special solutions of the third and ﬁfth painleve equations ... · solutions satisfy linear...

Special solutions of the third and fifth Painleve equationsand vortex solutions of the complex Sine-Gordon equations

Peter A ClarksonSchool of Mathematics, Statistics and Actuarial Science

University of Kent, Canterbury, CT2 7NF, [email protected]

“Symmetry, Separation, Super-integrability and Special Functions”University of Minnesota, Minneapolis, September 2010

Outline1. Introduction

2. Rational and special function solutions of third Painleve equation

d2w

dz2=

1

w

(dw

dz

)2

− 1

z

dw

dz+ αw2 + βz + γw3 +

δ

w

3. Rational and special function solutions of the fifth Painleve equation

d2w

dz2=

(1

2w+

1

w − 1

)(dw

dz

)2

− 1

z

dw

dz+

(w − 1)2

z2

(αw +

β

w

)+γw

z+δw(w + 1)

w − 1

4. Vortex solutions of the complex Sine-Gordon I equation

∇2ψ +(∇ψ)2ψ

1− |ψ|2+ ψ(1− |ψ|2) = 0, ∇ψ = (ψx, ψy)

5. Vortex solutions of the complex Sine-Gordon II equation

∇2ψ +(∇ψ)2ψ

2− |ψ|2+ 1

2ψ(1− |ψ|2)(2− |ψ|2) = 0, ∇ψ = (ψx, ψy)

Symmetry, Separation, Super-integrability and Special Functions, University of Minnesota, Minneapolis, September 2010 2

Classical Special Functions• Airy, Bessel, Whittaker, Kummer, hypergeometric functions• Special solutions in terms of rational and elementary functions (for certain values of

the parameters)

• Solutions satisfy linear ordinary differential equations and linear difference equa-tions

• Solutions related by linear recurrence relations

Painleve Transcendents — Nonlinear Special Functions• Special solutions such as rational solutions, algebraic solutions and special function

solutions (for certain values of the parameters)

• Solutions satisfy nonlinear ordinary differential equations and nonlinear differenceequations

• Solutions related by nonlinear recurrence relations


Painleve Equations

d2w

dz2= 6w2 + z PI

d2w

dz2= 2w3 + zw + α PII

d2w

dz2=

1

w

(dw

dz

)2

− 1

z

dw

dz+αw2 + β

z+ γw3 +

δ

wPIII

d2w

dz2=

1

2w

(dw

dz

)2

+3

2w3 + 4zw2 + 2(z2 − α)w +

β

wPIV

d2w

dz2=

(1

2w+

1

w − 1

)(dw

dz

)2

− 1

z

dw

dz+

(w − 1)2

z2

(αw +

β

w

)PV

+γw

z+δw(w + 1)

w − 1d2w

dz2=

1

2

(1

w+

1

w − 1+

1

w − z

)(dw

dz

)2

−(

1

z+

1

z − 1+

1

w − z

)dw

dzPVI

+w(w − 1)(w − z)

z2(z − 1)2

{α +

βz

w2+γ(z − 1)

(w − 1)2+δz(z − 1)

(w − z)2

}where α, β, γ and δ are arbitrary constants.


History of the Painleve Equations• Derived by Painleve, Gambier and colleagues in the late 19th/early 20th centuries.

• Studied in Minsk, Belarus by Erugin, Lukashevich, Gromak et al. since 1950’s;much of their work is published in the journal Diff. Eqns., translation of Diff. Urav..

• Barouch, McCoy, Tracy & Wu [1973, 1976] showed that the correlation functionof the two-dimensional Ising model is expressible in terms of solutions of PIII.

• Ablowitz & Segur [1977] demonstrated a close connection between completely in-tegrable PDEs solvable by inverse scattering, the so-called soliton equations, suchas the Korteweg-de Vries equation and the nonlinear Schrodinger equation, andthe Painleve equations.

• Flaschka & Newell [1980] introduced the isomonodromy deformation method(inverse scattering for ODEs), which expresses the Painleve equation as the compat-ibility condition of two linear systems of equations and are studied using Riemann-Hilbert methods. Subsequent developments by Deift, Fokas, Its, Zhou, . . .• Algebraic and geometric studies of the Painleve equations by Okamoto in 1980’s.

