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Phil. Trans. R. Soc. A (2008) 366, 1025–1054
doi:10.1098/rsta.2007.2060
Published online 26 June 2007
Real-valued algebro-geometric solutions
of the Camassa–Holm hierarchy
B Y F RITZ G ESZTESY 1, *
AND
H ELGE H OLDEN 2
1
Department of Mathematics, University of Missouri, Columbia,
MO 65211, USA
2
Department of Mathematical Sciences, Norwegian University of Science and
Technology, 7491 Trondheim, Norway
We provide a detailed treatment of real-valued, smooth and bounded algebro-geometric
solutions of the Camassa–Holm (CH) hierarchy and describe the associated isospectral
torus. We employ Dubrovin-type equations for auxiliary divisors and certain aspects of
direct and inverse spectral theory for self-adjoint Hamiltonian systems. In particular, we
rely on Weyl–Titchmarsh theory for singular (canonical) Hamiltonian systems. We also
briefly discuss real-valued algebro-geometric solutions with a cusp behaviour. While we
focus primarily on the case of stationary algebro-geometric CH solutions, we note that
the time-dependent case subordinates to the stationary one with respect to isospectral
torus questions.
Keywords: Camassa–Holm hierarchy; real-valued algebro-geometric
solutions; isospectral tori; self-adjoint Hamiltonian systems;
Weyl–Titchmarsh theory
1. Introduction
In Gesztesy & Holden (2002) we provided a detailed treatment of the Camassa–
Holm (CH) hierarchy with special emphasis on its algebro-geometric solutions.
The first nonlinear partial differential equation of this hierarchy, the Camassa–
Holm equation, also known as the dispersive shallow water equation (Camassa &
Holm 1993) is given by
4ut K uxxt K2uuxxx K4ux uxx C 24uux Z 0;
ðx; tÞ 2R2
ð1:1Þ
(choosing a convenient scaling of x, t). For various aspects of local and global
existence, and uniqueness of solutions of (1.1), wave breaking phenomena,
soliton-type solutions (‘peakons’), complete integrability aspects, such as
infinitely many conservation laws, (bi-)Hamiltonian formalism, Bäcklund
transformations, infinite dimensional symmetry groups, etc., we refer to the
literature provided in Gesztesy & Holden (2002, 2003, ch. 5). The case of
spatially periodic solutions, the corresponding inverse spectral problem,
isospectral classes of solutions and quasi-periodicity of solutions with respect
* Author for correspondence ([email protected]).
One contribution of 15 to a Theme Issue ‘30 years of finite-gap integration’.
1025
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F. Gesztesy and H. Holden
to time are discussed in Constantin (1997a,b, 1998a,b) and Constantin &
McKean (1999). Moreover, algebro-geometric solutions of (1.1) and their
properties are studied in Alber & Fedorov (2000, 2001), Alber et al. (2001)
and Gesztesy & Holden (2002, 2003, ch. 5).
In §2, we recall the basic polynomial recursion formalism that defines the CH
hierarchy using a zero-curvature approach. Section 3 recalls the stationary CH
hierarchy and the associated algebro-geometric formalism. Section 4 provides a
brief summary of self-adjoint canonical systems as needed in this paper, and §5
finally discusses the principal result of this paper, the class of real-valued, smooth
and bounded algebro-geometric solutions of the CH hierarchy and the associated
isospectral torus. We also briefly discuss real-valued algebro-geometric solutions
with a cusp behaviour (cf. (5.80)).
We focus primarily on the case of stationary CH hierarchy solutions as the
time-dependent case subordinates to the stationary one with respect to
isospectral torus questions, a fact that is briefly commented on at the end of §5.
This paper should be viewed as a companion to our treatment (Gesztesy &
Holden 2002, 2003, ch. 5) of the CH hierarchy and we refer to it for background
material and pertinent references on the subject.
2. The CH hierarchy, recursion relations and hyperelliptic curves
In this section, we review the basic construction of the Camassa–Holm hierarchy
using a zero-curvature approach following (Gesztesy & Holden 2002, 2003, ch. 5).
Throughout this section, we suppose the following hypothesis (N0 Z Ng f0g).
Hypothesis 2.1. In the stationary case, we assume that
dm u
u 2CNðRÞ;
2LNðRÞ; m 2N0 :
dx m
In the time-dependent case (cf. (2.28)–(2.35)), we suppose
vm u
uð$; tÞ 2CNðRÞ;
ð$; tÞ 2LNðRÞ; m 2N0 ;
vx m
ð2:1Þ
t 2R;
ð2:2Þ
uðx; $Þ; uxx ðx; $Þ 2C 1 ðRÞ; x 2R:
We start by formulating the basic polynomial set-up. One defines ff[ g[2N0
recursively by
f0 Z 1;
f[ ;x ZK2Gð2ð4uK uxx Þf[K1;x C ð4ux K uxxx Þf[K1 Þ;
where G is given by
N
N
G : L ðRÞ/ L ðRÞ;
1
ðGvÞðxÞ Z
4
ð
dy eK2jxKyj vðyÞ;
x 2R;
[ 2N;
ð2:3Þ
v 2LNðRÞ:
R
ð2:4Þ
One observes that G is the resolvent of minus the one-dimensional Laplacian at
energy parameter equal to K4, i.e.
K1
d2
:
ð2:5Þ
G Z K 2 C4
dx
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The first coefficient reads as follows:
f1 ZK2u C c1 ;
ð2:6Þ
where c1 is an integration constant. Subsequent coefficients are non-local with
respect to u. At each level, a new integration constant, denoted by c[ , is
introduced. Moreover, we introduce coefficients fg[ g[2N0 and fh[ g[2N0 by
ð2:7Þ
g[ Z f[ C 12 f[ ;x ; h[ Z ð4uK uxx Þf[ K g[ C1;x ; [ 2N0 :
Explicitly, one computes
f0 Z 1;
f1 ZK2u C c1 ;
f2 Z 2u2 C 2G u 2x C 8u 2 C c1 ðK2uÞ C c2 ;
g0 Z 1;
g1 ZK2uK ux C c1 ;
g2 Z 2u 2 C 2uux C 2G u 2x C ux uxx C 8uux C 8u 2 C c1 ðK2uK ux Þ C c2 ;
h 0 Z 4u C 2ux ;
h 1 ZK2u 2x K4uux K8u2 K 2G ux uxxx C u 2xx C 2ux uxx C 8uuxx C 8u 2x C 16uux
ð2:8Þ
Cc1 ð4u C 2ux Þ; etc:
Given hypothesis 2.1, one introduces the 2!2 matrix U by
!
K1
1
; x 2R;
U ðz; xÞ Z
z K1 ð4uðxÞK uxx ðxÞÞ 1
and for each n 2N0 , the following 2!2 matrix Vn by
!
KGn ðz; xÞ
Fn ðz; xÞ
; n 2N0 ; z 2Cnf0g;
Vn ðz; xÞ Z
z K1 Hn ðz; xÞ Gn ðz; xÞ
ð2:9Þ
x 2R;
ð2:10Þ
assuming Fn, Gn and Hn to be polynomials of degree n with respect to z and C N
in x. Postulating the zero-curvature condition
KVn;x ðz; xÞ C ½U ðz; xÞ; Vn ðz; xÞ Z 0;
ð2:11Þ
Fn;x ðz; xÞ Z 2Gn ðz; xÞK 2Fn ðz; xÞ;
ð2:12Þ
zGn;x ðz; xÞ Z ð4uðxÞK uxx ðxÞÞFn ðz; xÞK Hn ðz; xÞ;
ð2:13Þ
Hn;x ðz; xÞ Z 2Hn ðz; xÞK 2ð4uðxÞK uxx ðxÞÞGn ðz; xÞ:
ð2:14Þ
one finds
From (2.12) to (2.14) one infers that
d
1 d
detðVn ðz; xÞÞ ZK
ðzGn ðz; xÞ2 C Fn ðz; xÞHn ðz; xÞÞ Z 0
dx
z dx
ð2:15Þ
and hence
z 2 Gn ðz; xÞ2 C zFn ðz; xÞHn ðz; xÞ Z R2nC2 ðzÞ;
Phil. Trans. R. Soc. A (2008)
ð2:16Þ
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F. Gesztesy and H. Holden
where the polynomial R 2nC2 of degree 2nC2 is x -independent,
2nC1
Y
ðz K Em Þ; E0 ; E1 ; .; E2n 2C; E2nC1 Z 0:
R2nC2 ðzÞ Z
ð2:17Þ
mZ0
Next, one makes the ansatz that Fn, Hn and Gn are polynomials of degree n,
related to the coefficients f[ , h[ and g[ by
n
n
n
X
X
X
fnK[ ðxÞz [ ; Gn ðz; xÞ Z
gnK[ ðxÞz [ ; Hn ðz; xÞ Z
hnK[ ðxÞz [ :
Fn ðz; xÞ Z
[ Z0
[ Z0
[ Z0
ð2:18Þ
Insertion of (2.18) into (2.12)–(2.14) then yields the recursion relations (2.3),
(2.4) and (2.7) for f[ and g[ for [Z0, ., n. For fixed n2N, we obtain the
recursion (2.7) for h[ for [Z0, ., nK1 and
hn Z ð4uK uxx Þfn :
ð2:19Þ
(When nZ0 one directly gets h0 Z ð4uK uxx Þ.) Moreover, taking zZ0 in (2.16)
yields
2n
Y
Em :
ð2:20Þ
fn ðxÞhn ðxÞ ZK
mZ0
In addition, one finds
hn;x ðxÞK 2hn ðxÞ C 2ð4uðxÞK uxx ðxÞÞgn ðxÞ Z 0;
n 2N0 :
ð2:21Þ
Using the relations (2.19) and (2.7) permits one to write (2.21) as
s-CHn ðuÞ Z ðuxxx K4ux Þfn K 2ð4uK uxx Þfn;x Z 0;
n 2N0 :
ð2:22Þ
Varying n 2N0 in (2.22) then defines the stationary CH hierarchy. We record
the first few equations explicitly
s-CH0 ðuÞ Z uxxx K4ux Z 0;
s-CH1 ðuÞ ZK2uuxxx K4ux uxx C 24uux C c1 ðuxxx K4ux Þ Z 0;
s-CH2 ðuÞ Z 2u2 uxxx K8uux uxx K40u 2 ux C 2ðuxxx K4ux ÞG u 2x C 8u 2
K8ð4uK uxx ÞGðux uxx C 8uux Þ
Cc1 ðK2uuxxx K4ux uxx C 24uux Þ C c2 ðuxxx K4ux Þ Z 0; etc: ð2:23Þ
By definition, the set of solutions of (2.22), with n ranging in N0, represents
the class of algebro-geometric CH solutions. If u satisfies one of the stationary CH
equations in (2.22) for a particular value of n, then it satisfies infinitely many
such equations of order higher than n for certain choices of integration constants
c[ . At times, it will be convenient to abbreviate (algebro-geometric) stationary
CH solutions u simply as CH potentials.
Using equations (2.12)–(2.14), one can also derive individual differential
equations for Fn and Hn. Focusing on Fn only, one obtains
Fn;xxx ðz; xÞK4ðz K1 ð4uðxÞK uxx ðxÞÞ C 1ÞFn;x ðz; xÞ
K 2z K1 ð4ux ðxÞK uxxx ðxÞÞFn ðz; xÞ Z 0
Phil. Trans. R. Soc. A (2008)
ð2:24Þ
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Real-valued algebro-geometric solutions
and
Kðz 2 =2ÞFn;xx ðz; xÞFn ðz; xÞ C ðz 2 =4ÞFn;x ðz; xÞ2 C z 2 Fn ðz; xÞ2
C zð4uðxÞK uxx ðxÞÞFn ðz; xÞ2 Z R2nC2 ðzÞ:
ð2:25Þ
Equation (2.25) leads to an explicit determination of the integration constants
c1 ; .; cn in the stationary CH equations (2.22) in terms of the zeros E0Z0,
E1 ; .; E2nC1 of the associated polynomial R 2nC2 in (2.17). In fact, one can prove
c[ Z c[ ðEÞ;
ð2:26Þ
[ Z 0; .; n;
where
c0 ðE Þ Z 1;
ck ðEÞZK
k
X
j1 ;.;j2nC1Z0
ð2j1 Þ! /ð2j2nC1 Þ!
