, described as follows. The manifold M2 is defined in the space C 4 with the coordinates ( 2 8 1 , 2 8 2 , 2 8 3 , Or2) by the equation 2 8 1 - - 2 8 2 "31-2 8 3 J1- 2 3 1 9 2 3 2 " 2 5 3 =
-2sinTrc~2.
In addition a copy of C P 1 must be glued to each of the singular points ( ( - 1 ) n+l, ( - 1 ) n, ( - 1 ) nq-1 , ~1 if- n). The manifold M1 is defined in C 5 with the coordinates (181,182,183,184, lss) by the equations
i-183 181 =
1 "3I- 1 8 2 9 1 8 3
182 =
183 =
,
i~ 1 3 5 =
184 =
i-182
' , 185 = I --~ 1 8 2 9 1 8 3
0 , 181 "~ 1 8 4 :
i,
for
i(l+
1
82 " 183),
1 .ql_ 1 8 2 9 183 =
for
1+1s2"183 r
0.
For details see papers [3, 4], where the asymptotics of solutions of/>2 and P1 are parametrized by the points of these manifolds. To state the result we also need the Bgcklund transformation for P2 [6], ~=+1,
1 + 2cra2 ~2=u2+2u~_a(2(u2)2+x2)
,
~~ I u2=u2+~((~2)2--(u2)2),
~2=--~--~2,
(9.)
where fi2(z2) is the solution of P2 corresponding to the coefficient &2Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 187, pp. 75-87, 1990. Original article submitted December 29, 1990. 460
1072-3374/95/7304-0.460512.50 (~) 1995 Plenum Publishing Corporation
The mentioned correspondence Pk ~ Mk (k = 1, 2) is such that we have
P~ ~ M~: : k ( ~ , - ~ , - ~ , - ~ )
I
2~4--k(X2,~2,~2,~2) 21ri n
27ri
e-'5--nu2,
(3)
= -:k(~,"~,~L~), :
(4)
2 8 k ( X 2 , U 2 , t ~ , Ot2 ) ,
4~rln 1
n = 0,+1,+2,... ,
: k ( ~ , a~, %, ,~) = : ~ ( ~ , u~, ~,~,~).
%) (6)
The bar in Eq. (4) and below denotes the complex conjugation. W h e n applying Eq. (5), one should use the relation 28k+3 = --2,.qk ( k = 0 , :kl, + 2 , . . . ). The function fi2 in (6) is defined by (2). For the other correspondence we h a v e 18k+2n(e--2rdn/S X l , e4rdn/5 Ul, e61rin/Sutl) -~- 1 8 k ( X l ,
t~l, t t l ) ,
(7)
l~;(X'1,Ul,~-41) = --1S5_k(Xl,Ul, t/,i).
(8)
W h e n applying Eq. (7), one should use the relation 18k+5 18k, which holds for any integer k. Let u~ be an arbitrary fixed solution of P2 corresponding to the coefficient a ~ If we step by step apply the Bs transformation (2) to this equation, taking ~r = ~rk = ( - 1 ) TM on the k-th step, we obtain the Sequence u~ of solutions of :P2 for the coefficient a2k = (--1)k(a ~ + k). Define the sequence ek to be that of real roots of the equation =
(--1)k(c~ ~ + k) Obviously, ek ~ 0 Theorem.
as
=
(9)
--4Sk 15 - fls~-z.
k--* oo.
For the even and odd subsequences as n -+ +oo the relations 0 --10 + %[2~](-6~.
~~
~2nX 1) =
~ --5 . + ~
+ 1](-6~;~L + ~ L + : I )
= s
UeV/X ~ ~ ~:, +
-'
o ( ~1 + 6 ), (~ >
J~- E 2 n + l u l ~
"~
O)
(I0)
o { s 2n+t)
(11)
hold, where u~V and u~ d are the solutions of the t~rst PainlevJ equation P1 to which there correspond, respectively, the points of M1 with coordinates 281 ~
281 ~
seve -ir~176, 2"~3
--1 3
od iTrc~~ 1S3 e ~ 283 ~
--
sere ir:a~ , 2 S 2 .=
= 1 2
18~de--ilr~~
ev - - i r a ~ ev ia'o~~ 181 e - - 184 e ,
~176176 ~.
282 = - - i o i
~176176 -~-1o4 ,~
(12)
.