Subsequent developments by Noumi, Umemura, Yamada, . . .• The Painleve equations are a chapter in the “Digital Library of Mathematical

Functions”, which is a rewrite/update of Abramowitz & Stegun’s “Handbook ofMathematical Functions” — see http://dlmf.nist.gov.


Some Properties of the Painleve Equations• PII–PVI have Backlund transformations which relate solutions of a given Painleve

equation to solutions of the same Painleve equation, though with different values ofthe parameters with associated Affine Weyl groups that act on the parameter space.• PII–PVI have rational and algebraic solutions for certain values of the parameters.• PII–PVI have special function solutions expressed in terms of the classical spe-

cial functions [PII: Airy Ai(z), Bi(z); PIII: Bessel Jν(z), Yν(z), Jν(z), Kν(z); PIV:parabolic cylinder Dν(z); PV: Whittaker Mκ,µ(z), Wκ,µ(z) [equivalently KummerM(a, b, z), U(a, b, z) or confluent hypergeometric 1F1(a; c; z)]; PVI: hypergeomet-ric 2F1(a, b; c; z)], for certain values of the parameters.• These rational, algebraic and special function solutions of PII–PVI can usually be

written in determinantal form, frequently as Wronskians.• PI–PVI can be written as a (non-autonomous) Hamiltonian system and the Hamilto-

nian satisfies a second-order, second-degree differential equation.• PI–PVI possess Lax pairs (isomonodromy problems).• PI–PVI form a coalescence cascade, also known as a degeneration diagram

PVI −→ PV −→ PIVy yPIII −→ PII −→ PI


Hamiltonian Form of PIII(Jimbo & Miwa [1981], Okamoto [1987])

The Hamiltonian associated with PIII is

HIII(q, p, z;α, β) = p2q2 − zpq2 − (β − 1)pq + zp + 12 (β − 2− α) zq

where p and q satisfy

zdq

dz=∂HIII

∂p= 2pq2 − zq2 − (β − 1)q + z

zdp

dz= −∂HIII

∂q= −2p2q + 2zpq + (β − 1)p− 1

2 (β − 2− α) z

Eliminating p then q = w satisfies PIII whilst eliminating q then letting

p(z) =z

1− y(x), x = z2

gives

d2y

dx2=

(1

2y+

1

y − 1

)(dy

dx

)2

− 1

x

dy

dx+

(y − 1)2

8x2

(Ay +

B

y

)− y

2x

with A = (α− β + 2)2 and B = −(α + β − 2)2, which is PV with δ = 0.


The Hamiltonian function

σ(z;α, β) = 12HIII(q, p, z;α, β) + 1

2pq + 18(β − 2)2 − 1

4z2

with

HIII(q, p, z;α, β) = p2q2 − zpq2 − (β − 1)pq + zp + 12 (β − 2− α) zq

satisfies the Jimbo-Miwa-Okamoto σ-equation(z

d2σ

dz2 −dσ

dz

)2

+

{4

(dσ

dz

)2

− z2

}(z

dσ

dz− 2σ

)− zα(β − 2)

dσ

dz

= 14

{α2 + (β − 2)2

}z2

Conversely the solutions of the Hamiltonian system are given by

q =2zσ′′ + 2(1− β)σ′ − αz

z2 − 4 (σ′)2 , p = σ′ + 1

2z


Some Applications of the Third Painleve Equation

d2w

dz2=

1

w

(dw

dz

)2

− 1

z

dw

dz+αw2 + β

z+ γw3 +

δ

w

• Scattering of electromagnetic radiation (Myers [1965])• Ising Model (Barouch, McCoy, Tracy & Wu [1973, 1976], McCoy, Perk & Shrock

[1983])• Exact solutions of Einstein’s equations (Leaute & Marchilhacy [1982, 1983, 1984])• General relativity (MacCullum [1983], Persides & Xanthopoulos [1988], Wills

[1989])• The study of polyelectrolytes in excess salt solution (McCaskill & Fackerell [1988])• Random Matrix Theory (Tracy & Widom [1993], Forrester & Witte [2002, 2006],

. . . )

• Two-dimensional polymers (Zamolodchikov [1994])• Surfaces with Harmonic Inverse Mean Curvature (Bobenko, Eitner & Kitaev [1997])• Stimulated Raman scattering (Fokas & Menyuk [1999])• Orthogonal polynomials (Chen & Its [2010])