2
2k
2 ðj1 !Þ /ðj2nC1 !Þ2 ð2j1 K1Þ /ð2j2nC1 K1Þ
j1C/Cj2nC1Zk
2nC1
!E j11 .E j2nC1
;
k 2N:
ð2:27Þ
Next, we turn to the time-dependent CH hierarchy. Introducing a deformation
parameter tn 2R into u (i.e. replacing u(x) by uðx; tn Þ), the definitions (2.9),
(2.10) and (2.18) of U, Vn and Fn, Gn and Hn, respectively, still apply. The
corresponding zero-curvature relation then reads
Utn ðz; x; tn ÞK Vn;x ðz; x; tn Þ C ½U ðz; x; tn Þ; Vn ðz; x; tn Þ Z 0;
n 2N0 ;
ð2:28Þ
which results in the following set of time-dependent equations:
4utn ðx; tn ÞK uxxtn ðx; tn ÞK Hn;x ðz; x; tn Þ C 2Hn ðz; x; tn Þ
K 2ð4uðx; tn ÞK uxx ðx; tn ÞÞGn ðz; x; tn Þ Z 0;
Fn;x ðz; x; tn Þ Z 2Gn ðz; x; tn ÞK 2Fn ðz; x; tn Þ;
ð2:29Þ
ð2:30Þ
zGn;x ðz; x; tn Þ Z ð4uðx; tn ÞK uxx ðx; tn ÞÞFn ðz; x; tn ÞK Hn ðz; x; tn Þ:
ð2:31Þ
Inserting the polynomial expressions for Fn, Hn and Gn into (2.30) and (2.31),
respectively, first yields recursion relations (2.3) and (2.7) for f[ and g[ for
[ Z 0; .; n. For fixed n2N, we obtain from (2.29) the recursion for h[ for
[ Z 0; .; nK1 and
hn Z ð4uK uxx Þfn :
ð2:32Þ
(When nZ0 one directly gets h0 Z ð4uK uxx Þ.) In addition, one finds
4utn ðx; tn ÞK uxxtn ðx; tn ÞK hn;x ðx; tn Þ C 2hn ðx; tn Þ
K 2ð4uðx; tn ÞK uxx ðx; tn ÞÞgn ðx; tn Þ Z 0;
n 2N0 :
ð2:33Þ
Using relations (2.19) and (2.32) permits one to write (2.33) as
CHn ðuÞ Z 4utn K uxxtn C ðuxxx K4ux Þfn K 2ð4uK uxx Þfn;x Z 0;
Phil. Trans. R. Soc. A (2008)
n 2N0 :
ð2:34Þ
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F. Gesztesy and H. Holden
Varying n2N0 in (2.34) then defines the time-dependent CH hierarchy. We
record the first few equations explicitly
CH0 ðuÞ Z 4ut 0 K uxxt 0 C uxxx K4ux Z 0;
CH1 ðuÞ Z 4ut1 K uxxt1 K 2uuxxx K4ux uxx C 24uux C c1 ðuxxx K4ux Þ Z 0;
2
CH2 ðuÞ Z 4ut2 K uxxt 2 C 2u2 uxxx K8uu
x uxx K40u ux
2
2
C2ðuxxx K4ux ÞG u x C 8u K8ð4uK uxx ÞGðux uxx C 8uux Þ
Cc1 ðK2uuxxx K4ux uxx C 24uux Þ C c2 ðuxxx K4ux Þ Z 0; etc:
ð2:35Þ
Up to an inessential scaling of the (x, t1) variables, CH1 ðuÞZ 0 with c1Z0
represents the Camassa–Holm equation as discussed in Camassa & Holm (1993).
3. The stationary algebro-geometric CH formalism
This section is devoted to a quick review of the stationary CH hierarchies and the
corresponding algebro-geometric formalism as derived in Gesztesy & Holden
(2002, 2003, ch. 5).
We start with the stationary hierarchy and suppose that u : R2/C satisfies
dm u
N
u 2CNðRÞ;
m 2N0
ð3:1Þ
m 2L ðRÞ;
dx
and assume (2.3), (2.7), (2.9)–(2.11) and (2.16)–(2.22), keeping n2N0 fixed.
Recalling (2.17),
2nC1
Y
R2nC2 ðzÞ Z
ðz K Em Þ; E0 ; E1 ; .; E2n 2C; E2nC1 Z 0;
ð3:2Þ
mZ0
we introduce the (possibly singular) hyperelliptic curve Kn of arithmetic genus n
defined by
Kn : F n ðz; yÞ Z y2 K R2nC2 ðzÞ Z 0:
ð3:3Þ
In the following, we will occasionally impose further constraints on the zeros Em
of R 2nC2 introduced in (3.2) and assume that
ð3:4Þ
E0 ; .; E2n 2Cnf0g; E2nC1 Z 0:
We compactify Kn by adding two points at infinity, PNC, PNK, with PNC sPNK, still
denoting its projective closure by Kn. Hence, Kn becomes a two-sheeted Riemann
surface of arithmetic genus n. Points P on Kn nfPNGg are denoted by PZ(z, y),
where yð$Þ denotes the meromorphic function on Kn satisfying F n(z, y)Z0.
For notational simplicity, we will usually tacitly assume that n2N (the case
nZ0 being trivial).
In the following, the roots of the polynomials Fn and Hn will play a special role
and, hence, we introduce on C!R
n
n
Y
Y
Fn ðz; xÞ Z ðz K mj ðxÞÞ; Hn ðz; xÞ Z h 0 ðxÞ ðz K nj ðxÞÞ;
ð3:5Þ
jZ1
jZ1
temporarily assuming
h 0 ðxÞ s0;
Phil. Trans. R. Soc. A (2008)
x 2R:
ð3:6Þ
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Real-valued algebro-geometric solutions
Moreover, we introduce
m^j ðxÞ Z ðmj ðxÞ;Kmj ðxÞGn ðmj ðxÞ; xÞÞ 2Kn ;
j Z 1; .; n;
n^j ðxÞ Z ðnj ðxÞ; nj ðxÞGn ðnj ðxÞ; xÞÞ 2Kn ;
j Z 1; .; n;
x 2R;
ð3:7Þ
x 2R
ð3:8Þ
and
P0 Z ð0; 0Þ:
ð3:9Þ
The branch of y($) near PNG is fixed according to
lim
jzðPÞj/N
P/PNG
yðPÞ
ZH1:
zðPÞGn ðzðPÞ; xÞ
ð3:10Þ
Due to assumption (3.1), u is smooth and bounded, and hence Fn ðz; $Þ and
Hn ðz; $Þ share the same property. Thus, one concludes
mj ; nk 2C ðRÞ;
ð3:11Þ
j; k Z 1; .; n;
taking multiplicities (and appropriate reordering) of the zeros of Fn and Hn into
account.
Next, we introduce the fundamental meromorphic function fð$; xÞ on Kn by
fðP; xÞ Z
yKzGn ðz; xÞ
zHn ðz; xÞ
Z
;
Fn ðz; xÞ
y C zGn ðz; xÞ
P Z ðz; yÞ 2Kn ;
x 2R:
ð3:12Þ
Assuming (3.4) and (3.6), the divisor ðfð$; xÞÞ of fð$; xÞ is given by
ðfð$; xÞÞ Z DP0 nðxÞ
^ KDPNC mðxÞ
^ ;
ð3:13Þ
taking into account (3.10). Here we abbreviated
m^ Z f^
m1 ; .; m^n g;
n^ Z f^
n1 ; .; n^n g 2sn Kn ;
ð3:14Þ
where sm Kn , m2N, denotes the mth symmetric product of Kn. If h 0 is permitted
to vanish at a point x 12N, then for xZx 1, the polynomial Hn ð$; x 1 Þ is at most of
degree nK1 (cf. (2.18)) and (3.13) is altered to
ðfð$; x 1 ÞÞ Z DP0 PNKn^1 ðx 1 Þ;.;^nnK1 ðx 1 Þ KDPNC mðx
^ 1Þ;
ð3:15Þ
that is, one of the n^j ðxÞ tends to PNK as x/x 1 (cf. also (3.36)). Analogously one
can discuss the case of several n^j approaching PNK. Since this can be viewed as a
limiting case of (3.13), we will henceforth not particularly distinguish the case
h 0s0 from the more general situation where h0 is permitted to vanish.
Given fð$; xÞ, one defines the associated Baker–Akhiezer vector Jð$; x; x 0 Þ on
Kn nfPNC ; PNK; P0 g by
!
j1 ðP; x; x 0 Þ
JðP; x; x 0 Þ Z
; P 2Kn nfPNC ; PNK; P0 g; ðx; x 0 Þ 2R2 ; ð3:16Þ
j2 ðP; x; x 0 Þ
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F. Gesztesy and H. Holden
where
ðx
0
0
dx fðP; x ÞKðx K x 0 Þ ;
j1 ðP; x; x 0 Þ Z exp Kð1=zÞ
ð3:17Þ
j2 ðP; x; x 0 Þ ZKj1 ðP; x; x 0 ÞfðP; xÞ=z:
ð3:18Þ
x0
Although J is formally the analogue of the Baker–Akhiezer vector of the
stationary CH hierarchy when compared to analogous definitions in the context
of the KdV or AKNS hierarchies, its actual properties in a neighbourhood of its
essential singularity feature characteristic differences to standard Baker–
Akhiezer vectors as discussed in Gesztesy & Holden (2002, 2003, ch. 5).
The basic properties of f and J then read as follows.
Lemma 3.1. Suppose (3.1), assume the nth stationary CH equation (2.22)
holds, and let P Z ðz; yÞ 2Kn nfPNC ; PNK; P0 g, ðx; x 0 Þ 2R2 . Then f satisfies the
Riccati-type equation
fx ðP; xÞKz K1 fðP; xÞ2 K 2fðP; xÞ C 4uðxÞK uxx ðxÞ Z 0;
ð3:19Þ
as well as
fðP; xÞfðP ; xÞ ZK
zHn ðz; xÞ
;
Fn ðz; xÞ
fðP; xÞ C fðP ; xÞ ZK2
fðP; xÞKfðP ; xÞ Z
ð3:20Þ
zGn ðz; xÞ
;
Fn ðz; xÞ
ð3:21Þ
2y
;
Fn ðz; xÞ
ð3:22Þ
while J fulfils
Jx ðP; x; x 0 Þ Z U ðz; xÞJðP; x; x 0 Þ;
ð3:23Þ
KyJðP; x; x 0 Þ Z zVn ðz; xÞJðP; x; x 0 Þ;
ð3:24Þ
Fn ðz; xÞ
j1 ðP; x; x 0 Þ Z
Fn ðz; x 0 Þ
1=2
ðx
0
0 K1
exp Kðy=zÞ
dx Fn ðz; x Þ
;
j1 ðP; x; x 0 Þj1 ðP ; x; x 0 Þ Z
Fn ðz; xÞ
;
Fn ðz; x 0 Þ
j2 ðP; x; x 0 Þj2 ðP ; x; x 0 Þ ZK
ð3:26Þ
Hn ðz; xÞ
;
zFn ðz; x 0 Þ
j1 ðP; x; x 0 Þj2 ðP ; x; x 0 ÞK j1 ðP ; x; x 0 Þj2 ðP; x; x 0 Þ Z
ð3:27Þ
Gn ðz; xÞ
;
Fn ðz; x 0 Þ
ð3:28Þ
2y
:
zFn ðz; x 0 Þ
ð3:29Þ
j1 ðP; x; x 0 Þj2 ðP ; x; x 0 Þ C j1 ðP ; x; x 0 Þj2 ðP; x; x 0 Þ Z 2
Phil. Trans. R. Soc. A (2008)
ð3:25Þ
x0
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In addition, as long as the zeros of Fn ð$; xÞ are all simple for x 2U, U 4R an open
interval, Jð$; x; x 0 Þ, x; x 0 2U, is meromorphic on Kn nfP0 g.
