# §32.11 Asymptotic Approximations for Real Variables

## §32.11(i) First Painlevé Equation

There are solutions of (32.2.1) such that

where

and and are constants.

There are also solutions of (32.2.1) such that

32.11.3.

Next, for given initial conditions and , with real, has at least one pole on the real axis. There are two special values of , and , with the properties , , and such that:

1. (a)

If , then for , where is the first pole on the negative real axis.

2. (b)

If , then oscillates about, and is asymptotic to, as .

3. (c)

If , then changes sign once, from positive to negative, as passes from to 0.

For illustration see Figures 32.3.1 to 32.3.4, and for further information see Joshi and Kitaev (2005), Joshi and Kruskal (1992), Kapaev (1988), Kapaev and Kitaev (1993), and Kitaev (1994).

## §32.11(ii) Second Painlevé Equation

Consider the special case of  with :

32.11.4

with boundary condition

32.11.5.

Any nontrivial real solution of (32.11.4) that satisfies (32.11.5) is asymptotic to , for some nonzero real , where denotes the Airy function (§9.2). Conversely, for any nonzero real , there is a unique solution of (32.11.4) that is asymptotic to as .

If , then exists for all sufficiently large as , and

where

and , are real constants. Connection formulas for and are given by

where is the gamma function (§5.2(i)), and the branch of the function is immaterial.

If , then

32.11.10.

If , then has a pole at a finite point , dependent on , and

32.11.11.

For illustration see Figures 32.3.5 and 32.3.6, and for further information see Ablowitz and Clarkson (1991), Bassom et al. (1998), Clarkson and McLeod (1988), Deift and Zhou (1995), Segur and Ablowitz (1981), and Suleĭmanov (1987). For numerical studies see Miles (1978, 1980) and Rosales (1978).

## §32.11(iii) Modified Second Painlevé Equation

Replacement of by in (32.11.4) gives

32.11.12

Any nontrivial real solution of (32.11.12) satisfies

where

with and arbitrary real constants.

In the case when

with , we have

where is a nonzero real constant. The connection formulas for are

## §32.11(iv) Third Painlevé Equation

For , with and ,

where is an arbitrary constant such that , and

32.11.26,

where and are arbitrary constants such that and . The connection formulas relating (32.11.25) and (32.11.26) are

32.11.27

See also Abdullaev (1985), Novokshënov (1985), Its and Novokshënov (1986), Kitaev (1987), Bobenko (1991), Bobenko and Its (1995), Tracy and Widom (1997), and Kitaev and Vartanian (2004).

## §32.11(v) Fourth Painlevé Equation

Consider  with and , that is,

32.11.29

and with boundary condition

32.11.30.

Any nontrivial solution of (32.11.29) that satisfies (32.11.30) is asymptotic to as , where is a constant. Conversely, for any there is a unique solution of (32.11.29) that is asymptotic to as . Here denotes the parabolic cylinder function (§12.2).

Now suppose . If , where

32.11.31

then has no poles on the real axis. Furthermore, if , then

Alternatively, if is not zero or a positive integer, then

where

and and are real constants. Connection formulas for and are given by

where

and the branch of the function is immaterial.

Next if , then

32.11.38,

and has no poles on the real axis.

Lastly if , then has a simple pole on the real axis, whose location is dependent on .

For illustration see Figures 32.3.732.3.10. In terms of the parameter that is used in these figures .