Application of PIII to Orthogonal Polynomials (Chen & Its [2010])Consider the orthogonal polynomials with respect to the weight

w(x; z) = xαe−x−z/x, x ∈ [0,∞), α > 0

so we seek polynomials Pn(x; z) which satisfy∫ 1

0

Pm(x; z)Pn(x; z)w(x; z) dx = hn(z)δm,n

Consequently they satisfy the three term recurrence relationxPn(x; z) = Pn+1(x; z) + an(z)Pn(x; z) + bn(z)Pn−1(x; z)

where an(z) and bn(z) are expressible in terms of PIII with(α, β, γ, δ) = (−2(2n + 1 + ν),−2ν, 1,−1)

Further if we define the Hankel determinantDn(z) = det (µj+k(z))n−1

j,k=0

where

µk(z) =

∫ ∞0

xν+ke−x−z/x dx = 2z(ν+k+1)/2Kν+k+1(2√z)

with Kν(z) the modified Bessel function, then

Hn(z) = zd

dzlnDn(z)

satisfies the Jimbo-Miwa-Okamoto σ-equation.


Some Applications of the Fifth Painleve Equation

d2w

dz2=

(1

2w+

1

w − 1

)(dw

dz

)2

− 1

z

dw

dz+

(w − 1)2

z2

(αw +

β

w

)+γw

z+δw(w + 1)

w − 1

• One-dimensional Bose gas – sine kernel (Jimbo, Miwa, Mori & Sato [1980])• Exact solutions of Einstein’s equations (Leaute & Marchilhacy [1982, 1983, 1984])• Ising Model (McCoy, Park & Shrock [1983])• Random Matrix Theory (Tracy & Widom [1994], Adler, Shiota & van Moerbeke

[1995], Baik [2002], Forrester & Witte [2002], . . . )

• Quantum correlation function of the XXZ antiferromagnet (Essler, Frahm, Its &Korepin [1996])• Surfaces with Harmonic Inverse Mean Curvature (Bobenko, Eitner & Kitaev [1997])• Nonlinear σ models (Hirayama & Shi [2002])• Entanglement in extended quantum systems (Casini, Fosco & Huerta [2005], Casini

& Huerta [2005, 2008])• Quantum transport (Osipov & Kanzieper [2008])• Orthogonal polynomials (Basor & Chen [2009], Basor, Chen & Ehrhardt [2009],

Chen & Dai [2010], Forrester & Ormerod [2010])


Application of PV to Orthogonal Polynomials (Chen & Dai [2010])Consider the orthogonal polynomials with respect to the Pollaczek-Jacobi weight

w(x; z) = xa(1− x)be−z/x, x ∈ [0, 1], a > 0, b > 0

so we seek polynomials Pn(x; z) which satisfy∫ 1

0

Pm(x; z)Pn(x; z)w(x; z) dx = hn(t)δm,n

Consequently they satisfy the three term recurrence relationxPn(x; z) = Pn+1(x; z) + an(z)Pn(x; z) + bn(z)Pn−1(x; z)

where an(z) and bn(z) are expressible in terms of PV with(α, β, γ, δ) =

(12(2n + 1 + a + b)2,−1

2b2, a,−1

2

)Further if we define the Hankel determinant

Dn(z) = det (µj+k(z))n−1j,k=0

where

µk(z) =

∫ 1

0

xk+a(1− x)be−z/x dx = Γ(1 + b)U(1 + b,−a− k, z) ez

where U(a, b, z) is the Kummer function, then

Hn(z) = zd

dzlnDn(z)

satisfies the Jimbo-Miwa-Okamoto σ-equation.


Classical Solutions of the Third Painleve Equation

d2w

dz2=

1

w

(dw

dz

)2

− 1

z

dw

dz+αw2 + β

z+ γw3 +

δ

w

Three Cases:1. If γδ 6= 0 then set γ = 1 and δ = −1, without loss of generality.

2. If γ 6= 0 and δ = 0 or γ = 0 and δ 6= 0.

3. If γ = 0 and δ = 0.


Classical Solutions of PIII

d2w

dz2=

1

w

(dw

dz

)2

− 1

z

dw

dz+αw2 + β

z+ w3 − 1

wPIII

Theorem• PIII with γ = −δ = 1 has rational solutions if and only if

ε1α + ε2β = 4n

with n ∈ Z and ε1 = ±1, ε2 = ±1, independently.

