A Toda Lattice in Dimension 2 and Nevanlinna Theory
It is shown how to study the 2-D Toda system for SU(n+1) using Nevanlinna theory of meromorphic functions and holomorphic curves. The results generalize recent results of Jost Wang and Chen-Li.
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irk-123456789-1064362016-09-29T03:02:13Z A Toda Lattice in Dimension 2 and Nevanlinna Theory Eremenko, A. It is shown how to study the 2-D Toda system for SU(n+1) using Nevanlinna theory of meromorphic functions and holomorphic curves. The results generalize recent results of Jost Wang and Chen-Li. 2007 Article A Toda Lattice in Dimension 2 and Nevanlinna Theory / A. Eremenko // Журнал математической физики, анализа, геометрии. — 2007. — Т. 3, № 1. — С. 39-46. — Бібліогр.: 16 назв. — англ. 1812-9471 http://dspace.nbuv.gov.ua/handle/123456789/106436 en Журнал математической физики, анализа, геометрии Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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It is shown how to study the 2-D Toda system for SU(n+1) using Nevanlinna theory of meromorphic functions and holomorphic curves. The results generalize recent results of Jost Wang and Chen-Li. |
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Eremenko, A. A Toda Lattice in Dimension 2 and Nevanlinna Theory Журнал математической физики, анализа, геометрии |
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Eremenko, A. |
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Eremenko, A. |
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A Toda Lattice in Dimension 2 and Nevanlinna Theory |
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A Toda Lattice in Dimension 2 and Nevanlinna Theory |
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A Toda Lattice in Dimension 2 and Nevanlinna Theory |
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A Toda Lattice in Dimension 2 and Nevanlinna Theory |
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A Toda Lattice in Dimension 2 and Nevanlinna Theory |
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toda lattice in dimension 2 and nevanlinna theory |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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A Toda Lattice in Dimension 2 and Nevanlinna Theory / A. Eremenko // Журнал математической физики, анализа, геометрии. — 2007. — Т. 3, № 1. — С. 39-46. — Бібліогр.: 16 назв. — англ. |
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Журнал математической физики, анализа, геометрии |
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AT eremenkoa atodalatticeindimension2andnevanlinnatheory AT eremenkoa todalatticeindimension2andnevanlinnatheory |
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2025-07-07T18:29:16Z |
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2025-07-07T18:29:16Z |
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1837013880963858432 |
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Journal of Mathematical Physics, Analysis, Geometry
2007, vol. 3, No. 1, pp. 39�46
A Toda Lattice in Dimension 2 and Nevanlinna Theory
A. Eremenko
Purdue University
150 N University Str., West Lafayette IN 47907-2067, USA
E-mail:eremenko@math.purdue.edu
Received July 27, 2006
It is shown how to study the 2-D Toda system for SU(n+1) using Nevan-
linna theory of meromorphic functions and holomorphic curves. The results
generalize recent results of Jost�Wang and Chen�Li.
Key words: 2D Toda lattice, meromorphic functions, Nevanlinna theory.
Mathematics Subject Classi�cation 2000: 35Q58.
We consider the 2-dimensional open Toda system for SU(n+ 1):
�
1
2
�uj = �euj�1 + 2euj � euj+1 ; 1 � j � n; u0 = un+1 = 0; (1)
where uj are smooth functions in the complex plane C, and n is a positive integer.
This system was recently studied by several authors (see the reference list in
[8]). When n = 1 we obtain the Liouville equation
��u1 = 4eu1 : (2)
J. Jost and G. Wang [8] classi�ed all solutions of (1) satisfying the condition
Z
C
euj <1; 1 � j � n: (3)
In this paper, such classi�cation will be given for a larger class of solutions, namely
those that satisfy
B(r) =
Z
jzj�r
eu1 = O(rK) for some K > 0: (4)
Supported by NSF grants DMS-0100512 and DMS-0244547.
c
A. Eremenko, 2007
A. Eremenko
This will also give another proof of the result of Jost and Wang. Following their
suggestion in [8, p. 278], we apply the Nevanlinna theory of holomorphic curves.
It is well known, and goes back to Liouville [9], that the general solution of
(2) has the form
u = u1 = log
2jf 0j2
(1 + jf j2)2
; (5)
where f : C ! P
1 is a meromorphic function with no critical points (that is
f 0(z) 6= 0; z 2 C and f has no multiple poles).
The general solution of the Toda system can be similarly described in terms
of holomorphic curves C ! P
n. System (1) appears for the �rst time in the
works on the value distribution of holomorphic curves [1, 14, 15]. In [2], Calabi
proved that every solution of (1) comes from a holomorphic curve; this result will
be stated precisely below.