(13)
Here (2sl, 2s2,2ss, a ~ are the coordinates of the point of M2 corresponding to the original solution u~ Solutions of the Painlev~ equations can be uniquely determined by their asymptotic properties. To uniquely determine most of the solutions it suffices to specify the leading term of the asymptotics on the special rays in the complex xk-plane. In particular, for P1 these are the rays arg X 1 = 71"~- 2~rk/5, and for P2 these are the rays arg x2 = ~r + 2 r k / 6 . Using the results [3, 4] it is not difficult to restate our theorem in terms of asymptotic parameters of the solutions u2(x2) and u l ( x t ) on the indicated rays. For each of these rays there also exist special one-parameter classes of solutions of the Painlev~ equations, whose unique characterization requires specifying the leading term of the asymptotics on another ray. These solutions are also called separatrix solutions, since they correspond to the complex curves on the m o n o d r o m y data manifold that separate the domains corresponding to general solutions with different types of asymptotics on the initial ray. The case of general solutions presents no difficulties: iterating a general solution u2(x2) of P2 by the indicated rule, we always come to a general solution u l ( x t ) of P1. That is why we consider the case of separatrix solutions as an illustration. Here we use some results of [7]. We shall specify the separatrix 461
solutions only on one ray, the ray where they separate asymptotics. Besides the leading term we give one more t e r m of the asymptotics, which is also leading in a sense, since it is the major term containing the p a r a m e t e r of the solution. The readers who are not convinced by our arguments here can consider the result to be qualitative, and the additional asymptotics of the separatrix solution on any other ray can be easily determined by referring to the m o n o d r o m y data and results of [3, 4]. The solutions of the equation P2 which are separatrix for x2 --* - o o are specified by the condition ([7]) I--2,~i
-2,5 3 ~ O.
As easily seen from the definition of M2, it is necessary that
e -iTrt'(a~
231 =
233 =
e i~ra(a~+ll2),
o"
= 4-1,
(14)
232 being arbitrary. These solutions are described by the asymptotic relation
V2(t2) ~---Uas(~2) JI- a2 e-V~t2-(trotg+l/2)l~
where v~(t2) = a / V ~ +
a 2
a~ ~
(15)
(1 -4- O ( t 2 1 ) ) ,
+ O(t~-2) is the function containing no parameters except a ~ and
1 2-5/43-~/2 e - ~ l~
F (aa~ + 1) (sin(Tr(~~
-7Y
232).
(16)
Here F(z) is the gamma-function. Thus if a ~ # 71 + n, these equations determine two (for cr = +1 and = - 1 ) one-parameter families. In the case a ~ ~1 + n, relations (14) give one and the same result for a = -t-1! Hence we have one separatrix family, whose asymptotics is specified by the following condition on (see (16)):
(17) There exists one exceptional case: a ~ = 89+ n and ~_s2 -- ( - 1 ) n. In this case we have one "legitimate" solution such that a satisfies (17), and, moreover, Eq. (16) implies a2 = 0. Also, another additional oneparameter family of solutions occurs, which can be expressed in terms of the Airy functions [8]. It is to parametrize this family that we glued C P 1. This family has the same asymptotics (15) with az E C, but instead of (17) cr must satisfy the condition a(89 + n) + 89_< 0. The point at infinity which we must glue to M2 corresponds to the "legitimate" solution, which also can be expressed in terms of the Airy functions. It follows from our theorem that (see (19.), (13)) 1-4-132.1s3=1
-- 281 . 2 3 3 = 0 .
To this constraint on the coordinates of M1 there also corresponds a one-parameter family of separatrix (for xl -+ --oo) solutions of P1. The coordinates of these solutions are 1S2=lS3=i,
lSS=0,
lsl +
ls4=i,
(18)
and the solutions themselves have the asymptotics ua(Xl) =
v1(tl) = w,s(tl) + 462
tl =
5/',
tl - ,
al e_2(3/2)114t , (1 + O(t~1)),
(19)
where was(tl) = 1 / v ~ + O(t~ 2 ) contains no parameter at all. The parameter al is calculated by 1
(2)
1'8 "
al -- ~
1$1 -- 134 2
(20)