Next, we recall the Dubrovin-type equations for mj. Since in the remainder of
this section, we will frequently assume Kn to be non-singular, we list all
restrictions on Kn in this case,
Em 2Cnf0g;
Em sEm 0 ;
for m sm 0 ; m; m 0 Z 0; .; 2n C 1; E 2nC1 Z 0:
ð3:30Þ
Lemma 3.2. Suppose (3.1) and the nth stationary CH equation (2.22) holds
~ m 4R. Moreover, suppose
subject to the constraint (3.30) on an open interval U
~ m . Then
that the zeros mj, jZ1, ., n, of Fn($) remain distinct and non-zero on U
{m^j }jZ1, ., n defined by (3.7), satisfies the following first-order system of
differential equations:
mj;x ðxÞ Z 2
n
yð^
mj ðxÞÞ Y
ðmj ðxÞK m[ ðxÞÞ K1 ;
mj ðxÞ
j Z 1; .; n;
~m:
x 2U
ð3:31Þ
[ Z1
[ sj
Next, assume Kn to be non-singular and introduce the initial condition
f^
mj ðx 0 ÞgjZ1;.;n 3Kn ;
ð3:32Þ
for some x 0 2R, where mj ðx 0 Þ s0, jZ1, ., n, are assumed to be distinct. Then
there exists an open interval Um 4R, with x 0 2Um , such that the initial value
problem (3.31) and (3.32) has a unique solution fm^j gjZ1;.;n 3Kn satisfying
m^j 2C NðUm ; Kn Þ;
j Z 1; .; n;
ð3:33Þ
and mj, jZ1, ., n, remain distinct and non-zero on Um.
Combining the polynomial approach in §2 with (3.5) yields trace formulae for
the CH invariants. For simplicity, we just record two simple cases.
Lemma 3.3. Suppose (3.1), assume the nth stationary CH equation (2.22)
holds, and let x 2R. Then
uðxÞ Z
n
X
1X
1 2nC1
mj ðxÞK
E ;
2 jZ1
4 mZ0 m
1
0
!
n
C Y
B 2nC1
C
B Y
Em C
m ðxÞ K2 :
4uðxÞK uxx ðxÞ ZKB
A jZ1 j
@
mZ0
Ems0
Next, we turn to asymptotic properties of f and jj, jZ1, 2.
Phil. Trans. R. Soc. A (2008)
ð3:34Þ
ð3:35Þ
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1034
F. Gesztesy and H. Holden
Lemma 3.4. Suppose (3.1), assume the nth stationary CH equation (2.22)
holds, and let P Z ðz; yÞ 2Kn nfPNC ; PNK; P0 g, x2R. Then
fðP; xÞ Z
8
< K2z K1 K 2uðxÞ C ux ðxÞ C OðzÞ; P / PNC ;
z/0 :
2uðxÞ C ux ðxÞ C OðzÞ;
0
ð3:36Þ
z Z z K1 ;
P / PNK;
11=2
B 2nC1
C
B Y
C
fðP; xÞ Z B
Em C
z/0 @
A
fn ðxÞ K1 z C Oðz2 Þ;
P / P0 ;
z Z z 1=2 ;
ð3:37Þ
P / PNG;
z Z 1=z;
ð3:38Þ
mZ0
Ems0
and
j1 ðP; x; x 0 Þ Z expðGðx K x 0 ÞÞð1 C OðzÞÞ;
z/0
j2 ðP; x; x 0 Þ Z expðGðx K x 0 ÞÞ
z/0
8
< K2 C OðzÞ;
:
P / PNC ;
z Z 1=z;
2
ð2uðxÞ C ux ðxÞÞz C Oðz Þ;
P / PNK;
ð3:39Þ
0
1
11=2
0
C
B
B 2nC1
ð
C
B 1 x
B Y
0
K
dx B
Em C
j1 ðP; x; x 0 Þ Z expB
C
B
B
z/0
A
@ z x0
@ mZ0
0 K1
fn ðx Þ
C
C
C Oð1ÞC
C;
A
ð3:40Þ
Ems0
P / P0 ;
zZz
1=2
;
0
j2 ðP; x; x 0 Þ Z Oðz
K1
z/0
1
11=2
0
C
B
B 2nC1
ð
C
B 1 x
B Y
0B
B
ÞexpBK
dx B
Em C
C
A
@ z x0
@ mZ0
0 K1
fn ðx Þ
C
C
C Oð1ÞC
C; ð3:41Þ
A
Ems0
P / P0 ;
z Zz
1=2
:
Since the representations of f and u in terms of the Riemann theta function
associated with Kn (assuming Kn to be non-singular) are not explicitly needed in
this paper, we omit the corresponding details and refer to the detailed treatment
in Gesztesy & Holden (2002, 2003, ch. 5) instead.
Finally, we will recall that solvability of the Dubrovin equations (3.31) on
Um 4R in fact implies equation (2.22) on Um.
Phil. Trans. R. Soc. A (2008)
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Real-valued algebro-geometric solutions
1035
Theorem 3.1. Fix n2N, assume (3.30), and suppose that f^
mj gjZ1;.;n satisfies
the stationary Dubrovin equations (3.31) on an open interval Um 4R such that mj,
jZ1, ., n, remain distinct and non-zero on Um. Then u 2CNðUm Þ defined by
n
X
1X
1 2nC1
mj ðxÞK
E ;
ð3:42Þ
uðxÞ Z
2 jZ1
4 mZ0 m
satisfies the nth stationary CH equation (2.22), i.e.
s-CHn ðuÞ Z 0 on Um :
ð3:43Þ
4. Basic facts on self-adjoint Hamiltonian systems
We now turn to the Weyl–Titchmarsh theory for singular Hamiltonian
(canonical) systems and briefly recall the basic material needed in §5. This
material is standard and can be found, for instance, in Kogan & Rofe-Beketov
(1974), Hinton & Shaw (1981, 1983, 1984), Clark & Gesztesy (2002) and Lesch &
Malamud (2003), and references therein.
Hypothesis 4.1. (i) Define the 2!2 matrix
!
0 K1
JZ
;
1 0
and suppose aj;k ; bj;k 2L1loc ðRÞ, j, kZ1, 2, and AðxÞZ ðaj;k ðxÞÞj;kZ1;2 R 0, BðxÞZ
ðbj;k ðxÞÞj;kZ1;2 Z BðxÞ for a.e. x2R. We consider the Hamiltonian system
J J 0 ðz; xÞ Z ðzAðxÞ C BðxÞÞJðz; xÞ; z 2C;
ð4:1Þ
for a.e. x2R, where z plays the role of the spectral parameter, and where
ð4:2Þ
Jðz; xÞ Z ðj1 ðz; xÞj2 ðz; xÞÞu ; jj ðz; $Þ 2ACloc ðRÞ; j Z 1; 2:
Here ACloc ðRÞ denotes the set of locally absolutely continuous functions on R and
the M and M u denote the adjoint and transpose of a matrix M, respectively.
(ii) For all non-trivial solutions J of (4.1), we assume the definiteness
hypothesis (cf. Atkinson 1964, §9.1)
ðd
dx Jðz; xÞ AðxÞJðz; xÞO 0;
ð4:3Þ
c
on every interval ðc; dÞ 3R, c!d.
A simple example of a Hamiltonian system satisfying (4.3) is obtained when
!
wðxÞ 0
AðxÞ Z
;
0
0
for some weight function w 2L1loc ðRÞ, wO0 a.e. on R, and b2;2 ðxÞO 0 a.e. on R
(cf. §5). Hypothesis 4.1 (ii) clearly holds in this case.
Next, we introduce the vector space ðKN% a! b%NÞ
ð b
2
2
LA ðða; bÞÞ Z f : ða; bÞ/ C measurable dxðfðxÞ; AðxÞfðxÞÞC2 !N ; ð4:4Þ
a
P2
where ðf; jÞC2 Z jZ1 fj jj denotes the standard scalar product in C2. Fix a point
x02R. Then the Hamiltonian system (4.1) is said to be in the limit point case at
N (respectively, KN) if for some (and hence for all) z 2CnR, precisely one
Phil. Trans. R. Soc. A (2008)
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F. Gesztesy and H. Holden
solution of (4.1) lies in L2A ððx 0 ;NÞÞ (respectively, L2A ððKN; x 0 ÞÞ). (By the analogue
of Weyl’s alternative, if (4.1) is not in the limit point case atGN, all solutions of
(4.1) lie in L2A ððx 0 ;GNÞÞ for all z 2C. In the latter case, the Hamiltonian system
(4.1) is said to be in the limit circle case at GN.)
To simplify matters for the remainder of this section, we will always suppose
the limit point case at GN from now on.
Hypothesis 4.2. Assume hypothesis 4.1 and suppose that the Hamiltonian
system (4.1) is in the limit point case at GN.
An elementary example of a Hamiltonian system satisfying hypothesis 4.2 is given
by the case where all entries of A and B are essentially bounded on R (cf. §5).
When considering the Hamiltonian system (4.1) on the half-line ½x 0 ;NÞ
(respectively, ðKN; x 0 ), a self-adjoint (separated) boundary condition at the
point x0 is of the type
aJðx 0 Þ Z 0;
where aZ ða1 a2 Þ 2C
aa Z I ;
1!2
ð4:5Þ
satisfies
aJ a Z 0 ðequivalently; ja1 j2 C ja2 j2 Z 1; Imða2 a1 Þ Z 0Þ:
ð4:6Þ
In particular, the boundary condition (4.5) (with a satisfying (4.6)) is equivalent
to a1 j1 ðx 0 ÞC a2 j2 ðx 0 ÞZ 0 with a1 =a2 2R if a2 s0 and a2 =a1 2R if a1 s0. The
special case a0 Z ð 1 0 Þ will be of particular relevance in §5. Due to our limit
point assumption at GN in hypothesis 4.2, no additional boundary condition at
GN needs to be introduced when considering (4.1) on the half-lines ½x 0 ;NÞ and
ðKN; x 0 . The resulting full-line and half-line Hamiltonian systems are said to be
self-adjoint on R, ½x 0 ;NÞ and ðKN; x 0 , respectively (assuming of course a
boundary condition of the type (4.5) in the two half-line cases).
Next, we digress a bit and briefly turn to Herglotz functions and their
representations in terms of measures, the focal point of Weyl–Titchmarsh theory
(and hence spectral theory) of self-adjoint Hamiltonian systems.
Definition 4.1. Any analytic map m : CC/ CC is called a Herglotz function
(here CCZ fz 2CjImðzÞO 0g). Similarly, any analytic map M : CC/ Ck!k ,
k 2N, is called a k!k matrix-valued Herglotz function if ImðM ðzÞÞR 0 for all
z 2CC.