Strictly speaking, the present paper contains no new results. Its purpose is to
translate some old results of value distribution theory to the language of PDE,
and thus to bring these results to the attention of a wider audience. We begin
with the simpler case of the Liouville equation.
1. Liouville Equation
We recall that the order of a meromorphic function f can be de�ned by the for-
mula
� = lim sup
r!1
logA(r; f)
log r
; (6)
where
A(r; f) =
1
�
Z
jzj�r
jf 0j2
(1 + jf j2)2
:
If u is related to f by (5) and satis�es (4), then
� � K: (7)
So (5) establishes a bijective correspondence between solutions of the Liouville
equation satisfying (4) and meromorphic functions of �nite order without cri-
tical points. This class of meromorphic functions was completely described by
F. Nevanlinna in [10]. It coincides with the set of all solutions of di�erential
equations
f 000
f 0
�
3
2
�
f 00
f 0
�2
= P; (8)
where P is an arbitrary polynomial.
40 Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 1
A Toda Lattice in Dimension 2 and Nevanlinna Theory
The expression in the left hand side of this equation is called the Schwarzian
derivative. It is well known (after Schwarz) that (8) is equivalent to the linear
di�erential equation
w00 +
1
2
Pw = 0: (9)
More precisely, solutions f of (8) are ratios w1=w2 of two linearly independent
solutions of (9), and every such ratio is a solution of (8). This can be veri�ed by
a direct computation.
The order � of the meromorphic function f is the same as the orders of all
nonzero solutions of (9), and it is given by the formula
� = (degP )=2 + 1; if P 6= 0:
If P = 0, then all solutions of (8) are fractional-linear functions.
So we have the following recipe for writing all solutions of (2) satisfying (4).
Theorem 1. Let u be a solution of the Liouville equation (2) that satis�es (4).
Then
u = log
2jf 0j2
(1 + jf j2)2
;
where f = w1=w2, and the wi are two linearly independent solutions of
w00 + (1=2)Pw = 0, where P is a polynomial of degree at most 2(K � 1).
In the opposite direction, if P is a polynomial of degree d � 0 and u and f are
de�ned as above, then u is a solution of (2) satisfying (4) with K = d=2 + 1.
The well-known asymptotic behavior of solutions of the equation (9) permits
to derive precise asymptotic formulas for solutions u of (2). We only consider
some special cases. First we infer that the order of growth � of the function B in
(4) can only assume a discrete sequence of values: 0; 1; 3=2; 2; 5=2 : : : :
If � = 0, then P = 0 and f is fractional-linear. This case was studied by
W.X. Chen and G. Li [3]. They noticed a curious fact that spherical derivative of
any fractional-linear function always has rotational symmetry about some center.
If � = 1, then P = const, and f is a fractional-linear function of eaz , where
a 6= 0 is a complex number. So we obtain a family of elementary solutions of (1).
If � = 3=2, then degP = 1, and u can be expressed in terms of the Airy
function.
In the general case, we have the following asymptotic behavior. Let P (z) =
azd + : : :, where d = degP . The Stokes lines of the equation (9) are de�ned by
Im(
p
azd=2+1) = 0:
They break the complex plane into d+2 sectors of opening 2�=(d+2). In each of
these sectors, u(z) tends to�1, while on the Stokes lines it grows like (d=2) log jzj.
Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 1 41
A. Eremenko
2. Toda System
In this paper, by a holomorphic curve we mean a holomorphic map
f : C! P
n;
where Pn is the complex projective space of dimension n. In homogeneous coor-
dinates, a holomorphic curve is described by a vector-function
F : C! C
n+1; F = (f0; f1; : : : ; fn)
such that fj are entire functions without common zeros. These entire functions
are de�ned up to a common multiple which is an entire function without zeros
in the plane. To each holomorphic curve correspond derived curves de�ned in
homogeneous coordinates as
Fk : C! C
nk ; nk =
�
n+ 1
k + 1
�
; 1 � k � n;
Fk = f ^ f 0 ^ : : : ^ f (k):
It will be convenient to set
F0 = F; and F�1 = 1: (10)
Notice that Fn : C ! C is an entire function; it is equal to the Wronskian
determinant of f0; : : : ; fn.
f is linearly nondegenerate, that is its image is not contained in any hyper-
plane, if and only if Fk 6= 0 for 1 � k � n.
The following relations are sometimes called �local Pl�ucker formulas� (see [1,
8, 13, 15]).