Thus, for s ~ # 89+ n, comparing Eqs. (12), (13), (14) we see that: The even iterations of the separatrix solutions of P2 for a = +1 have the presentation (10), where u~V(xl ) is the separatrix solution of P1. The coordinates of this solution on the manifold M1 are given by Eqs. (12) and (18), and t h e n the asymptotics as xl ~ - o o is determined by Eqs. (19), (20). The odd iterations of the separatrix solutions of P2 for a = +1 do not have the presentation (11). The even iterations of the separatrix solutions of P2 for a = - 1 do not have the presentation (10). Finally, the odd iterations of the separatrix solutions of P2 for a = - 1 have the presentation (11), in which w~d(xl) is the separatrix solution of P1. The coordinates of this solution are determined by Eqs. (13) and (18) and its asymptotics is, of course, given by Eqs. (19) and (20). Now let us consider the case s0 __--'12-~ ~2, 2Sl = 2s3 = (--1) n+l, 2s2 # (--1) n. In this case for each 2s2 there is one separatrix solution, whose iterations do not have the presentation (10), (11). Finally, we analyzed the case c~~ = ~+n,1 2Sl = -2s2 = 2sa = ( , 1 ) n+l not quite rigorously. As mentioned above, in this case all the separatrix solutions can be described in terms of the Airy functions. One of them is of the type discussed in the preceding paragraph; its iterations also do not seem to have the presentation (10), (11). Besides this solution there is an additional "glued" family of solutions. The asymptotics of this family is also given by Eq. (15), with the sign of a determined by the condition or(} + n ) + 89< 0. The result of iterating these solutions has the presentation (10), (11). The solutions u~ d and u~v have asymptotics (19), where the parameter al d/ev is connected with the parameter a2 by the relation a2 = - b V ~ 6 - 5 / s
e -a(1/2+n)l~
a~
r(89
1
where the upper (lower) sign is taken for al d (respectively, a~V). We also consider the case where the rational solution of P2 is taken as the "initial" solution u~ It is well known that any such solution can be obtained from the trivial solution u2(x2) - 0 of P2 for ~2 = 0 by iterating the Bs transformations (2). To these solutions correspond the coordinates 281 = 2 32 = 2 83 ~ 0,
O~0 = m ,
where m is an integer equal to the number of iterations. Both even and odd iterations have presentations (10), (11), respectively, where Ul(Xl) = u~d(xl) = u~V(xl) is the solution of the equation P1 with the coordinates 1,92 = 183 = 0,
151 = 134 = 185 ~ i.
This solution of P1 has a specific asymptotic behavior: for ]Xlt ---* oo, arg x~ E (re/5; 9rr/5), it has no singularities and the asymptotics is given by the series U l ( X l ) = -- - - ~
-[- O ( ( - - . T 1 ) - - 2 ) .
Further, this solution has an infinite number of poles in the sector - z r / 5 < arg X 1 < 7/'/5, and the leading term of its asymptotics is described there by the modulated Weierstrass 79-function. The proof of this theorem is carried out based on the isomonodromic deformation technique [I0] with necessary modifications due to handling the triple turning point. For the justification of such a procedure, see [5]. We also omit some details, which are obvious from our viewpoint. First we change the variables in the equation
(21) 463
whose isomonodromic deformation equation is the second Painlev6 transcendent P2 ([19]). We make the substitution (1) and also, to facilitate the WKB-analysis, substitutions ---p-sA,
p--[~[,
~=arg~.
(22)
T h e n Eq. (21) takes the form - O-15
- i (4A 2 - 4e -1~
+ 408e-4/~u 1 +
pa2e2i~(2(ua) 2 + xl)) 03-
-- (4e-si~/~ q-4t76ei~l,,~ -- (ae-15i~ Ar.p12e-3i~)} ) o.2 _ 2p9 e-i~u~rl } ff2.
(23)
Using the explicit expression for the squared eigenvalues of system (23), /.t2_ ~
16 (~2 _ e_10i~) 3 )~2
8p12e2i~( )~2 _e_lOi~) ( Xl Jr-e -l~ ~) + k.
-[-4/918e-2icp ((ui)2 --2,1 (2(Ul)2 -Jl-Xl .at-~)) -[./724e-6i~ ( ~f12 _ _
el0i~(2(u 1)2 + xa)2 ) ,
(24)
we easily find the asymptotic behavior of the turning points, ~a,2,3 = e - 5 i ~ + O(p6),
A4,s,8 = - e - 5 i ~ + O(p~),
and construct the limit Stokes graph for different values of q0 = arge (see Fig. 1).
@
@
FIG. 1 454
(25)
We define the function #(A) on the plane with the cuts (,,~1;~2), (~3;)~4), (~5; ~6). Note that, as e ~ O, the cuts (A1; A2) and (As; An) degenerate into points and the cut (A3; A4) remains nondegenerate. We fix the branch of the square root by the condition > i (4A 2 - 6e - 1 0 i ~ )
~()0 ~,-.oo§
on the upper sheet. Consider the WKB-approximation in the canonical domains, as A ~ oo (cf. [11]),
q W K B ( A ) = T(A)exp
/] } h(v)dv
(26)
,
where A(A)=-p-15#aa-diag(T-l~-~T)=Aao'a+AoI,
T(A) = a3 = - 4 i ( ~ 2 - e - 1 ~
a~+~, ia2--at
ia2-at 1
'
- 4ip 6 ~-'i~721 - ip 1~ d ~ ( 2 ( ~ 1 )
2 + ~1),
a2 _~ --4e--5i~(,~2 _ e - 1 ~ al =-2pae-i'au~l .