Herglotz functions are characterized by a representation of the form
ðN
mðzÞ Z a C bz C
duðlÞððlKzÞ K1 Klð1 C l2 Þ K1 Þ; z 2CnR;
ð4:7Þ
KN
ðN
a 2R;
duðlÞð1 C l2 Þ K1 !N;
bR 0;
ð4:8Þ
KN
uððl; mÞ Z lim lim
dY0 3Y0
1
p
ð mCd
dn Imðmðn C i3ÞÞ;
ð4:9Þ
lCd
in the following sense: every Herglotz function admits a representation of the
type (4.7) and (4.8) and, conversely, any function of the type (4.7) and (4.8) is a
Herglotz function. Moreover, local singularities and zeros of m are necessarily
Phil. Trans. R. Soc. A (2008)
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Real-valued algebro-geometric solutions
located on the real axis and at most of first order in the sense that
uðflgÞ Z limðuðl C 3ÞKuðlK3ÞÞ ZKlim i3mðl C i3ÞR 0;
3Y0
3Y0
lim i3mðl C i3Þ K1 R 0;
l 2R;
l 2R:
3Y0
ð4:10Þ
ð4:11Þ
In particular, isolated poles of m are simple and located on the real axis, the
corresponding residues being negative. Analogous results hold for matrix-valued
Herglotz functions (Gesztesy & Tsekanovskii 2000 and references therein).
For subsequent purpose in §5, we also note that K1/z is a Herglotz function
and compositions of Herglotz functions remain Herglotz functions. In addition,
diagonal elements of a matrix-valued Herglotz function are Herglotz functions.
Returning to Hamiltonian systems on half-lines satisfying hypotheses 4.1 and
4.2, we now denote by JGðz; x; x 0 Þ the unique solution of (4.1) satisfying
JGðz; $; x 0 Þ 2L2A ð½x 0 ;GNÞÞ, z 2CnR, normalized by j1;Gðz; x 0 ; x 0 ÞZ 1. Then the
half-line Weyl–Titchmarsh function mGðz; xÞ, associated with the Hamiltonian
system (4.1) on ½x;GNÞ and the fixed boundary condition a0 Z ð 1 0 Þ at the point
x 2R, is defined by
mGðz; xÞ Z j2;Gðz; x; x 0 Þ=j1;Gðz; x; x 0 Þ;
z 2CnR; Gx R x 0 :
ð4:12Þ
The actual normalization of JGðz; x; x 0 Þ was chosen for convenience only and is
clearly immaterial in the definition of m Gðz; xÞ in (4.12).
One easily verifies that m Gðz; xÞ satisfies the following Riccati-type differential
equation:
m 0 ðz; xÞ C ½b2;2 ðxÞ C a2;2 ðxÞzmðz; xÞ2
C ½b1;2 ðxÞ C b2;1 ðxÞ C ða1;2 ðxÞ C a2;1 ðxÞÞzmðz; xÞ C b1;1 ðxÞ C a1;1 ðxÞz Z 0:
ð4:13Þ
Finally, the 2!2 Weyl–Titchmarsh matrix M ðz; xÞ associated with the
Hamiltonian system (4.1) on R is then defined in terms of the half-line Weyl–
Titchmarsh functions mGðz; xÞ by
M ðz; xÞ Z ðMj;j 0 ðz; xÞÞj;j 0Z1;2 ;
z 2CnR;
ð4:14Þ
M1;1 ðz; xÞ Z ½mKðz; xÞK mCðz; xÞ K1 ;
M1;2 ðz; xÞ Z M2;1 ðz; xÞ Z 2 K1 ½mKðz; xÞK mCðz; xÞ K1 ½mKðz; xÞ C mCðz; xÞ;
M2;2 ðz; xÞ Z ½mKðz; xÞK mCðz; xÞ K1 mKðz; xÞmCðz; xÞ:
ð4:15Þ
One verifies that M ðz; xÞ is a 2!2 matrix-valued Herglotz function. We emphasize
that for any fixed x 02R, M(z, x 0) contains all the spectral information of the selfadjoint Hamiltonian system (4.1) on R (assuming hypotheses 4.1 and 4.2).
Phil. Trans. R. Soc. A (2008)
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F. Gesztesy and H. Holden
5. Real-valued algebro-geometric CH solutions and the associated
isospectral torus
In our final and principal section, we study real-valued algebro-geometric solutions of
the CH hierarchy associated with curves Kn whose affine part is non-singular and
determine the isospectral manifold of smooth and bounded CH solutions. We focus on
the stationary case as this is the primary concern in this context and briefly comment
on the time-dependent case at the end of this section.
To study the direct spectral problem, we first introduce the following
assumptions.
Hypothesis 5.1. Suppose
ð5:1Þ
E0 ! E1 !/! E2n ! E2nC1 Z 0
and let u be a real-valued solution of the nth stationary CH equation (2.22),
s-CHn ðuÞ Z 0
ð5:2Þ
(i.e. u is a particular algebro-geometric CH potential), satisfying
u 2CNðRÞ; vkx u 2LNðRÞ; k Z 0; 1; 2;
4uK uxx O 0:
We start by noticing that the basic stationary equation (3.23),
J Z ðj1 ; j2 Þu ;
Jx ðz; xÞ Z U ðz; xÞJðz; xÞ;
ðz; xÞ 2C !R;
ð5:3Þ
ð5:4Þ
ð5:5Þ
is equivalent to the following Hamiltonian (canonical) system:
~ z ; xÞ;
~ x ð~
z ; xÞ Z ½~
z AðxÞ C BðxÞJð~
JJ
where
JZ
AðxÞ Z
0 K1
1
~ Z ðj
~1 ; j
~2 Þu ;
J
ð~
z ; xÞ 2C !R;
ð5:6Þ
!
0
;
4uðxÞK uxx ðxÞ
0
0
0
~ z ; xÞ Z Jðz; xÞ;
Jð~
z~ ZK1=z;
!
0
K1
K1
1
;
BðxÞ Z
ð5:7Þ
!
;
x 2R:
ð5:8Þ
In particular, due to assumptions (5.3) and (5.4), the Hamiltonian system (5.6)
satisfies hypotheses 4.1 and 4.2. Explicitly, the Hamiltonian system (5.6) boils
down to
~1;x ð~
~2 ð~
~1 ð~
j
z ; xÞ Z j
z ; xÞKj
z ; xÞ;
~2;x ð~
~1 ð~
~2 ð~
j
z ; xÞ ZK~
z ð4uðxÞK uxx ðxÞÞj
z ; xÞ C j
z ; xÞ;
ð5:9Þ
ðz; xÞ 2C !R
ð5:10Þ
~2 results in a particular case of the weighted Sturm–
and upon eliminating j
Liouville problem
1
d d
K p
Cq ;
ð5:11Þ
r
dx dx
Phil. Trans. R. Soc. A (2008)
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1039
Real-valued algebro-geometric solutions
of the type
~1 ð~
~1 ð~
~1:xx ð~
z ; xÞ C j
z ; xÞ Z z~ð4uðxÞK uxx ðxÞÞj
z ; xÞ;
Kj
ðz; xÞ 2C !R;
ð5:12Þ
with ‘weight’ r Z ð4uK uxx Þ and constant coefficients pZqZ1.
Introducing
n
S Z g ½E2[ ; E2[ C1 ;
ð5:13Þ
[ Z0
we define
R2nC2 ðlÞ1=2 Z R2nC2 ðlÞ1=2 8
K1
>
>
>
>
>
< ðK1ÞnCj
!
>
ðK1Þn
>
>
>
>
:
iðK1ÞnCjC1
for l 2ðE2nC1 ;NÞ;
for l 2ðE2jK1 ; E2j Þ; j Z 1; .; n;
for l 2ðKN; E0 Þ;
l 2R ð5:14Þ
for l 2ðE2j ; E2jC1 Þ; j Z 0; .; n;
and
R2nC2 ðlÞ1=2 Z lim R2nC2 ðl C i3Þ1=2 ;
3Y0
l 2S
ð5:15Þ
and analytically continue R2nC2 ð$Þ1=2 to CnS. We also note the property
R2nC2 ð
z Þ1=2 Z R2nC2 ðzÞ1=2 :
ð5:16Þ
For notational convenience, we will occasionally call ðE2jK1 ; E2j Þ, j Z 1; .; n,
spectral gaps and E2jK1 ; E2j the corresponding spectral gap endpoints.
Next, we introduce the cut plane
ð5:17Þ
P Z CnS
and the upper, respectively, lower sheets PG of Kn by
PG Z fðz;GR2nC2 ðzÞ1=2 Þ 2Kn jz 2Pg
ð5:18Þ
with the associated charts
P Z ðz;GR2nC2 ðzÞ1=2 Þ1 z:
zG : PG/ P;
ð5:19Þ
The two branches JGðz; x; x 0 Þ of the Baker–Akhiezer vector JðP; x; x 0 Þ in
(3.16) are then given by
JGðz; x; x 0 Þ Z JðP; x; x 0 Þ;
P Z ðz; yÞ 2PG;
JG Z ðj1;G; j2;GÞu
ð5:20Þ
and one infers from (3.38) that
j1;Gðz; $; x 0 Þ 2L2 ððx 0 ;HNÞÞ;
for jzj sufficiently large:
ð5:21Þ
Thus, introducing
~Gð~
z ; x; x 0 Þ Z JHðz; x; x 0 Þ;
J
Phil. Trans. R. Soc. A (2008)
~G Z ðj
~1;G; j
~2;GÞu ;
J
z~ ZK1=z
ð5:22Þ
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1040
F. Gesztesy and H. Holden
and the two branches fGðz; xÞ of fðP; xÞ on PG by
fGðz; xÞ Z fðP; xÞ;
P Z ðz; yÞ 2PG;
ð5:23Þ
~Gð~
z ; xÞ
one infers from (4.12) and (5.21) that the Weyl–Titchmarsh functions m
associated with the self-adjoint Hamiltonian system (5.6) on the half-lines ½x;GNÞ
and the Dirichlet boundary condition indexed by a0 Z ð 1 0 Þ at the point x2R
are given by
~2;Gð~
~1;Gð~
~Gð~
m
z ; xÞ Z j
z ; x; x 0 Þ=j
z ; x; x 0 Þ Z j2;Hðz; x; x 0 Þ=j1;Hðz; x; x 0 Þ
Z ðK1=zÞfHðz; xÞ; z 2CnS:
ð5:24Þ
More precisely, (5.21) yields (5.24) only for sufficiently large jzj. However, since
~Gð$; xÞ are analytic in C\R, and by (3.12), fGð$; xÞ are
by general principles m
analytic in CnS, one infers (5.24) by analytic continuation. In particular, (5.21)
extends to all z 2CnS, i.e.
j1;Gðz; $; x 0 Þ 2L2 ððx 0 ;HNÞÞ; z 2CnS:
ð5:25Þ
Next, we mention a useful fact concerning a special class of Herglotz functions
closely related to the problem at hand. The result must be well known to experts,
but since we could not quickly locate a proof in the literature, we provide the
simple contour integration argument below.
1=2
Lemma 5.1. Let PN be a monic polynomial of degree N. Then PN =R2nC2 is a
Herglotz function if and only if one of the following alternatives applies:
(i) NZn and
Pn ðzÞ Z
n
Y
ðz K aj Þ;
aj 2½E 2jK1 ; E 2j ;
j Z 1; .; n:
ð5:26Þ
jZ1
1=2
If (5.26) is satisfied, then Pn =R 2nC2 admits the Herglotz representation
ð
Pn ðzÞ
1
jPn ðlÞjdl
1
Z
; z 2CnS:
ð5:27Þ
1=2
1=2
p S jR2nC2 ðlÞ j lKz
R2nC2 ðzÞ
(ii) N Z nC 1 and
n
Y
PnC1 ðzÞ Z ðz K b[ Þ;
b0 2ðKN; E0 ;
bj 2½E2jK1 ; E2j ;
j Z 1; .; n:
[ Z0
ð5:28Þ
If (5.28) is satisfied, then
admits the Herglotz representation
!
ð
PnC1 ðzÞ
PnC1 ðiÞ
1
jPnC1 ðlÞjdl
1
l
C
;
Z Re
K
p S jR2nC2 ðlÞ1=2 j lKz 1 C l2
R2nC2 ðzÞ1=2
R2nC2 ðiÞ1=2
ð5:29Þ
z 2CnS:
1=2
PnC1 =R2nC2
Proof. Since Herglotz functions are O(z) as jzj/Nand cannot vanish faster than
O(1/z) as jzj/N, we can confine ourselves to the range N 2fn; nC 1; nC 2g. We
start with the case NZn and employ the following contour integration approach.
