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32 Painlevé TranscendentsProperties

§32.10 Special Function Solutions

Contents

§32.10(i) Introduction

For certain combinations of the parameters, PIIPVI have particular solutions expressible in terms of the solution of a Riccati differential equation, which can be solved in terms of special functions defined in other chapters. All solutions of PIIPVI that are expressible in terms of special functions satisfy a first-order equation of the form

32.10.1 (w)n+j=0n-1Fj(w,z)(w)j=0,

where Fj(w,z) is polynomial in w with coefficients that are rational functions of z.

§32.10(ii) Second Painlevé Equation

PII has solutions expressible in terms of Airy functions (§9.2) iff

32.10.2 α=n+12,

with n. For example, if α=12ε, with ε=±1, then the Riccati equation is

32.10.3 εw=w2+12z,

with solution

32.10.4 w(z;12ε)=-εϕ(z)/ϕ(z),

where

32.10.5 ϕ(z)=C1Ai(-2-1/3z)+C2Bi(-2-1/3z),

with C1, C2 arbitrary constants.

Solutions for other values of α are derived from w(z;±12) by application of the Bäcklund transformations (32.7.1) and (32.7.2). For example,

32.10.6 w(z;32)=Φ-12Φ2+z,
32.10.7 w(z;52)=12Φ2+z+2zΦ2+Φ+z24Φ3+2zΦ-1,

where Φ=ϕ(z)/ϕ(z), with ϕ(z) given by (32.10.5).

More generally, if n=1,2,3,, then

32.10.8 w(z;n+12)=z(ln(τn(z)τn+1(z))),

where τn(z) is the n×n determinant

32.10.9 τn(z)=|ϕ(z)ϕ(z)ϕ(n-1)(z)ϕ(z)ϕ′′(z)ϕ(n)(z)ϕ(n-1)(z)ϕ(n)(z)ϕ(2n-2)(z)|,

and

32.10.10 w(z;-n-12)=-w(z;n+12).

§32.10(iii) Third Painlevé Equation

If γδ0, then as in §32.2(ii) we may set γ=1 and δ=-1. PIII then has solutions expressible in terms of Bessel functions (§10.2) iff

32.10.11 ε1α+ε2β=4n+2,

with n, and ε1=±1, ε2=±1, independently. In the case ε1α+ε2β=2, the Riccati equation is

32.10.12 zw=ε1zw2+(αε1-1)w+ε2z.

If αε1, then (32.10.12) has the solution

32.10.13 w(z)=-ε1ϕ(z)/ϕ(z),

where

32.10.14 ϕ(z)=zν(C1Jν(ζ)+C2Yν(ζ)),

with ζ=ε1ε2z, ν=12αε1, and C1, C2 arbitrary constants.

For examples and plots see Milne et al. (1997). For determinantal representations see Forrester and Witte (2002) and Okamoto (1987c).

§32.10(iv) Fourth Painlevé Equation

PIV has solutions expressible in terms of parabolic cylinder functions (§12.2) iff either

32.10.15 β=-2(2n+1+εα)2,

or

32.10.16 β=-2n2,

with n and ε=±1. In the case when n=0 in (32.10.15), the Riccati equation is

32.10.17 w=ε(w2+2zw)-2(1+εα),

which has the solution

32.10.18 w(z)=-εϕ(z)/ϕ(z),

where

32.10.19 ϕ(z)=(C1U(a,2z)+C2V(a,2z))exp(12εz2),

with a=α+12ε, and C1, C2 arbitrary constants. When a+12 is zero or a negative integer the U parabolic cylinder functions reduce to Hermite polynomials (§18.3) times an exponential function; thus

32.10.20 w(z;-m,-2(m-1)2)=-Hm-1(z)Hm-1(z),
m=1,2,3,,

and

32.10.21 w(z;-m,-2(m+1)2)=-2z+Hm(z)Hm(z),
m=0,1,2,.

If 1+εα=0, then (32.10.17) has solutions

32.10.22 w(z)={2exp(z2)π(C-erfc(z)),ε=1,2exp(-z2)π(C-erfc(z)),ε=-1,

where C is an arbitrary constant and erfc is the complementary error function (§7.2(i)).

For examples and plots see Bassom et al. (1995). For determinantal representations see Forrester and Witte (2001) and Okamoto (1986).

§32.10(v) Fifth Painlevé Equation

If δ0, then as in §32.2(ii) we may set δ=-12. PV then has solutions expressible in terms of Whittaker functions (§13.14(i)), iff

32.10.23 a+b+ε3γ=2n+1,

or

32.10.24 (a-n)(b-n)=0,

where n, a=ε12α, and b=ε2-2β, with εj=±1, j=1,2,3, independently. In the case when n=0 in (32.10.23), the Riccati equation is

32.10.25 zw=aw2+(b-a+ε3z)w-b.

If a0, then (32.10.25) has the solution

32.10.26 w(z)=-zϕ(z)/(aϕ(z)),

where

32.10.27 ϕ(z)=C1Mκ,μ(ζ)+C2Wκ,μ(ζ)ζ(a-b+1)/2exp(12ζ),

with ζ=ε3z, κ=12(a-b+1), μ=12(a+b), and C1, C2 arbitrary constants.

For determinantal representations see Forrester and Witte (2002), Masuda (2004), and Okamoto (1987b).

§32.10(vi) Sixth Painlevé Equation

PVI has solutions expressible in terms of hypergeometric functions (§15.2(i)) iff

32.10.28 a+b+c+d=2n+1,

where n, a=ε12α, b=ε2-2β, c=ε32γ, and d=ε41-2δ, with εj=±1, j=1,2,3,4, independently. If n=1, then the Riccati equation is

32.10.29 w=aw2z(z-1)+(b+c)z-a-cz(z-1)w-bz-1.

If a0, then (32.10.29) has the solution

32.10.30 w(z) =ζ-1aϕ(ζ)ϕζ,
ζ =11-z,

where

32.10.31 ϕ(ζ)=C1F(b,-a;b+c;ζ)+C2ζ-b+1-cF(-a-b-c+1,-c+1;2-b-c;ζ),

with C1, C2 arbitrary constants.

Next, let Λ=Λ(u,z) be the elliptic function (§§22.15(ii), 23.2(iii)) defined by

32.10.32 u=0Λtt(t-1)(t-z),

where the fundamental periods 2ϕ1 and 2ϕ2 are linearly independent functions satisfying the hypergeometric equation

32.10.33 z(1-z)2ϕz2+(1-2z)ϕz-14ϕ=0.

Then PVI, with α=β=γ=0 and δ=12, has the general solution

32.10.34 w(z;0,0,0,12)=Λ(C1ϕ1+C2ϕ2,z),

with C1, C2 arbitrary constants. The solution (32.10.34) is an essentially transcendental function of both constants of integration since PVI with α=β=γ=0 and δ=12 does not admit an algebraic first integral of the form P(z,w,w,C)=0, with C a constant.

For determinantal representations see Forrester and Witte (2004) and Masuda (2004).