�log jFkj2 = 4
jFk�1j2jFk+1j2
jFkj4
; 0 � k � n� 1: (11)
Here j j is the Euclidean norm. As Fn is an entire function, we have
�log jFnj2 = 0: (12)
These relations (11) and (12) hold in the classical sense, that is outside the zeros
of Fk.
For a given holomorphic curve f : C! P
n we set for k = 1; : : : ; n:
uk = log jFk�2j2 � 2 log jFk�1j2 + log jFkj2 + log 2: (13)
Notice that the uk do not change if all fk are multiplied by a common factor, so
(u1; : : : ; uk) depend only on the holomorphic curve f rather than the choice of its
homogeneous representation.
42 Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 1
A Toda Lattice in Dimension 2 and Nevanlinna Theory
It is veri�ed by simple computation using (11), (12) and our conventions (10)
that these functions uk satisfy the Toda system (1).
E. Calabi [2] proved the converse statement: every solution of the system (1)
in the plane arises from a holomorphic curve f : C! P
n, satisfying
Fk(z) 6= 0; z 2 C; 1 � k � n; (14)
via (13).
Conditions (14) are equivalent to
Fn(z) 6= 0; z 2 C: (15)
This can be proved by a computation in local coordinates, see, for example
[13]. The following arguments are taken from Petrenko's book [11] (see also [5]).
Holomorphic curves that satisfy (15) are called unrami�ed.
Proposition 1. The class of unrami�ed holomorphic curves f : C ! P
n
coincides with the class of curves that have homogeneous representations of the
form F = (f0; : : : ; fn), where f1; : : : ; fn is a basis of solutions of a di�erential
equation
w(n+1) + Pnw
(n) + : : :+ P0w = 0; (16)
where Pj are arbitrary entire functions.
P r o o f. If f is unrami�ed we can choose a homogeneous representation
where Fn = W (f0; : : : ; fn) � 1: Then
�����������
w(n+1) w(n) : : : w
f
(n+1)
0 f
(n)
0 : : : f0
f
(n+1)
1 f
(n)
1 : : : f1
: : : : : : : : : : : :
f
(n+1)
n f
(n)
n : : : fn
�����������
= 0
is the required equation.
In the opposite direction, suppose that (f0; : : : ; fn) is a fundamental system
of solutions of (16). If the Wronskian W = W (f0; : : : ; fn) has a zero, W (z0) = 0,
then the columns of the matrix of W (z0) are linearly dependent, and we obtain
a nontrivial linear combination g of f0; : : : ; fn such that g(k)(z0) = 0 for k =
0; : : : ; n. As g is a solution of the same equation (16), we conclude from the
uniqueness theorem that g � 0, but then W � 0 which is impossible because
f0; : : : ; fn are linearly independent. This proves the Prop. 1.
So we obtained a parametrization of all smooth solutions of the Toda system
in the plane: every solution has the form (13) where Fk are the derived curves
Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 1 43
A. Eremenko
of a curve f whose coordinates form a basis of solutions of a linear di�erential
equation with entire coe�cients.
Now we turn our attention to condition (4). We recall that the order of
a holomorphic curve can be de�ned by equation (6), where
A(r; f) =
1
2�
Z
jzj�r
�log jF0j =
1
�
Z
jzj�r
jF1j2
jF0j4
:
The geometric meaning of A(r; f) is the normalized area of the disc jzj � r with
respect to the pull-back of the Fubini-Study area form via f .
Let f be a holomorphic curve associated with a solution of the Toda system via
(13). Then B(r) = 2�A(r; f), and condition (4) implies that f is of �nite order,
� � K. It is known that a holomorphic curve f of �nite order has a homogeneous
representation F = (f0; : : : ; fn) where the fj are entire functions of �nite order,
in fact maximum of their orders equals the order of f (see, for example, [4]).
Let us �x such a representation (f0; : : : ; fn). By Proposition 1, these entire
functions constitute a basis of solutions of a di�erential equation
w(n+1) + Pnw
(n) + : : :+ P0w = 0; (17)
Now we use a theorem of M. Frei [6]:
Proposition 2. If a linear di�erential equation of the form (17) has n +
1 linearly independent solutions of �nite order, then all Pj, j = 0; : : : ; n, are
polynomials.
The converse is also true: all solutions of the equation (17) with polynomial
coe�cients have �nite order.
There is a simple algorithm which permits to �nd orders of solutions of (17).
We follow [16].