Consider a closed oriented contour GR;3 , which consists of the clockwise oriented
semicircle C3 Z fz 2Cjz Z E0 K3 expðKiaÞ;Kp=2% a% p=2g centred at E0, the
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Real-valued algebro-geometric solutions
1041
straight line LCZ fz 2CCjz Z E0 C x C i3; 0% x % Rg (oriented from left to
right), the following part of the anticlockwise oriented circle of radius ðR2 C 32 Þ1=2
centred at E0, CR Z fz 2Cjz Z E0 C ðR2 C 32 Þ1=2 expðibÞ, arctanð3=RÞ% b% 2pK
arctanð3=RÞg, and the straight line LKZ fz 2CKjz Z E0 C x Ki3; 0% x % Rg
(oriented from right to left). Then, for 3O0 small enough and RO0 sufficiently
large, one infers
ð
Pn ðzÞ
1
1
Pn ðzÞ
1
1
Pn ðlÞdl
Z
dz Z
:
1=2
1=2
1=2
3Y0;R[N p S lKz iR
2pi GR;3 zKz R2nC2 ðzÞ
R2nC2 ðzÞ
2nC2 ðlÞ
ð5:30Þ
#
Here we used (5.14) to compute the contributions of the contour integral along [E0, R]
in the limit 3Y0 and note that the integral over CR tends to zero as R[N since
Pn ðzÞ
R2nC2 ðzÞ1=2
Z Oðjzj K1 Þ:
ð5:31Þ
z/N
Next, utilizing the fact that Pn is monic and using (5.14) again, one infers that
Fn ðlÞdl=½iR2nC2 ðlÞ1=2 represents a positive measure supported on S if and only if Pn
has precisely one zero in each of the intervals ½E2jK1 ; E2j , j Z 1; .; n. In other words,
Pn ðlÞ
1=2
iR2nC1 ðlÞ
Z
jPn ðlÞj
jR2nC1 ðlÞ1=2 j
R 0 on S;
ð5:32Þ
if and only if Pn has precisely one zero in each of the intervals ½E2jK1 ; E2j , j Z 1; .; n.
The Herglotz representation (4.7) and (4.8) then finishes the proof of (5.27).
In the case where NZnC1, the proof of (5.28) follows along similar lines
taking into account the additional residues at Gi inside GR;3 , which are
responsible for the real part on the right-hand side of (5.29).
1=2
Finally, in the case NZnC2, assume that PnC2 =R2nC2 is a Herglotz function.
Then necessarily,
ð0
PnC2 ðzÞ
Z a C bz C
duðlÞðlKzÞ K1 ; z 2CnS;
ð5:33Þ
E0
R2nC2 ðzÞ1=2
for some a2R, bR0, and some finite (positive) measure u supported on [E0, 0],
since
lim ImðPnC2 ðlÞR2nC2 ðl C i3Þ K1=2 Þ Z 0;
3Y0
for lO E2nC2 Z 0 and l! E0 :
ð5:34Þ
In particular, (5.33) implies
PnC2 ðzÞR2nC2 ðzÞ K1=2 Z bz C Oð1Þ;
jzj/N
bR 0:
ð5:35Þ
However, by (5.14), one immediately infers
PnC2 ðlÞR2nC2 ðlÞ K1=2 Z Kl C Oð1Þ:
l[N
This contradiction dispenses with the case NZnC2.
Phil. Trans. R. Soc. A (2008)
ð5:36Þ
&
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1042
F. Gesztesy and H. Holden
Now we are in position to state the following result concerning the half-line
and full-line Weyl–Titchmarsh functions associated with the self-adjoint
~ Gð~
Hamiltonian system (5.6). We denote by m
z ; xÞ the Weyl–Titchmarsh
m-functions corresponding to (5.6) associated with the half-lines ðx;GNÞ and
the Dirichlet boundary condition indexed by a0 Z ð 1 0 Þ at the point x2R, and
~ ð~
by M
z ; xÞ the 2!2 Weyl–Titchmarsh matrix corresponding to (5.6) on R (cf.
(4.12), (4.14) and (4.15)). Moreover, S0 denotes the open interior of S and the
real part of a matrix M is defined as usual by ReðM ÞZ ðM C M Þ=2.
Theorem 5.1. Assume hypothesis 5.1 and let ðz; xÞ 2R !ðCnSÞ, z~ZK1=z.
Then
GR2nC2 ðzÞ1=2 C zGn ðz; xÞ
~Gð~
m
z ; xÞ Z
;
zFn ðz; xÞ
R2nC2 ðiÞ1=2
~Gð~
z ; xÞ Z 1GRe
m
iFn ði; xÞ
1
G
p
!
C
ð5:37Þ
n
X
Gn ðmj ðxÞ; xÞð1H3j ðxÞÞ
dFn ðmj ðxÞ; xÞ=dz
jZ1
1
z K mj ðxÞ
jR2nC2 ðlÞ1=2 jdl
1
l
;
K
jlFn ðl; xÞj
lKz 1 C l2
S
ð
ð5:38Þ
where 3j ðxÞ 2f1;K1g, j Z 1; .; n, is chosen such that
Gn ðmj ðxÞ; xÞ3j ðxÞ
R 0;
dFn ðmj ðxÞ; xÞ=dz
ð5:39Þ
j Z 1; .; n:
Moreover,
~ ð~
M
z ; xÞ Z
KHn ðz; xÞ zGn ðz; xÞ
K1
2R2nC2 ðzÞ1=2
~ ð~
~ ði; xÞÞ C
M
z ; xÞ Z ReðM
zGn ðz; xÞ
ð
zFn ðz; xÞ
!
;
1
l
K
dUðl; xÞ
;
lKz 1 C l2
S
ð5:40Þ
ð5:41Þ
where
Uðl; xÞ Z
1
2piR2nC2 ðlÞ
1=2
Hn ðl; xÞ
KlGn ðl; xÞ
KlGn ðl; xÞ
KlFn ðl; xÞ
!
;
l 2S0 :
ð5:42Þ
The essential spectrum of the half-line Hamiltonian systems (5.6) on ½x;GNÞ (with
any self-adjoint boundary condition at x) as well as the essential spectrum of the
Hamiltonian system (5.6) on R is purely absolutely continuous and given by
K1
g KE 2lK1 ;KE 2K1
[ C1 g KE 2n ;N :
nK1
ð5:43Þ
[ Z0
The spectral multiplicities are simple in the half-line cases and of uniform
multiplicity two in the full-line case.
Phil. Trans. R. Soc. A (2008)
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1043
Real-valued algebro-geometric solutions
Proof. Equation (5.37) follows from (3.12), (5.14) and (5.24). Equation (5.40)
is then a consequence of (3.20)–(3.22), (4.14), (4.15), (5.24) and (5.37). Different
self-adjoint boundary conditions at the point x lead to different half-line
Hamiltonian systems whose Weyl–Titchmarsh functions are related by a linear
fractional transformation (cf. Clark & Gesztesy 2002) that leads to the
invariance of the essential spectrum with respect to the boundary condition
at x. In order to prove the Herglotz representation (5.38), one can follow the
corresponding computation for Schrödinger operators with algebro-geometric
potentials in Levitan (1987, §8.1). For this purpose, one first notes that by (5.29)
also R2nC2 ðzÞ1=2 =½zFn ðz; xÞ is a Herglotz function. A contour integration as in the
proof of lemma 5.1 then proves
!
n
X
jR2nC2 ðmj ðxÞÞ1=2 j
R2nC2 ðzÞ1=2
R2nC2 ðiÞ1=2
1
Z Re
C
zFn ðz; xÞ
iFn ði; xÞ
m ðxÞjdFn ðmj ðxÞ; xÞ=dzj z K mj ðxÞ
jZ1 j
1
C
p
ð
jR2nC2 ðlÞ1=2 jdl
1
l
;
K
jlFn ðl; xÞj
lKz 1 C l2
S
ð5:44Þ
!
n
X
Gn ðmj ðxÞ; xÞ3j ðxÞ
R2nC2 ðzÞ1=2
R2nC2 ðiÞ1=2
1
Z Re
K
zFn ðz; xÞ
iFn ði; xÞ
dFn ðmj ðxÞ; xÞ=dz z K mj ðxÞ
jZ1
C
1
p
jR2nC2 ðlÞ1=2 jdl
1
l
:
K
jlFn ðl; xÞj
lKz 1 C l2
S
ð
ð5:45Þ
The only difference compared to the corresponding argument in the proof of lemma
5.1 concerns additional (approximate) semicircles of radius 3 centred at each mj(x),
j Z 1; .; n, in the upper and lower complex half-planes. Whenever
mj ðxÞ 2ðE2jK1 ; E2j Þ, the limit 3Y0 picks up a residue contribution displayed in the
sum over j in (5.44). This contribution vanishes, however, if mj ðxÞ 2fE2jK1 ; E2j g.
In this case, dFn ðmj ðxÞ; xÞ=dz s0 by (4.10) and R2nC2 ðmj ðxÞÞZ 0 by (2.17).
Equation (5.45) then follows from (3.7) and the sign of 3j(x) must be chosen
according to (5.39) in order to guarantee non-positive residues in (5.45) (cf. (4.10)).
Next, we apply the Lagrange interpolation formula. If QnK1 is a polynomial of
degree nK1, then
QnK1 ðzÞ Z Fn ðzÞ
n
X
QnK1 ðmj Þ
1
;
dFn ðmj Þ=dz z K mj
jZ1
z 2C:
ð5:46Þ
Since Fn and Gn are monic polynomials of degree n, we can apply (5.46) to
QnK1ZGnKFn and obtain
n
X
Gn ðmj ðxÞ; xÞ
Gn ðz; xÞ
1
Z1C
:
dFn ðmj ðxÞ; xÞ=dz z K mj ðxÞ
Fn ðz; xÞ
jZ1
Phil. Trans. R. Soc. A (2008)
ð5:47Þ
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1044
F. Gesztesy and H. Holden
Insertion of (5.47) into (5.37) then yields
n
Gn ðmj ðxÞ; xÞ½3j ðxÞ C ð1K 3j ðxÞÞ
GR2nC2 ðzÞ1=2 X
1
~Gð~
C
C1
z ; xÞ Z
m
z K mj ðxÞ
zFn ðz; xÞ
dFn ðmj ðxÞ; xÞ=dz
jZ1
ð5:48Þ
and hence (5.38) follows by inserting (5.45) into (5.48). Equations (5.41) and (5.42)
are clear from the matrix analogue of (4.9).
The statement (5.43) for the essential half-line spectra then follows from the fact
~G (as a function of z) is
that the measure in the Herglotz representation (5.38) of m
supported on the set S in (5.13), with a strictly positive density on the open interior
S0 of S. The transformation z /K1=z then yields (5.43) and since half-line spectra
with a regular endpoint x have always simple spectra, this completes the proof of our
half-line spectral assertions. The full-line case follows in exactly the same manner
since the corresponding 2!2 matrix-valued measure U in the Herglotz
~ (as a function of z) also has support S and rank equal
representation (5.41) of M
0
to 2 on S .
&
Returning to direct spectral theory, we note that the two spectral problems
~
(5.6) on R associated with the vanishing of the first and second component of J
at x, respectively, are clearly self-adjoint since they correspond to the choices
aZ ð 1 0 Þ and aZ ð 0 1 Þ in (4.5). Hence, a comparison with (3.5), (3.26) and
(3.27) necessarily yields fmj ðxÞgjZ1;.;n ; fnj ðxÞgjZ1;.;n 3R. Thus, we will assume
the convenient eigenvalue ordering
mj ðxÞ! mjC1 ðxÞ; nj ðxÞ! njC1 ðxÞ; for j Z 1; .; nK1; x 2R:
ð5:49Þ
~
The zeros of j1 ð$; xÞ belong to the Dirichlet spectral problem associated with the
Hamiltonian system (5.12) (respectively, the weighted Sturm–Liouville problem
(5.12)) on R. A comparison with (3.26) then relates the zeros mj ðx 1 Þ, j Z 1; .; n,
of Fn ð$; x 1 Þ in (3.5) to the Dirichlet spectrum of (5.6) (respectively, (5.12)) on R.