• PIII with γ = −δ = 1 has solutions in terms of the solution of the Riccati equation

zw′ = ε1zw2 + (αε1 − 1)w + ε2z

if and only ifε1α + ε2β = 4n + 2

with n ∈ Z and ε1 = ±1, ε2 = ±1, independently. The Riccati equation has solution

w(z) = −ε1ϕ′(z)/ϕ(z)

where

ϕ(z) = zν {C1Jν(ζ) + C2Yν(ζ)} , ν = 12αε2, ζ =

√ε1ε2 z

with Jν(ζ) and Yν(ζ) Bessel functions.


PIII — Associated Special PolynomialsTheorem (PAC [2003], Kajiwara [2003])

Suppose that Sn(z;µ) satisfies the recursion relation

Sn+1Sn−1 = −z[SnS

′′n − (S ′n)

2]− SnS ′n + (z + µ)S2

n

with S−1(z;µ) = S0(z;µ) = 1. Then

wn = w(z;αn, βn, 1,−1) = 1 +d

dzlnSn−1(z;µ− 1)

Sn(z;µ)≡ Sn(z;µ− 1)Sn−1(z;µ)

Sn(z;µ)Sn−1(z;µ− 1)

satisfies PIII

w′′n =(w′n)2

wn− w′n

z+αnw

2n + βnz

+ w3n −

1

wnwith αn = 2n + 2µ− 1 and βn = 2n− 2µ + 1.

The first few polynomials, which are monic polynomials of degree 12n(n + 1), are

S1(z;µ) = ζ

S2(z;µ) = ζ3 − µS3(z;µ) = ζ6 − 5µζ3 + 9µζ − 5µ2

S4(z;µ) = ζ10 − 15µζ7 + 63µζ5 − 225µζ3 + 315µ2ζ2 − 175µ3ζ + 36µ2

with ζ = z − µ.


Determinantal form of Rational Solutions of PIIITheorem (Kajiwara & Masuda [1999])

Let ϕk(z;µ) be the polynomial defined by∞∑k=0

ϕk(z;µ)λk = (1 + λ)µ exp(zλ)

and τn(z;µ) be the n× n determinant given by the Wronskian

τn(z;µ) =W(ϕ1, ϕ3, . . . , ϕ2n−1) =

∣∣∣∣∣∣∣∣ϕ1 ϕ3 · · · ϕ2n−1

ϕ′1 ϕ′3 · · · ϕ′2n−1... ... . . . ...ϕ

(n−1)1 ϕ

(n−1)3 · · · ϕ(n−1)

2n−1

∣∣∣∣∣∣∣∣with τ−1(z;µ) = τ0(z;µ) = 1, then

wn = w(z;αn, βn, 1,−1) =τn(z;µ− 1)τn−1(z;µ)

τn(z;µ)τn−1(z;µ− 1)

satisfies PIII with

αn = 2n + 2µ− 1, βn = 2n− 2µ + 1, γn = 1, δn = −1

• Here ϕk(z;µ) = L(µ−k)k (−z), with L(m)

k (ζ) the associated Laguerre polynomial.


Determinantal form of Bessel Function Solutions of PIII

Theorem (Okamoto [1987], Masuda [2004])Let τn(z; ν) by the n× n determinant

τn(z; ν) =

∣∣∣∣∣∣∣∣∣ψν ψ

(1)ν . . . ψ

(n−1)ν

ψ(1)ν ψ

(2)ν . . . ψ

(n)ν

... ... . . . ...ψ

(n−1)ν ψ

(n)ν . . . ψ

(2n−2)ν

∣∣∣∣∣∣∣∣∣ , ψ(k)ν =

(z

d

dz

)kψν,

where

ψν(z) =

{C1Jν(z) + C2Yν(z), if ε = 1,

C1Iν(z) + C2Kν(z), if ε = −1,

with Jν(z), Yν(z), Iν(z) and Kν(z) Bessel functions and C1 and C2 arbitrary constants,then

wn(z; ν) = ετn+1(z; ν)τn(z; ν + 1)

τn+1(z; ν + 1)τn(z; ν),

for n ≥ 1, satisfies PIII with

αn = −2ε(n + ν − 1), βn = −2(n− ν), γn = 1, δn = −1


Classical Solutions of the Fifth Painleve Equation

d2w

dz2=

(1

2w+

1

w − 1

)(dw

dz

)2

− 1

z

dw

dz+

(w − 1)2

z2

(αw +

β

w

)+γw

z+δw(w + 1)

(w − 1)

Two Cases:1. If δ 6= 0 then set δ = −1

2, without loss of generality.

