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Movable algebraic singularities of second-order ordinary differential equations
Any nonlinear equation of the form y ″ = ∑ n = 0 N a n ( z ) y n has a solution with leading behavior proportional to ( z − z 0 ) − 2 / ( N − 1 ) about a point z 0 , where the coefficients a n are analytic at z 0 and a N ( z 0 ) ≠ 0 . Equations are considered for which each possible leading term of...
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| Published in: | Journal of mathematical physics 2009-02, Vol.50 (2), p.023509-023509-18 |
|---|---|
| Main Authors: | , |
| Format: | Article |
| Language: | English |
| Subjects: | |
| Citations: | Items that this one cites Items that cite this one |
| Online Access: | Get full text |
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| Summary: | Any nonlinear equation of the form
y
″
=
∑
n
=
0
N
a
n
(
z
)
y
n
has a solution with leading behavior proportional to
(
z
−
z
0
)
−
2
/
(
N
−
1
)
about a point
z
0
, where the coefficients
a
n
are analytic at
z
0
and
a
N
(
z
0
)
≠
0
. Equations are considered for which each possible leading term of this form extends to a Laurent series solution in fractional powers of
z
−
z
0
. For these equations we show that the only movable singularities that can be reached by analytic continuation along finite-length curves are of the algebraic type just described. This generalizes results of Shimomura [“On second order nonlinear differential equations with the quasi-Painlevé property II,” RIMS Kokyuroku
1424, 177 (2005)]. The possibility that these algebraic singularities could accumulate along infinitely long paths ending at a finite point is considered. Smith [“On the singularities in the complex plane of the solutions of
y
″
+
y
′
f
(
y
)
+
g
(
y
)
=
P
(
x
)
,” Proc. Lond. Math. Soc.
3, 498 (1953)] showed that such singularities do occur in solutions of a simple equation outside this class. |
|---|---|
| ISSN: | 0022-2488 1089-7658 |
| DOI: | 10.1063/1.3068414 |