Movable Singularities of Solutions of Ordinary Differential

S.9.1. Movable Singularities of Solutions of Ordinary Differential

Equations

1 ◦ . The connection between the structure of differential equations and singularities of their solutions was established more than a hundred years ago. The singularities of solutions of linear ordinary differential equations are completely determined by singularities of the coefficients of the equations. Since the position of such singularities does not depend on integration constants, they are called fixed singularities. In the case of nonlinear equations, their solutions may also possess movable singularities, whose position depends on the initial conditions (integration constants).

Below, we give simplest examples of first-order ordinary differential equations and their solutions with movable singularities.

Equation

Type of singularity of the solution u 2 ′

Solution

z =− u

u = 1/(z − z 0 )

movable pole

u ′ z =1

/u

u=2 z−z 0 algebraic branch point

u = ln(z − z 0 )

logarithmic branch point

essentially singular point Algebraic branch points, logarithmic branch points, and essentially singular points are called “critical

u ′ z =− u ln 2 u

u = exp[1/(z − z 0 )]

singular points.”

2 ◦ . In 1884, L. L. Fuchs established the following fact: the first-order nonlinear differential equation

u ′ z = R(z, u),

where the function R is rational in the second argument and analytic with respect to the first, admits solutions without movable critical points (other than movable poles) only if it coincides with the

general Riccati equation u ′ z = A 0 ( z) + A 1 ( z)u + A 2 ( z)u 2 .

* Section S.9 was written by V. G. Baydulov and V. A. Gorodtsov.

u ′′ zz = R(z, u, u ′ z ),

where R = R(z, u, w) is a rational function of u and w and is analytic in z, were classified by P. Painlev´e (1900) and B. Gambier (1910). These authors showed that all equations of this form whose solutions have no movable critical points (other than fixed singular points and movable poles) can be divided into 50 classes. The equations of 44 out of these classes can be integrated by quadrature or their order can be reduced. The remaining 6 classes, in canonical form, are irreducible and are called Painlev´e equations (their solutions are called Painlev´e transcendents).

4 ◦ . The first Painlev´e equation has the form

u ′′ zz =6 u 2 + z.

The equation has a movable pole z 0 ; in its neighborhood, the solutions can be represented by the series

5 C, a = 0, a 6 = 1 300 z 2 0 , where z 0 and

a 2 =− 1 10 z 0 , a 3 =− 1 6 , a 4 =

C are arbitrary constants; the coefficients a n ( n ≥ 7) are uniquely determined by z 0 and C. The second Painlev´e equation is expressed as

u 3 ′′

zz =2 u + zu + a.

In a neighborhood of the movable pole z 0 , its solutions admit the following expansions:

4 C, b = 1 72 z 0 ( m + 3α),

b 5 = 1 3024 2 (27 + 81 α −2 3 z 0 ) m + 108α − 216Cz 0 ,

C are arbitrary constants; and the coefficients b n ( n ≥ 6) are uniquely determined by z 0 and C. More detailed information about the Painlev´e equations can be found in the literature cited at the end of this subsection. It should be observed that the solution of the fourth Painlev ´e equation has a movable pole, while the solutions of the third, the fifth, and the sixth Painlev ´e equations have fixed logarithmic branch points.

where m= ■ 1; z 0 and

Remark. In 1888, S. V. Kowalevskaya succeeded in integrating the equations of motion of a rigid body having a fixed point and subject to gravity, in a case previously unknown. She examined solutions of a system of six first-order nonlinear

ordinary differential equations. Solutions were sought in the form of series expansions in powers of each unknown quantity with movable poles,

u = (z − z 0 ) − n a 0 + a 1 ( z−z 0 )+ ··· .

The generality of the solution was ensured by a suitable (corresponding to the order of the system) number of arbitrary

coefficients in the expansions and the free parameter z 0 .

It should be mentioned that the studies of S. V. Kowalevskaya preceded the works of Painlev ´e on the classification of second-order ordinary differential equations, where similar expansions were used.

References for Subsection S.9.1: V. V. Golubev (1950), G. M. Murphy (1960), A. R. Its and V. Yu. Novokshenov (1986), M. Tabor (1989), V. I. Gromak and N. A. Lukashevich (1990), A. R. Chowdhury (2000), V. I. Gromak (2002), A. D. Polyanin and V. F. Zaitsev (2003).