2. If δ = 0 and γ 6= 0, when it is equivalent to PIII.


Classical Solutions of PV

d2w

dz2=

(1

2w+

1

w − 1

)(dw

dz

)2

− 1

z

dw

dz+

(w − 1)2(αw2 + β)

z2w+γw

z− w(w + 1)

2(w − 1)PV

Theorem• PV with δ = −1

2, has solutions in terms of Whittaker functions Mκ,µ(z), Wκ,µ(z), orequivalently Kummer functions M(a, b, z), U(a, b, z), or confluent hypergeometricfunctions 1F1(a; c; z), if and only if

ε1

√2α + ε2

√−2β + ε3γ = 2n + 1

where n ∈ Z, with εj = ±1, j = 1, 2, 3, independently.

• PV with δ = −12, has a rational solution if and only if one of the following holds with

m,n ∈ Z and µ an arbitrary constant.

(i), α = 12(m + n + 1 + µ)2, β = −1

2(m− n)2 and γ = −µ,(ii), α = 1

2(m− n)2, β = −12(m + n + 1 + µ)2 and γ = µ,

(iii), α = 18µ

2, β = −18(µ− 2m + 2n)2 and γ = −m− n,

(iv), α = 18(µ− 2m + 2n)2, β = −1

8µ2 and γ = m + n,

(v), α = 18(2m + 1)2, β = −1

8(2n + 1)2 and γ = m− n− µ, with µ 6= 0.


Determinantal form of Whittaker Function Solutions of PV

Theorem (Okamoto [1987], Masuda [2004])Let

τ (i,j)n (z) =

∣∣∣∣∣∣∣∣∣∣ϕ

(0)i,j (z) ϕ

(1)i,j (z) . . . ϕ

(n−1)i,j (z)

ϕ(1)i,j (z) ϕ

(2)i,j (z) . . . ϕ

(n)i,j (z)

... ... . . . ...

ϕ(n−1)i,j (z) ϕ

(n)i,j (z) . . . ϕ

(2n−2)i,j (z)

∣∣∣∣∣∣∣∣∣∣, ϕ

(k)i,j (z) =

(z

d

dz

)kϕi,j(z)

where

ϕi,j(z) = A 1F1(a + i; c + j; z) + Bz1−c−j1F1(a− c + 1 + i− j; 2− c− j; z)

with 1F1(a; c; z) the confluent hypergeometric function and A, B arbitrary constants,so ϕi,j(z) satisfies

zd2ϕi,j

dz2 + (c + j − z)dϕi,jdz− (a + i)ϕi,j = 0

Then

w(z;α, β, γ, δ) = −(

c− ac− a− 1

)n τ (0,0)n (z) τ

(1,1)n+1 (z)

τ(1,0)n (z) τ

(0,1)n+1 (z)

is a solution of PV for

(α, β, γ, δ) =(

12(c− a)2,−1

2(a + n)2, n + 1− c,−12

)Symmetry, Separation, Super-integrability and Special Functions, University of Minnesota, Minneapolis, September 2010 20

Determinantal form of Rational Solutions of PV Cases (i), (ii)Theorem (Thomas & PAC [2010])

Define

Wn(f ) =W(f,

df

dz, . . . ,

dn−1f

dzn−1

)whereW (f1, f2, . . . , fn) is the Wronskian, then

w(i)m,n(z) =

Wn

(L

(µ)m (z)

)Wn+1

(L

(µ+1)m−1 (z)

)Wn

(L

(µ)m−1(z)

)Wn+1

(L

(µ+1)m (z)

)w(ii)m,n(z) =

Wn

(L

(µ)m−1(z)

)Wn+1

(L

(µ+1)m (z)

)Wn

(L

(µ)m (z)

)Wn+1

(L

(µ+1)m−1 (z)

)with L(α)

m (z) the associated Laguerre polynomial

L(µ)m (z) =

z−µ ez

m!

