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32 Painlevé TranscendentsProperties

§32.11 Asymptotic Approximations for Real Variables

Contents

§32.11(i) First Painlevé Equation

There are solutions of (32.2.1) such that

32.11.1 w(x)=-16|x|+d|x|-1/8sin(ϕ(x)-θ0)+o(|x|-1/8),
x-,

where

32.11.2 ϕ(x)=(24)1/4(45|x|5/4-58d2ln|x|),

and d and θ0 are constants.

There are also solutions of (32.2.1) such that

32.11.3 w(x)16|x|,
x-.

Next, for given initial conditions w(0)=0 and w(0)=k, with k real, w(x) has at least one pole on the real axis. There are two special values of k, k1 and k2, with the properties -0.45142 8<k1<-0.45142 7, 1.85185 3<k2<1.85185 5, and such that:

  1. (a)

    If k<k1, then w(x)>0 for x0<x<0, where x0 is the first pole on the negative real axis.

  2. (b)

    If k1<k<k2, then w(x) oscillates about, and is asymptotic to, -16|x| as x-.

  3. (c)

    If k2<k, then w(x) changes sign once, from positive to negative, as x passes from x0 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 PII with α=0:

32.11.4 w′′=2w3+xw,

with boundary condition

32.11.5 w(x)0,
x+.

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

If |k|<1, then wk(x) exists for all sufficiently large |x| as x-, and

32.11.6 wk(x)=d|x|-1/4sin(ϕ(x)-θ0)+o(|x|-1/4),

where

32.11.7 ϕ(x)=23|x|3/2-34d2ln|x|,

and d (0), θ0 are real constants. Connection formulas for d and θ0 are given by

32.11.8 d2=-π-1ln(1-k2),
32.11.9 θ0=32d2ln2+phΓ(1-12id2)+14π(1-2sign(k)),

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

If |k|=1, then

32.11.10 wk(x)sign(k)12|x|,
x-.

If |k|>1, then wk(x) has a pole at a finite point x=c0, dependent on k, and

32.11.11 wk(x)sign(k)(x-c0)-1,
xc0+.

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 w by iw in (32.11.4) gives

32.11.12 w′′=-2w3+xw.

Any nontrivial real solution of (32.11.12) satisfies

32.11.13 w(x)=d|x|-1/4sin(ϕ(x)-χ)+O(|x|-5/4ln|x|),
x-,

where

32.11.14 ϕ(x)=23|x|3/2+34d2ln|x|,

with d (0) and χ arbitrary real constants.

In the case when

32.11.15 χ+32d2ln2-14π-phΓ(12id2)=nπ,

with n, we have

32.11.16 w(x)kAi(x),
x+,

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

32.11.17 d2=π-1ln(1+k2),
sign(k)=(-1)n.

In the generic case

32.11.18 χ+32d2ln2-14π-phΓ(12id2)nπ,

we have

32.11.19 w(x)=σ12x+σρ(2x)-1/4cos(ψ(x)+θ)+O(x-1),
x+,

where σ, ρ (>0), and θ are real constants, and

32.11.20 ψ(x)=232x3/2-32ρ2lnx.

The connection formulas for σ, ρ, and θ are

32.11.21 σ=-sign(s),
32.11.22 ρ2=π-1ln((1+|s|2)/|2s|),
32.11.23 θ=-34π-72ρ2ln2+ph(1+s2)+phΓ(iρ2),

where

32.11.24 s=(exp(πd2)-1)1/2exp(i(32d2ln2-14π+χ-phΓ(12id2))).

§32.11(iv) Third Painlevé Equation

For PIII, with α=-β=2ν () and γ=-δ=1,

32.11.25 w(x)-1-λΓ(ν+12)2-2νx-ν-(1/2)e-2x,
x+,

where λ is an arbitrary constant such that -1/π<λ<1/π, and

32.11.26 w(x)Bxσ,
x0,

where B and σ are arbitrary constants such that B0 and |σ|<1. The connection formulas relating (32.11.25) and (32.11.26) are

32.11.27 σ=(2/π)arcsin(πλ),
32.11.28 B=2-2σΓ2(12(1-σ))Γ(12(1+σ)+ν)Γ2(12(1+σ))Γ(12(1-σ)+ν).

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 PIV with α=2ν+1 () and β=0, that is,

32.11.29 w′′=(w)22w+32w3+4xw2+2(x2-2ν-1)w,

and with boundary condition

32.11.30 w(x)0,
x+.

Any nontrivial solution of (32.11.29) that satisfies (32.11.30) is asymptotic to hU2(-ν-12,2x) as x+, where h (0) is a constant. Conversely, for any h (0) there is a unique solution wh(x) of (32.11.29) that is asymptotic to hU2(-ν-12,2x) as x+. Here U denotes the parabolic cylinder function (§12.2).

Now suppose x-. If 0h<h*, where

32.11.31 h*=1/(π1/2Γ(ν+1)),

then wh(x) has no poles on the real axis. Furthermore, if ν=n=0,1,2,, then

32.11.32 wh(x)h2nx2nexp(-x2),
x-.

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

32.11.33 wh(x)=-23x+43d3sin(ϕ(x)-θ0)+O(x-1),
x-,

where

32.11.34 ϕ(x)=133x2-43d23ln(2|x|),

and d (>0) and θ0 are real constants. Connection formulas for d and θ0 are given by

32.11.35 d2 =-143π-1ln(1-|μ|2),
32.11.36 θ0 =13d23ln3+23πν+712π+phμ+phΓ(-23i3d2),

where

32.11.37 μ=1+(2ihπ3/2exp(-iπν)/Γ(-ν)),

and the branch of the ph function is immaterial.

Next if h=h*, then

32.11.38 wh*(x)-2x,
x-,

and wh*(x) has no poles on the real axis.

Lastly if h>h*, then wh(x) has a simple pole on the real axis, whose location is dependent on h.

For illustration see Figures 32.3.732.3.10. In terms of the parameter k that is used in these figures h=23/2k2.

See also Wong and Zhang (2009a).