The correspondence between each mj and the related spectral point of the
Dirichlet problem (5.6) (respectively, (5.12)) on R is of course effected by the
~2 ð$; x 1 Þ do not belong
transformation z /K1=z. In contrast to this, the zeros of j
to the Neumann spectrum associated with the Hamiltonian system (5.6)
(respectively, the weighted Sturm–Liouville problem (5.12)) on R. In fact, by
~2 ð$; x 1 Þ correspond to a mixed boundary condition of the
(5.9), zeros of j
~
~
type j1;x ðx 1 ÞC j1 ðx 1 ÞZ 0. By (3.27), this relates the zeros nj ðx 1 Þ, j Z 1; .; n, of
Hn ð$; x 1 Þ in (3.5) to the spectrum of (5.6) (respectively, (5.12)) on R
~1;x ðx 1 ÞC j
~1 ðx 1 ÞZ 0.
corresponding to the self-adjoint boundary condition j
Combining lemma 5.1 with the Herglotz property of the 2!2 Weyl–
~ ð$; xÞ then yields the following refinement of theorem 3.2.
Titchmarsh matrix M
Theorem 5.2. Assume hypothesis 5.1. Then fm^j gjZ1;.;n , with the projections
mj ðxÞ, j Z 1; .; n, the zeros of Fn ð$; xÞ in (3.5), satisfies the first-order system of
differential equations (3.31) on UmZR and
m^j 2CNðR; Kn Þ;
j Z 1; .; n:
ð5:50Þ
Moreover,
mj ðxÞ 2½E2jK1 ; E2j ;
Phil. Trans. R. Soc. A (2008)
j Z 1; .; n;
x 2R:
ð5:51Þ
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Real-valued algebro-geometric solutions
1045
In particular, m^j ðxÞ changes sheets whenever it hits E 2jK1 or E 2j and its projection
mj(x) remains trapped in ½E 2jK1 ; E 2j for all j Z 1; .; n and x2R. The analogous
statements apply to n^j ðxÞ and one infers
ð5:52Þ
nj ðxÞ 2½E 2jK1 ; E 2j ; j Z 1; .; n; x 2R:
~
Proof. Since M ð$; xÞ is a 2!2 Herglotz matrix, its diagonal elements are
Herglotz functions. Thus,
Hn ðz; xÞ
KzFn ðz; xÞ
~ 1;1 ð~
~ 2;2 ð~
M
z ; xÞ Z
; M
z ; xÞ Z
; z~ ZK1=z
ð5:53Þ
1=2
2R2nC2 ðzÞ
2R2nC2 ðzÞ1=2
are Herglotz functions (the left-hand sides with respect to z~, the right-hand sides
with respect to z) and the interlacing properties (5.51) and (5.52) then follow
from (5.28) and (5.26).
&
Remark 5.1. Combining the interlacing property (5.51) with (2.18), (2.19) and
(2.20) yields (cf. also (3.35))
!
!
2n
n
Y
Y
4uðxÞK uxx ðxÞ ZK
ð5:54Þ
Em
mj ðxÞK2 O 0; x 2R;
mZ0
jZ1
in accordance with (5.4). Moreover, since by (5.52) the vj(x) also remain trapped
in the intervals ½E2jK1 ; E 2j for all x2R, none of the n^j can reach PNK and hence
h 0 Z 4uC 2ux s0 on R (cf. the discussion surrounding (3.15)). Actually,
h 0 ðxÞO 0;
x 2R;
ð5:55Þ
1=2
since Hn ð$; xÞ=R2nC2 is a Herglotz function (cf. (5.27)).
Remark 5.2. The zeros mj ðxÞ 2ðE2jK1 ; E2j Þ, j Z 1; .; n of Fn ð$; xÞ, which are
related to eigenvalues of the Hamiltonian system (5.6) on R associated with the
~1 ðxÞZ 0, in fact, are related to left and right half-line
boundary condition j
eigenvalues of the corresponding Hamiltonian system restricted to the half-lines
ðKN; x and ½x;NÞ, respectively. Indeed, by (5.22) and (5.25), depending on
whether m^j ðxÞ 2PC or m^j ðxÞ 2PK, mj(x) is related to a left or right half-line
~1 ðxÞZ 0. A careful
eigenvalue associated with the Dirichlet boundary condition j
investigation of the sign of the right-hand sides of the Dubrovin equations (3.30)
(combining (5.1), (5.14) and (5.18)), then proves that the mj(x) related to right
(respectively, left) half-line eigenvalues of the Hamiltonian system (5.6)
~1 ðxÞZ 0, are strictly monotone
associated with the boundary condition j
increasing (respectively, decreasing) with respect to x, as long as the mj stay
away from the right (respectively, left) endpoints of the corresponding spectral
gaps ðE2jK1 ; E 2j Þ. Here we purposely avoided the limiting case where some of the
mk(x) hit the boundary of the spectral gaps, mk ðxÞ 2fE 2kK1 ; E 2k g, since the halfline eigenvalue interpretation is lost as there is no L2 ððx;GNÞÞ2 eigenfunction
~
~1 ðxÞZ 0 in this case. In fact, whenever an eigenvalue mk(x) hits a
JðxÞ
satisfying j
spectral gap endpoint, the associated point m^j ðxÞ on Kn crosses over from one
sheet to the other (equivalently, the corresponding left half-line eigenvalue
becomes a right half-line eigenvalue and vice versa) and, accordingly, strictly
increasing half-line eigenvalues become strictly decreasing half-line eigenvalues
and vice versa. In particular, using the appropriate local coordinate ðz K E2k Þ1=2
(respectively, ðz K E2kK1 Þ1=2 ) near E2k (respectively, E2kK1), one verifies that
mk(x) does not pause at the endpoints E2k and E2kK1.
Phil. Trans. R. Soc. A (2008)
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1046
F. Gesztesy and H. Holden
Next, we turn to the inverse spectral problem and determine the isospectral
manifold of real-valued, smooth and bounded CH solutions.
Our basic assumptions then will be the following.
Hypothesis 5.2. Suppose
E0 ! E1 !/! E2n ! E2nC1 Z 0;
fix x 02R and assume that the initial data
ð5:56Þ
f^
mj ðx 0 Þ Z ðmj ðx 0 Þ;Kmj ðx 0 ÞGn ðmj ðx 0 Þ; x 0 ÞÞgjZ1;.;n 3Kn ;
ð5:57Þ
for the Dubrovin equations (3.31) are constrained by
mj ðx 0 Þ 2½E2jK1 ; E2j ;
ð5:58Þ
j Z 1; .; n:
Theorem 5.3. Assume hypothesis 5.2. Then the Dubrovin initial value problem
(3.31), (5.57) and (5.58) has a unique solution fm^j gjZ1;.;n 3Kn satisfying
m^j 2CNðR; Kn Þ;
ð5:59Þ
j Z 1; .; n;
and the projections mj remain trapped in the intervals ½E2jK1 ; E2j , j Z 1; .; n, for
all x2R,
mj ðxÞ 2½E2jK1 ; E2j ;
j Z 1; .; n;
x 2R:
ð5:60Þ
Moreover, u defined by the trace formula (3.34), i.e.
uðxÞ Z
n
X
1X
1 2nC1
mj ðxÞK
E ;
2 jZ1
4 mZ0 m
x 2R;
ð5:61Þ
satisfies hypothesis 5.1, i.e.
u 2CNðRÞ;
ð5:62Þ
u is real -valued;
vkx u 2LNðRÞ;
k 2N0 ;
ð5:63Þ
x 2R
ð5:64Þ
s-CHn ðuÞ Z 0 on R;
ð5:65Þ
4uK uxx O 0;
and the nth stationary CH equation
with integration constants c[ in (5.65) given by c[ Z c[ ðEÞ, [ Z 1; .; n, according
to (2.26) and (2.27).
Proof. Given initial data constrained by mj ðx 0 Þ 2ðE2jK1 ; E2j Þ, j Z 1; .; n, one
1=2
concludes from the Dubrovin equations (3.31) and the sign properties of R2nC2 on
the intervals ½E2kK1 ; E2k , k Z 1; .; n, described in (5.14), that the solution mj(x)
remains in the interval ½E2jK1 ; E2j as long as m^j ðxÞ stays away from the branch
points ðE2jK1 ; 0Þ; ðE2j ; 0Þ. In case m^j hits such a branch point, one can use the local
chart around (Em,0), with local coordinate zZ sðz K Em Þ1=2 , s 2f1;K1g,
m 2f2j K1; 2jg, to verify (5.59) and (5.60). Relations (5.61)–(5.63) are then
evident from (5.59), (5.60) and
jvkx mj ðxÞj% Ck ;
Phil. Trans. R. Soc. A (2008)
k 2N0 ;
j Z 1; .; n;
x 2R:
ð5:66Þ
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Real-valued algebro-geometric solutions
1047
In the course of the proof of theorem 3.1 presented in Gesztesy & Holden
(2002, 2003, §5.3), one constructs the polynomial formalism (Fn, Gn, Hn, R 2nC2,
etc.) and then obtains identity (3.35) as an elementary consequence.
The latter immediately proves (5.64). Finally, (5.65) follows from theorem 3.1
(with UmZR).
&
Corollary 5.1. Fix fEm gmZ0;.;2nC1 3R and assume the ordering (5.56). Then
the isospectral manifold of smooth and bounded real-valued solutions u 2
CNðRÞh LNðRÞ of s-CHn ðuÞZ 0 is given by a real n-dimensional torus Tn.
Proof. The discussion in remark 5.2 and theorem 5.3 shows that the motion of
each m^j ðxÞ on Kn proceeds topologically on a circle and is uniquely determined by
the initial data m^k ðx 0 Þ, k Z 1; .; n. More precisely, the initial data
m^j ðx 0 Þ Z ðmj ðx 0 Þ; yð^
mj ðx 0 ÞÞÞ Z ðmj ðx 0 Þ;Kmj ðx 0 ÞGn ðmj ðx 0 Þ; x 0 ÞÞ;
mj ðx 0 Þ 2½E 2jK1 ; E 2j ;
j Z 1; .; n
ð5:67Þ
are topologically equivalent to data of the type
ðmj ðx 0 Þ; sj ðx 0 ÞÞ 2½E 2jK1 ; E 2j !fC;Kg;
j Z 1; .; n;
ð5:68Þ
the sign of sj ðx 0 Þ depending on m^j ðx 0 Þ 2PG. If some of the mk ðx 0 Þ 2fE 2kK1 ; E 2k g,
then the determination of the sheet PG and hence the sign sk ðx 0 Þ in (5.68)
becomes superfluous and is eliminated from (5.68). Indeed, since by (2.16),
mj ðx 0 Þ2 Gn ðmj ðx 0 Þ; x 0 Þ Z R2nC2 ðmj ðx 0 ÞÞ;
ð5:69Þ
Gn ðmj ðx 0 Þ; x 0 Þ is determined up to a sign unless mj ðx 0 Þ hits a spectral gap
endpoint E2jK1 ; E2j in which case Gn ðmj ðx 0 Þ; x 0 ÞZ R2nC2 ðmj ðx 0 ÞÞZ 0 and the sign
ambiguity disappears. The n data in (5.68) (properly interpreted if
mj ðx 0 Þ 2fE2jK1 ; E2j g) can be identified with circles. Since the latter are
independent of each other, the isospectral manifold of real-valued, smooth and
bounded solutions of s-CHn ðuÞZ 0 is given by a real n-dimensional torus Tn.