dm

dzm(zm+µ e−z

), m ≥ 0

are rational solution of PV respectively for(α(i)m,n, β

(i)m,n, γ

(i)m,n, δ

(i)m,n

)=(

12(m + n + 1 + µ)2,−1

2(m− n)2,−µ,−12

)(α(ii)m,n, β

(ii)m,n, γ

(ii)m,n, δ

(ii)m,n

)=(

12(m− n)2,−1

2(m + n + 1 + µ)2, µ,−12


PV — Generalized Umemura PolynomialsTheorem (Masuda, Ohta & Kajiwara [2001])

Suppose that Um,n(z;µ) satisfies the recursion relations

Um+1,nUm−1,n = 8z[Um,nU

′′m,n −

(U ′m,n

)2 ]+ 8Um,nU

′m,n + (z + 2µ− 2− 6m + 2n)U 2

m,n

Um,n+1Um,n−1 = 8z[Um,nU

′′m,n −

(U ′m,n

)2 ]+ 8Um,nU

′m,n + (z − 2µ− 2 + 2m− 6n)U 2

m,n

with

U−1,−1(z;µ) = U−1,0(z;µ) = U0,−1(z;µ) = U0,0(z;µ) = 1

Then

w(iii)m,n(z;µ) = w

(z;α(iii)

m,n, β(iii)m,n, γ

(iii)m,n, δ

(iii)m,n

)= − Um,n−1(z;µ)Um−1,n(z;µ)

Um−1,n(z;µ− 2)Um,n−1(z;µ + 2)

is a rational solution of PV for(α(iii)m,n, β

(iii)m,n, γ

(iii)m,n, δ

(iii)m,n

)=(

18µ

2,−18(µ− 2m + 2n)2,−m− n,−1

2

)and

w(v)m,n(z;µ) = w

(z;α(v)

m,n, β(v)m,n, γ

(v)m,n, δ

(v)m,n

)= −Um,n−1(z;µ + 1)Um,n+1(z;µ− 1)

Um−1,n(z;µ− 1)Um+1,n(z;µ + 1)

is a rational solution of PV for(α(v)m,n, β

(v)m,n, γ

(v)m,n, δ

(v)m,n

)=(

18(2m + 1)2,−1

8(2n + 1)2,m− n− µ,−12


Theorem (Masuda, Ohta & Kajiwara [2001])Let ϕk(z;µ) = L

(µ)k (1

2z) and ψk(z;µ) = L(µ)k (−1

2z), with ϕk(z;µ) = ψk(z;µ) = 0

for k < 0, and where L(µ)k (x) is the associated Laguerre polynomial. Also define

Sm,n(z;µ) =

∣∣∣∣∣∣∣∣∣∣∣∣

ψ1(z;µ) · · · ψ−m+2(z;µ) ψ−m+1(z;µ) · · · ψ−m−n+2(z;µ)... . . . ... ... . . . ...

ψ2m−1(z;µ) · · · ψm(z;µ) ψm−1(z;µ) · · · ψm−n(z;µ)ϕn−m(z;µ) · · · ϕn+1(z;µ) ϕn(z;µ) · · · ϕ2n−1(z;µ)

... . . . ... ... . . . ...ϕ−n−m+2(z;µ) · · · ϕ−n+1(z;µ) ϕ−n+2(z;µ) · · · ϕ1(z;µ)

∣∣∣∣∣∣∣∣∣∣∣∣then

w(iii)m,n(z;µ) = w

(z;α(iii)

m,n, β(iii)m,n, γ

(iii)m,n, δ

(iii)m,n

)= − Sm,n−1(z;µ)Sm−1,n(z;µ)

Sm−1,n(z;µ− 2)Sm,n−1(z;µ + 2)


(iii)m,n, γ

(iii)m,n, δ

(iii)m,n

)=(

18µ

2,−18(µ− 2m + 2n)2,−m− n,−1

2

)and

w(v)m,n(z;µ) = w

(z;α(v)

m,n, β(v)m,n, γ

(v)m,n, δ

(v)m,n

)=

2n + 1

2m + 1

Sm,n−1(z;µ + 1)Sm,n+1(z;µ− 1)

Sm−1,n(z;µ− 1)Sm+1,n(z;µ + 1)