Suppose that Pj 6= 0 for at least one j 2 [0; n]. Plot in the plane the points
with coordinates (k;deg Pk � k), for 0 � k � n+ 1, degPn+1 = 0. Let C be the
convex hull of these points, and � the part of the boundary @C, which is visible
from above. This polygonal line � is called the Newton diagram of the equation
(17). Then the orders of solutions are among the negative slopes of segments
of �. Let �� be the negative slope of a segment of �, which has the largest
absolute value among all negative slopes of segments of �. According to a result
of K. P�oschl [12], a solution of exact order � always exists. Then � is the order
of the holomorphic curve de�ned by (17), and we have
� = max
0�k�n
n+ 1� k + degPk
n+ 1� k
: (18)
Now we state our �nal result.
44 Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 1
A Toda Lattice in Dimension 2 and Nevanlinna Theory
Theorem 2. Every solution u of the Toda system in C that satis�es (4) is
of the from (13), where Fk are the derived curves of a holomorphic curve whose
homogeneous coordinates form a basis of solutions of the equation (17) with poly-
nomial coe�cients. The degrees of these coe�cients, when substituted to (18),
give � � K, where K is the constant from (4).
Every basis of solutions of any equation (17) de�nes a solution (uk) of the
Toda system by the above rule. This solution satis�es (4) with K = �, where � is
de�ned by (18).
Some special cases are:
1. If P0 = : : : = Pn = 0 in (17), then a basis of solutions is (1; z; : : : ; zn) which
is the rational normal curve. Thus we recover the main result of the paper [8].
2. Suppose that all Pj are constants. Then solutions of (17) are generalized
exponential sums, and we obtain a class of explicit solutions of (1).
The asymptotic behavior of solutions of (1) is more complicated in the general
case n > 1 than for n = 1.
The Author thanks Misha Sodin who introduced him to the subject of Toda
systems, supplied with the references and made valuable comments.
References
[1] L. Ahlfors, The Theory of Meromorphic Curves. � Acta Soc. Sci. Fennicae. Nova
Ser. A. 3 (1941), No. 4, 31.
[2] E. Calabi, Metric Riemann Surfaces, in: Contributions to the Theory of Riemann
Surfaces. Princeton Univ. Press, Princeton, NJ. � Ann. Math. Stud. (1953), No. 30,
77�85.
[3] W.X. Chen and G. Li, Classi�cation of Solutions of Some Nonlinear Elliptic Equa-
tions. � Duke Math. J. 63 (1991). No. 3, 615�622.
[4] A. Eremenko, Extremal Holomorphic Curves for Defect Relations. � J. d'Analyse
math. 74 (1998), 307�323.
[5] A. Eremenko, Entire and Meromorphic Solutions of Ordinary Di�erential Equations.
� In: Complex Analysis. I. Encyclop�dia of Math. Sci. 85. Springer, New York,
1997.
[6] M. Frei, Sur l'Ordre des Solutions Enti�eres d'une �Equations Di��erentielle Lin�eaire.
� C.R. Acad. Sci. 236 (1953), 38�40.
[7] A. Goldberg, Some Questions of Value Distribution Theory. Gos. Izdat. Fiz.-Mat.
Lit., Moscow, 1960. (Russian)
[8] J. Jost and G. Wang, Classi�cation of Solutions of a Toda System in R2. � Int.
Math. Res. Notices (2002) No. 6, 277�290.
Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 1 45
A. Eremenko
[9] J. Liouville, Sur l'�Equation aux Di��erences Partielles
d
2
dudv
log� �
�
2a2
= 0. �
J. Math. Pures Appl. 18 (1853), 71�72.
[10] F. Nevanlinna, �Uber eine Klasse Meromorpher Funktionen, 7-e Congr�es des Math-
maticiens Scandinaves. Oslo, 1929. A.W. Broggers, Oslo, 1930.
[11] V. Petrenko, Entire Curves. Vyshcha Shkola, Kharkov, 1984. (Russian)
[12] K. P�oschl, �Uber Anwachsen und Nullstellenverteilung der ganzen transzendenten
L�osungen linearer Di�erentialgleichungen. I. II. � J. Reine Angew. Math. 199,
(1958), 121�138; 200 (1958), 129�138.
[13] B. Shabat, Distribution of Values of Holomorphic Mappings. AMS Transl. 61.
Providence, RI, 1985.
[14] H. Weyl and J. Weyl, Meromorphic Curves. � Ann. Math. (2) 39 (1938), 516�538.
[15] H. Weyl, Meromorphic Functions and Analytic Curves. Ser. Ann. Math. Stud.
Princeton, 1943.
[16] H. Wittich, Neuere Untersuchungen �uber eindeutige analytische Funktionen.
Springer, Berlin, 1955.
46 Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 1
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