&
Remark 5.3.
(i) For simplicity, we only focused on the case 4uK uxx O 0. The opposite case
4uK uxx ! 0 is completely analogous and results in a reflection of Em,
mZ 0; .; 2nC 1, and mj ðxÞ; nj ðxÞ, j Z 1; .; n, about zZ0, etc.
(ii) The time-dependent case also offers nothing new. Higher-order CHr flows
drive each m^j ðx; tr Þ around the same circles as in the stationary case in
complete analogy to the familiar KdV case.
In summary, one observes that the reality problem for smooth and bounded
solutions of the CH hierarchy, assuming the ordering (5.56) (respectively, the
one obtained upon reflection with respect to zZ0), parallels that of the
KdV hierarchy with the basic self-adjoint Lax operator (the one-dimensional
Schrödinger operator) replaced by the self-adjoint Hamiltonian system (5.6).
Phil. Trans. R. Soc. A (2008)
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1048
F. Gesztesy and H. Holden
The following result was found in response to a query of Igor Krichever who
inquired about the significance of the eigenvalue ordering (5.56) (or the one
obtained upon reflection at zZ0). As it turns out, such an ordering is crucial if
one is interested in smooth algebro-geometric solutions on R.
Theorem 5.4. Suppose
Em ! EmC1 ; m Z 0; .2n;
and
E2j0K1 Z 0 ðrespectively; E2j0 Z 0Þ for some j0 2f1; .; ng:
ð5:70Þ
Fix x02R and assume that the initial data
f^
mj ðx 0 Þ Z ðmj ðx 0 Þ;Kmj ðx 0 ÞGn ðmj ðx 0 Þ; x 0 ÞÞgjZ1;.;n 3Kn ;
ð5:71Þ
for the Dubrovin equations (3.31) are constrained by
mj ðx 0 Þ 2½E2jK1 ; E2j ;
j 2f1; .; ngnfj0 g;
and
mj0 ðx 0 Þ 2ðE2j0K1 ; E2j0 ðrespectively; mj0 ðx 0 Þ 2½E2j0K1 ; E2j0 ÞÞ:
ð5:72Þ
Then there exists a set Um3R of the type
Um Z Rnfxk gk2Z ;
xk ! xkC1 ;
k 2Z;
lim xk ZKN;
kYKN
lim xk ZN;
k[N
ð5:73Þ
such that the Dubrovin initial value problem (3.31), (5.57) and (5.58) has a unique
solution fm^j gjZ1;.;n 3Kn satisfying
m^j 2CNðUm ; Kn Þ;
j Z 1; .; n
ð5:74Þ
and the projections mj remain trapped in the intervals ½E2jK1 ; E2j , j Z 1; .; n, for
all x2Um,
ð5:75Þ
mj ðxÞ 2½E2jK1 ; E2j ; j Z 1; .; n; x 2Um :
Moreover, u defined by the trace formula (3.34), i.e.
n
X
1X
1 2nC1
mj ðxÞK
E ; x 2Um ;
uðxÞ Z
2 jZ1
4 mZ0 m
ð5:76Þ
satisfies
u 2CNðUm Þ;
u is real -valued;
4uK uxx O 0;
ð5:77Þ
x 2Um ;
ð5:78Þ
on Um ;
ð5:79Þ
and the nth stationary CH equation
s-CHn ðuÞ Z 0
with integration constants c[ in (5.65) given by c[ Z c[ ðEÞ, [ Z 1; .; n, according
of the type
to (2.26) and (2.27). At each xk, u x exhibits a singularity
ð5:80Þ
ux ðxÞ Z Ck ðx K xk Þ K1=3 C o ðx K xk ÞK1=3 ; Ck s0; k 2Z:
x/xk
In particular, u ;C 1 ðRÞ, ux ;LNðRÞ. The isospectral manifold corresponding to
(5.70)–(5.72) is then given by TnK1!R.
Phil. Trans. R. Soc. A (2008)
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1049
Real-valued algebro-geometric solutions
Proof. One can follow the proof of theorem 5.3 and corollary 5.1 with one
important twist, though, since the right-hand side of the Dubrovin equation of mj0
blows up as mj0 / E2j0K1 ðrespectively; E2j0 Þ. For notational convenience and
without loss of generality, we may assume E1Z0 (and hence j0Z1) in the
following. Recalling the Dubrovin equations (3.31), one verifies that its solutions
are smooth with respect to x as long as m1 stays away from E1Z0. Varying x2R,
the sign restrictions on m1;x in terms of the right-hand side of the corresponding
equation in (3.31) eventually accelerate m1 into E1Z0 as x tends to some xk2R,
and we now analyse what happens to all mj for x in a neighbourhood of xk.
Recalling the local coordinate sz 1=2 , sZG1, near E1Z0 and hence introducing
z1 ðxÞ Z sðKm1 ðxÞÞ1=2 ; for m1 ðxÞ sufficiently close to E1 Z 0 as x / xk ð5:81Þ
d2 2K , the Dubrovin equation for m
and the corresponding point m^1 Z Kz
j
n
1
becomes for x near xk,
n
2 Y
yðKzd
1 ðxÞ Þ
z1;x ðxÞ Z
ðKz1 ðxÞ2 K m[ ðxÞÞ K1 ;
3
z1 ðxÞ
[ Z2
Z C1 z1 ðxÞ K2 ð1 C oð1ÞÞ;
ð5:82Þ
x/xk
mj;x ðxÞ Z 2
n
yðm^j ðxÞÞ Y
ðmj ðxÞKm[ ðxÞÞK1 ;
mj ðxÞ
j Z 2; .; n;
ð5:83Þ
[ Z1
[ sj
for some constant C1s0. (Here we implicitly assume that no other mj, j Z 2; .; n
simultaneously hits E2jK1 or E2j as x/xk. Otherwise one simply resorts to the
proper local coordinate for such a mj. We omit the details.) To treat the
singularity of z1;x as z1 ðxÞ/ 0 for x/xk, we now resort to a well-known trick
described, for instance, in Hille (1976, theorem 3.2.2) in the context of scalar
first-order differential equations. Instead of looking for solutions z1 , mj as
functions of x, we now look for x Z xðz1 Þ, m~j Z m~j ðz1 Þ as functions of z1 , where we
denote m~j ðz1 ÞZ mj ðxÞ, j Z 2; .; n. Then (5.82) and (5.83) turn into
n
Y
z31
x 0 ðz1 Þ Z
ðKz21 K~
m[ ðz1 ÞÞ;
ð5:84Þ
d
2
yðKz1 Þ [ Z2
n
n
Y
yðm^
~j ðz1 ÞÞ z31 Y
ðKz21 K~
m[ ðz1 ÞÞ
ð~
mj ðz1 ÞK~
m[ ðz1 ÞÞ K1 ;
m~j;z1 ðxÞ Z 2 d
m
~
ðz
Þ
2
j 1 [ Z2
y Kz1
[ Z1
[ sj
j Z 2; .; n:
ð5:85Þ
Since the right-hand sides in (5.84) and (5.85) are holomorphic with respect to
the n variables z1 ; m~2 ; .; m~n for z1 near zero and m~j near ½E2jK1 ; E2j , j Z 2; .; n,
equations (5.84) and (5.85) yield solutions x; m~2 ; .; m~n holomorphic with respect
to z1 near z1 Z 0. In particular, since
x 0 ðz1 Þ Z z21 =C1 C Oðz31 Þ;
z1/0
Phil. Trans. R. Soc. A (2008)
mj0 ðz1 Þ Z Cj z21 C Oðz31 Þ;
z1/0
j Z 2; .; n;
ð5:86Þ
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1050
F. Gesztesy and H. Holden
for some constants Cj s0, j Z 1; .; n, one obtains
xðz1 Þ Z xk C z31 =ð3C1 Þ C Oðz41 Þ;
ð5:87Þ
z1/0
m~j ðz1 Þ Z m~j ð0Þ C Cj z31 =3 C Oðz41 Þ Z mj ðxk Þ C Cj z31 =3 C Oðz41 Þ:
z1/0
ð5:88Þ
Thus, inverting xðz1 Þ, one observes
z1 ðxÞ Z ½3C1 ðx K xk Þ1=3 C Oððx K xk Þ2=3 Þ;
ð5:89Þ
x/xk
mj ðxÞ Z mj ðxk Þ C C1 Cj ðx K xk Þ C Oððx K xk Þ4=3 Þ;
j Z 2; .; n
ð5:90Þ
Z Kð2=3Þð3C1 Þ2=3 ðx K xk Þ K1=3 C oððx K xk Þ K1=3 Þ
ð5:91Þ
x/xk
and hence,
m1;x ðxÞ Z K2z1 ðxÞz1;x ðxÞ
x/xk
and
ux ðxÞ Z ð1=2Þ
n
X
mj;x ðxÞ
jZ1
Z Kð1=3Þð3C1 Þ2=3 ðx K xk Þ K1=3 C oððx K xk Þ K1=3 Þ:
x/xk
ð5:92Þ
The singular behaviour (5.91) and (5.92) repeats itself after each revolution of m1
around its circle and occurs whenever m1 passes again through E1Z0, giving rise
to the exceptional set fxk gk2Z in (5.73). Hence, mj 2C 1 ðRÞ, j Z 2; .; n, while
m1;x ZKz1 ðxÞ2 blows up whenever x approaches an element of fxk gk2Z . The rest of
the discussion follows as in theorem 5.3 and corollary 5.1. Since m1 ðx 0 ÞZ E1 Z 0 is
not an admissible initial condition in (5.72), one point must be removed from the
circle associated to m1, which topologically results in R instead of S 1 and hence in
the non-compact isospectral manifold TnK1!R.
&
Thus, smooth algebro-geometric CH solutions require E0Z0 or E2nC1Z0.
Finally, we briefly turn to the time-dependent case.
Hypothesis 5.3. Suppose that u : R2 / C satisfies
uð$; tÞ 2CNðRÞ;
vm u
ð$; tÞ 2LNðRÞ;
vx m
uðx; $Þ; uxx ðx; $Þ 2C 1 ðRÞ;
x 2R:
m 2N0 ;
t 2R;
ð5:93Þ
The basic problem in the analysis of algebro-geometric solutions of the CH
hierarchy consists in solving the time-dependent rth CH flow with initial data a
stationary solution of the nth equation in the hierarchy. More precisely, given
n2N0, consider a solution u ð0Þ of the nth stationary CH equation s-CHn ðu ð0Þ ÞZ 0
associated with Kn and a given set of integration constants fc[ g[Z1;.;n 3C. Next,
let r2N0; we intend to construct a solution u of the rth CH flow CHr(u)Z0 with
uðt0;r ÞZ uð0Þ for some t0;r 2R. To emphasize that the integration constants in the
definitions of the stationary and the time-dependent CH equations are
Phil. Trans. R. Soc. A (2008)
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1051
Real-valued algebro-geometric solutions
independent of each other, we indicate this by adding a tilde on all the time~r, H
~ r , f~s , g~s ,
dependent quantities. Hence, we shall employ the notation V~ r , F~ r , G
h~s , c~s , etc. in order to distinguish them from Vn, Fn, Gn, Hn, f[, g[, h[, c[, etc. in
the following. In addition, we will follow a more elaborate notation inspired by
Hirota’s t-function approach and indicate the individual rth CH flow by a
separate time variable tr2R.
Summing up, we are seeking a solution u of
g r ðuÞ Z 4ut K uxxt C ðuxxx K4ux Þf~r K 2ð4uK uxx Þf~r;x Z 0;
CH
r
r
uðx; t0;r Þ Z u ð0Þ ðxÞ;
ð5:94Þ
x 2R;
s-CHn ðuð0Þ Þ Z ðuxxx K4ux Þfn K 2ð4uK uxx Þfn;x Z 0;
ð5:95Þ
for some t0;r 2R, n; r 2N0 , where u satisfies (5.93).
