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32 Painlevé TranscendentsProperties

§32.8 Rational Solutions

Contents

§32.8(i) Introduction

PIIPVI possess hierarchies of rational solutions for special values of the parameters which are generated from “seed solutions” using the Bäcklund transformations and often can be expressed in the form of determinants. See Airault (1979).

§32.8(ii) Second Painlevé Equation

Rational solutions of PII exist for α=n() and are generated using the seed solution w(z;0)=0 and the Bäcklund transformations (32.7.1) and (32.7.2). The first four are

32.8.1 w(z;1)=-1/z,
32.8.2 w(z;2)=1z-3z2z3+4,
32.8.3 w(z;3)=3z2z3+4-6z2(z3+10)z6+20z3-80,
32.8.4 w(z;4)=-1z+6z2(z3+10)z6+20z3-80-9z5(z3+40)z9+60z6+11200.

More generally,

32.8.5 w(z;n)=z(ln(Qn-1(z)Qn(z))),

where the Qn(z) are monic polynomials (coefficient of highest power of z is 1) satisfying

32.8.6 Qn+1(z)Qn-1(z)=zQn2(z)+4(Qn(z))2-4Qn(z)Qn′′(z),

with Q0(z)=1, Q1(z)=z. Thus

32.8.7 Q2(z) =z3+4,
Q3(z) =z6+20z3-80,
Q4(z) =z10+60z7+11200z,
Q5(z) =z15+140z12+2800z9+78400z6-3 13600z3-62 72000,
Q6(z) =z21+280z18+18480z15+6 27200z12-172 48000z9+14488 32000z6+1 93177 60000z3-3 86355 20000.

Next, let pm(z) be the polynomials defined by pm(z)=0 for m<0, and

32.8.8 m=0pm(z)λm=exp(zλ-43λ3).

Then for n2

32.8.9 w(z;n)=z(ln(τn-1(z)τn(z))),

where τn(z) is the n×n determinant

32.8.10 τn(z)=|p1(z)p3(z)p2n-1(z)p1(z)p3(z)p2n-1(z)p1(n-1)(z)p3(n-1)(z)p2n-1(n-1)(z)|.

For plots of the zeros of Qn(z) see Clarkson and Mansfield (2003).

§32.8(iii) Third Painlevé Equation

Special rational solutions of PIII are

32.8.11 w(z;μ,-μκ2,λ,-λκ4)=κ,
32.8.12 w(z;0,-μ,0,μκ)=κz,
32.8.13 w(z;2κ+3,-2κ+1,1,-1)=z+κz+κ+1,

with κ, λ, and μ arbitrary constants.

In the general case assume γδ0, so that as in §32.2(ii) we may set γ=1 and δ=-1. Then PIII has rational solutions iff

32.8.14 α±β=4n,

with n. These solutions have the form

32.8.15 w(z)=Pm(z)/Qm(z),

where Pm(z) and Qm(z) are polynomials of degree m, with no common zeros.

For examples and plots see Milne et al. (1997); also Clarkson (2003a). For determinantal representations see Kajiwara and Masuda (1999).

§32.8(iv) Fourth Painlevé Equation

Special rational solutions of PIV are

32.8.16 w1(z;±2,-2) =±1/z,
32.8.17 w2(z;0,-2) =-2z,
32.8.18 w3(z;0,-29) =-23z.

There are also three families of solutions of PIV of the form

32.8.19 w1(z;α1,β1)=P1,n-1(z)/Q1,n(z),
32.8.20 w2(z;α2,β2)=-2z+(P2,n-1(z)/Q2,n(z)),
32.8.21 w3(z;α3,β3)=-23z+(P3,n-1(z)/Q3,n(z)),

where Pj,n-1(z) and Qj,n(z) are polynomials of degrees n-1 and n, respectively, with no common zeros.

In general, PIV has rational solutions iff either

32.8.22 α =m,
β =-2(1+2n-m)2,

or

32.8.23 α =m,
β =-2(13+2n-m)2,

with m,n. The rational solutions when the parameters satisfy (32.8.22) are special cases of §32.10(iv).

For examples and plots see Bassom et al. (1995); also Clarkson (2003b). For determinantal representations see Kajiwara and Ohta (1998) and Noumi and Yamada (1999).

§32.8(v) Fifth Painlevé Equation

Special rational solutions of PV are

32.8.24 w(z;12,-12μ2,κ(2-μ),-12κ2)=κz+μ,
32.8.25 w(z;12,κ2μ,2κμ,μ)=κ/(z+κ),
32.8.26 w(z;18,-18,-κμ,μ)=(κ+z)/(κ-z),

with κ and μ arbitrary constants.

In the general case assume δ0, so that as in §32.2(ii) we may set δ=-12. Then PV has a rational solution iff one of the following holds with m,n and ε=±1:

  1. (a)

    α=12(m+εγ)2 and β=-12n2, where n>0, m+n is odd, and α0 when |m|<n.

  2. (b)

    α=12n2 and β=-12(m+εγ)2, where n>0, m+n is odd, and β0 when |m|<n.

  3. (c)

    α=12a2, β=-12(a+n)2, and γ=m, with m+n even.

  4. (d)

    α=12(b+n)2, β=-12b2, and γ=m, with m+n even.

  5. (e)

    α=18(2m+1)2, β=-18(2n+1)2, and γ.

These rational solutions have the form

32.8.27 w(z)=λz+μ+(Pn-1(z)/Qn(z)),

where λ, μ are constants, and Pn-1(z), Qn(z) are polynomials of degrees n-1 and n, respectively, with no common zeros. Cases (a) and (b) are special cases of §32.10(v).

For examples and plots see Clarkson (2005). For determinantal representations see Masuda et al. (2002). For the case δ=0 see Airault (1979) and Lukaševič (1968).

§32.8(vi) Sixth Painlevé Equation

Special rational solutions of PVI are

32.8.28 w(z;μ,-μκ2,12,12-μ(κ-1)2)=κz,
32.8.29 w(z;0,0,2,0) =κz2,
32.8.30 w(z;0,0,12,-32) =κ/z,
32.8.31 w(z;0,0,2,-4) =κ/z2,
32.8.32 w(z;12(κ+μ)2,-12,12(μ-1)2,12κ(2-κ))=zκ+μz,

with κ and μ arbitrary constants.

In the general case, PVI has rational solutions if

32.8.33 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, and at least one of a, b, c or d is an integer. These are special cases of §32.10(vi).