In the usual way we can check that its relative error is of the order 0(615A -5) and its asymptotics differs from the canonical one [9, 10] only in the right-hand diagonal factor. In the vicinity of triple turning points (for the sake of definiteness we consider the point )~1,2,3 = e-si~ + O(p6)) we introduce a new local variable ~ by ~=e-5i~--p6e
i~
= e -5i~~ 1--
66ff 2
,
(27)
and in Eq. (23) we perform the gauge transformation
,=wz,
(2s)
with W=(I+ia2)al
( 26-3/2
i~3/2 r ) (I - in2).
The introduced function Z(~) satisfies the equation OZ
0r
= (b3aa + b2cr2 + blO'l -[- boI)Z,
(29)
where
b3 = 4~ 4 + 2(Ul) 2 + Xl -{- fl -]- ~E 6 ~4Ztl -- ~2 (721)2 -{b2 = --i(4r
+ 2(Ul) 2 + Xl -~- ~) -~- 2 66
(
~4Ul -- ~2
(
(721)2 .~_
2-- ~
-{- -l -- ~ - 6 ~ 2 ) -{-
~2(~4 + fl)) 1 -- 89
] +
8(1 -- 166~2) '
ie12 ~4(~4 + fl) 8(1 -- 1-6n~2~'2 ;
465
bx = - 2 ~ u l -
1 ~,
1 bo = - 2--~"
For e = 0, system (29) coincides with the linear system associated with the equation P1 [12, 4]. Thus as the special function which sews together different WKB-approximations near the triple turning point we must take the q~-functions of the first Painlev6 equation. This assertion follows from the fact that, for one thing, if we take the q2-function of the equation P2 as the local solution of Eq. (29), then the error has the order O(e6~ s) (the proof foUows easily from arguments quite similar to those used in [11] and several other papers). For another thing, the relative error o f the WKB-approximation in the domain ~ = O(r 0 < (~ < 1/2, does not exceed some O(~(7/2-~)). Finally, we easily verify that W - 1 T = ~1 ~ 3 a a / 2 -(I + O ( u l ( - 2 ) + O(r and so the local and WKB-approximations in the considered domain can be sewed together with the power precision. This sewing allows us to obtain the leading t e r m of the asymptotics (as ~ ~ 0) of the Stokes multipliers 28k by expressing the latter in terms of the Stokes multipliers lSk. The formal computations in the main order in r lead us to ~he following result: for s -+ 0, 0 < arg e < ~r/30, we have 1 - 2s1 9 2 s 3 = 1 + ls2 9 1s3, 2S3 = 182 e2A, 281 = --I,S3 e - 2 A , (30) where A=
A3(T)dv+i
f3+x2f
+
ffS+(x,+fl)( A0--~Az 2 a
One can easily compute this phase integral using (26) and (24); it suffices to expand #(A) in powers of and take the terms that do not w n i s h after integrating and taking the limit as e ~ O. We omit the elementary computations and state the result: ~ -"+OO
"~2~i (C--10-4-88
i
diag
"~"(4 ~5 2t-(X1+ ~)~) "(1+ 0(fl6~2)) --~-0(~--1), T -1
T
,
dA = --~ log ~ . I + O
+0
,
and so the phase integral is A = -~- a2 + o(1).
(31)
The q-function (associated with P2) that we constructed above by sewing together the approximate solutions is accurate only up to the power of ~. Therefore, for the Stokes multipliers 2sk, expressions (30) are certainly valid for IRe A I < const, or
[Im ~21 ~ const, i.e., for
cp = a r g e = O. 466
Thus, we have obtained the proof for the case of the odd iterations (see (9)). The part concerning the even iterations is obtained by using symmetries (5) and (7). Translated by O. A. Ivaxlov. Literature Cited
Ordinary Differential Equations, Longmans, Green, London (1926). A. V. Kitaev (this volume). A. A. Kapaev, Teor. Mat. Fiz., TT, No. 3, 323-332 (1988). A. A. Kapaev, Differents. Uravn., 24, No. 10, 1684--1695 (1988). A. V. Kitaev, Zapiski Nauch. Sere. LOMI, 179, 101-109 (1989). N. A. Lukashevich, Differents. Uravn., T, No. 6, 1124-1125 (1971). A. A. Kapaev, (Self-review of the Candidate Thesis), LGU, Leningrad (1987). V. I. Gromak, Different. Uravn., 14, No. 12, 2131-2135 (1978). H. Flaschka and A. C. Newell, Commun. Math. Phys., 76, No. 1, 65-116 (1980). A. R. It-s (Its) and V. Yu. Novokshenov, Lecture Notes in Math., 1191, Springer-Verlag (1986). M. V. Fedoryuk, Asymptotic Methods for Ordinary Differential Equation, [in Russian], Moscow (1983). M. Jimbo and T. Miwa, Physica D., 2, No. 3, 407-448 (1981).
1. E. L. Ince,
2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.
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