We pause for a moment to reflect on the pair of equations (5.94) and (5.95): as
it turns out (cf. Gesztesy & Holden 2002, 2003, §5.4), it represents a dynamical
system on the set of algebro-geometric solutions isospectral to the initial value
u(0). By isospectral we here allude to the fact that for any fixed tr 2R, the
solution uð$; tr Þ of (5.94) and (5.95) is a stationary solution of (5.95),
s-CHn ðuð$; tr ÞÞ Z ðuxxx ð$; tr ÞK4ux ð$; tr ÞÞfn ð$; tr Þ
K 2ð4uð$; tr ÞK uxx ð$; tr ÞÞfn;x ð$; tr Þ Z 0;
ð5:96Þ
associated with the fixed underlying algebraic curve Kn. Put differently, uð$; tr Þ is
an isospectral deformation of u(0) with tr the corresponding deformation
parameter. In particular, uð$; tr Þ traces out a curve in the set of algebrogeometric solutions isospectral to u(0).
Thus, relying on this isospectral property of the CH flows, we will go a step
further and assume (5.95) not only at tr Z t0;r but also for all tr 2R. Hence, we
start with
ð5:97Þ
Ut ðz; x; tr ÞKV~ r;x ðz; x; tr Þ C ½U ðz; x; tr Þ; V~ r ðz; x; tr Þ Z 0;
r
KVn;x ðz; x; tr Þ C ½U ðz; x; tr Þ; Vn ðz; x; tr Þ Z 0;
where (cf. (2.18))
0
ðz; x; tr Þ 2C !R2 ;
1
K1
U ðz; x; tr Þ Z @
ð5:98Þ
1
A;
ð4uðx; tr ÞK uxx ðx; tr ÞÞ 1
0
1
~ r ðz; x; tr Þ F~ r ðz; x; tr Þ
KG
B
C
V~ r ðz; x; tr Þ Z @
A;
K1 ~
~
z H r ðz; x; tr Þ G r ðz; x; tr Þ
z
0
K1
KGn ðz; x; tr Þ
Fn ðz; x; tr Þ
V~ n ðz; x; tr Þ Z @
A
z K1 Hn ðz; x; tr Þ
Phil. Trans. R. Soc. A (2008)
1
Gn ðz; x; tr Þ
ð5:99Þ
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1052
F. Gesztesy and H. Holden
and
Fn ðz; x; tr Þ Z
n
X
n
Y
fnK[ ðx; tr Þz Z ðz K mj ðx; tr ÞÞ;
[ Z0
jZ1
Gn ðz; x; tr Þ Z
[
n
X
gnK[ ðx; tr Þz [ ;
ð5:100Þ
ð5:101Þ
[ Z0
Hn ðz; x; tr Þ Z
n
X
hnK[ ðx; tr Þz [ Z h 0 ðx; tr Þ
n
Y
ðz K nj ðx; tr ÞÞ;
ð5:102Þ
jZ1
[ Z0
h 0 ðx; tr Þ Z 4uðx; tr Þ C 2ux ðx; tr Þ;
F~ r ðz; x; tr Þ Z
r
X
ð5:103Þ
f~rKs ðx; tr Þz s ;
ð5:104Þ
g~rKs ðx; tr Þz s ;
ð5:105Þ
h~rKs ðx; tr Þz s ;
ð5:106Þ
sZ0
~ r ðz; x; tr Þ Z
G
r
X
sZ0
~ r ðz; x; tr Þ Z
H
r
X
sZ0
ð5:107Þ
h~0 ðx; tr Þ Z 4uðx; tr Þ C 2ux ðx; tr Þ;
~
~
for fixed n; r 2N0 . Here f[ ðx; tr Þ, f s ðx; tr Þ, g[ ðx; tr Þ, g~s ðx; tr Þ, h[ ðx; tr Þ and hs ðx; tr Þ
for [ Z 0; .; n, sZ0, ., r, are defined as in (2.3) and (2.7) with u(x) replaced by
u(x, tr), etc. and with appropriate integration constants. Explicitly, (5.97) and
(5.98) are equivalent to
~ r;x ðz; x; tr Þ C 2H
~ r ðz; x; tr Þ
4utr ðx; tr ÞK uxxtr ðx; tr ÞKH
~ r ðz; x; tr Þ Z 0;
K 2ð4uðx; tr ÞK uxx ðx; tr ÞÞG
ð5:108Þ
~ r ðz; x; tr ÞK 2F~ r ðz; x; tr Þ;
F~ r;x ðz; x; tr Þ Z 2G
ð5:109Þ
~ r;x ðz; x; tr Þ Z ð4uðx; tr ÞK uxx ðx; tr ÞÞF~ r ðz; x; tr ÞKH
~ r ðz; x; tr Þ
zG
ð5:110Þ
and
Fn;x ðz; x; tr Þ Z 2Gn ðz; x; tr ÞK 2Fn ðz; x; tr Þ;
ð5:111Þ
Hn;x ðz; x; tr Þ Z 2Hn ðz; x; tr ÞK 2ð4uðx; tr ÞK uxx ðx; tr ÞÞGn ðz; x; tr Þ;
ð5:112Þ
zGn;x ðz; x; tr Þ Z ð4uðx; tr ÞK uxx ðx; tr ÞÞFn ðz; x; tr ÞK Hn ðz; x; tr Þ:
ð5:113Þ
One observes that equations (2.3)–(2.25) apply to Fn, Gn, Hn, f[, g[ and h[, and
(2.3)–(2.8) and (2.18), with n replaced by r and c[ replaced by c~[ , apply to F~ r ,
~r, H
~ r , f~[ , g~[ and h~[ . In particular, the fundamental identity (2.16),
G
z 2 Gn ðz; x; tr Þ2 C zFn ðz; x; tr ÞHn ðz; x; tr Þ Z R2nC2 ðzÞ;
Phil. Trans. R. Soc. A (2008)
tr 2R;
ð5:114Þ
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Real-valued algebro-geometric solutions
1053
holds as in the stationary context and the hyperelliptic curve Kn is still given by
2nC1
Y
ðz K Em Þ:
ð5:115Þ
Kn : F n ðz; yÞ Z y2 K R2nC2 ðzÞ Z 0; R2nC2 ðzÞ Z
mZ0
Here we are still assuming (3.4), i.e.
ð5:116Þ
E0 ; .; E2n 2Cnf0g; E2nC1 Z 0:
The independence of (5.114) of tr2R can be interpreted as follows. The rth
KdV flow represents an isospectral deformation of the curve Kn in (5.115), in
particular, the branch points of Kn remain invariant under these flows
vtr Em Z 0; m Z 0; .; 2n C 1:
ð5:117Þ
Together with the comments following (5.95), this shows that isospectral torus
questions are conveniently reduced to the study of the stationary hierarchy of
CH flows since time-dependent solutions just trace out a curve in the isospectral
torus defined by the stationary hierarchy. This is of course in complete
agreement with other completely integrable 1C1-dimensional hierarchies such as
the KdV, Toda and AKNS hierarchies.
We are indebted to Mark Alber, Roberto Camassa, Adrian Constantin, Yuri Fedorov and Igor
Krichever for stimulating discussions. This research was supported in part by the Research Council of
Norway.
References
Alber, M. S. & Fedorov, Y. N. 2000 Wave solutions of evolution equations and Hamiltonian flows
on nonlinear subvarieties of generalized Jacobians. J. Phys. A 33, 8409–8425. (doi:10.1088/03054470/33/47/307)
Alber, M. S. & Fedorov, Y. N. 2001 Algebraic geometrical solutions for certain evolution equations
and Hamiltonian flows on nonlinear subvarieties of generalized Jacobians. Inverse Probl. 17,
1017–1042. (doi:10.1088/0266-5611/17/4/329)
Alber, M. S., Camassa, R., Fedorov, Yu. N., Holm, D. D. & Marsden, J. E. 2001 The complex
geometry of weak piecewise smooth solutions of integrable nonlinear PDE’s of shallow water
and dym type. Commun. Math. Phys. 221, 197–227. (doi:10.1007/PL00005573)
Atkinson, F. V. 1964 Discrete and continuous boundary problems. New York, NY: Academic Press.
Camassa, R. & Holm, D. 1993 An integrable shallow water equation with peaked solitons. Phys.
Rev. Lett. 71, 1661–1664. (doi:10.1103/PhysRevLett.71.1661)
Clark, S. & Gesztesy, F. 2002 Weyl–Titchmarsh M-function asymptotics, local uniqueness results,
trace formulas, and Borg-type theorems for Dirac operators. Trans. Am. Math. Soc. 354,
3475–3534. (doi:10.1090/S0002-9947-02-03025-8)
Constantin, A. 1997a A general-weighted Sturm–Liouville problem. Ann. Scuola Norm. Sup. Pisa
24, 767–782.
Constantin, A. 1997b On the spectral problem for the periodic Camassa–Holm equation. J. Math.
Anal. Appl. 210, 215–230. (doi:10.1006/jmaa.1997.5393)
Constantin, A. 1998a On the inverse spectral problem for the Camassa–Holm equation. J. Funct.
Anal. 155, 352–363. (doi:10.1006/jfan.1997.3231)
Constantin, A. 1998b Quasi-periodicity with respect to time of spatially periodic finite-gap
solutions of the Camassa–Holm equation. Bull. Sci. Math. 122, 487–494. (doi:10.1016/S00074497(99)80001-3)
Constantin, A. & McKean, P. 1999 A shallow water equation on the circle. Commun. Pure Appl.
Math. 52, 949–982. (doi:10.1002/(SICI )1097-0312(199908)52:8!949::AID-CPA3O3.0.CO;2-D)
Phil. Trans. R. Soc. A (2008)
Downloaded from http://rsta.royalsocietypublishing.org/ on July 28, 2017
1054
F. Gesztesy and H. Holden
Gesztesy, F. & Holden, H. 2002 Algebro-geometric solutions of the Camassa–Holm hierarchy. Rev.
Math. Iberoamericana 19, 73–142.
Gesztesy, F. & Holden, H. 2003 Soliton equations and their algebro-geometric solutions. Vol. I:
(1C1)-dimensional continuous models. Cambridge studies in advanced mathematics, vol. 79.
Cambridge, UK: Cambridge University Press.
Gesztesy, F. & Tsekanovskii, E. 2000 On matrix-valued Herglotz functions. Math. Nachr. 218,
61–138. (doi:10.1002/1522-2616(200010)218:1!61::AID-MANA61O3.0.CO;2-D)
Hille, E. 1976 Ordinary differential equations in the complex domain. New York, NY: Wiley.
Hinton, D. B. & Shaw, K. 1981 On Titchmarsh–Weyl M(l)-functions for linear Hamiltonian
systems. J. Differ. Equations 40, 316–342. (doi:10.1016/0022-0396(81)90002-4)
Hinton, D. B. & Shaw, K. 1983 Hamiltonian systems of limit point or limit circle type with both
endpoints singular. J. Differ. Equations 50, 444–464. (doi:10.1016/0022-0396(83)90071-2)
Hinton, D. B. & Shaw, K. 1984 On boundary value problems for Hamiltonian systems with two
singular points. SIAM J. Math. Anal. 15, 272–286. (doi:10.1137/0515022)
Kogan, V. I. & Rofe-Beketov, F. S. 1974 On square-integrable solutions of symmetric systems of
differential equations of arbitrary order. Proc. Roy. Soc. Ed. A 74, 1–40.
Lesch, M. & Malamud, M. 2003 On the deficiency indices and self-adjointness of symmetric
Hamiltonian systems. J. Differ. Equations 189, 556–615. (doi:10.1016/S0022-0396(02)00099-2)
Levitan, B. M. 1987 Inverse Sturm–Liouville problems. Utrecht, The Netherlands: VNU Science Press.
Phil. Trans. R. Soc. A (2008)