(v)m,n, γ

(v)m,n, δ

(v)m,n

)=(

18(2m + 1)2,−1

8(2n + 1)2,m− n− µ,−12


Determinantal form of Rational Solutions of PV Cases (iii)–(v)Theorem (Thomas & PAC [2010])

Let ϕk(z; a) = e−z/2L(a)k (1

2z) and ψk(z; a) = L(a)k (−1

2z), with L(a)k (x) the associ-

ated Laguerre polynomial. Also define the double Wronskian

Vm,n(z; θ) = exp(12mz)W(ϕ1, ϕ3, . . . , ϕ2m−1, ψ1, ψ3, . . . , ψ2n−1)

with a = θ −m− n. Then

w(iii)m,n(z; θ) = w

(z;α(iii)

m,n, β(iii)m,n, γ

(iii)m,n, δ

(iii)m,n

)= −Vm,n−1(z; θ + 1)Vm−1,n(z; θ − 1)

Vm−1,n(z; θ + 1)Vm,n−1(z; θ − 1)


(iii)m,n, γ

(iii)m,n, δ

(iii)m,n

)=(

18(θ −m + n)2,−1

8(θ + m− n)2,−m− n,−12

)and

w(v)m,n(z;µ) = w

(z;α(v)

m,n, β(v)m,n, γ

(v)m,n, δ

(v)m,n

)= − 2n + 1

2m + 1

Vm,n−1(z; θ)Vm,n+1(z; θ)

Vm−1,n(z; θ)Vm+1,n(z; θ)


(v)m,n, γ

(v)m,n, δ

(v)m,n

)=(

18(2m + 1)2,−1

8(2n + 1)2, θ,−12


Vortex Solutions of the Complex Sine-Gordon I Equation

∇2ψ +(∇ψ)2ψ

1− |ψ|2+ ψ(1− |ψ|2) = 0


Complex Sine-Gordon I equationThe 2-dimensional complex Sine-Gordon I equation

∇2ψ +(∇ψ)2ψ

1− |ψ|2+ ψ(1− |ψ|2) = 0

where ∇ψ = (ψx, ψy), is associated with the Lagrangian

ESG1 =

∫∫R2

{|∇ψ|2

1− |ψ|2+ 1− |ψ|2

}dx dy

Making the transformation

ψ(x, y) = cos(ϕ(x, y)) eiη(x,y), ψ(x, y) = cos(ϕ(x, y)) e−iη(x,y)

yields the Pohlmeyer-Regge-Lund model (Pohlmeyer [1976], Lund & Regge [1976])

∇2ϕ +cosϕ

sin3ϕ(∇η)2 = 1

2 sin(2ϕ)

sin(2ϕ)∇2η = 4∇ϕ •∇ηNote that setting η = 0 and rescaling ϕ yields the Sine-Gordon equation

∇2ϕ + sinϕ = 0


Complex Sine-Gordon I equationThe 2-dimensional complex Sine-Gordon I equation

∇2ψ +(∇ψ)2ψ

1− |ψ|2+ ψ(1− |ψ|2) = 0 (1)

where ∇ψ = (ψx, ψy), has a separable solution in polar coordinates

ψ(r, θ) = ϕn(r) einθ

where ϕn(r) satisfies

d2ϕndr2

+1

r

dϕndr

+ϕn

1− ϕ2n

{(dϕndr

)2

− n2

r2

}+ ϕn(1− ϕ2

n) = 0 (2)

This equation also arises in the study of the theory of entanglement in extended quantumsystems (Casini, Fosco & Huerta [2005], Casini & Huerta [2005, 2008]).


d2ϕndr2

+1

r

dϕndr

+ϕn

1− ϕ2n

{(dϕndr

)2

− n2

r2

}+ ϕn(1− ϕ2

n) = 0 (2)

This can be transformed into PV in two different ways•Making the transformation

ϕn(r) =1 + un(z)

1− un(z), with r = 1

2z

in (2) yields

d2un

dz2 =

(1

2un+

1

un − 1

)(dundz

)2

− 1

z

dundz

+n2(un − 1)2(u2

n − 1)

8z2un− un(un + 1)

2(un − 1)

which is PV with α = 18n

2, β = −18n

2, γ = 0 and δ = −12.

•Making the transformation

ϕn(r) =1√

1− vn(z), with r =

√z

in (2) yields

d2vndz2

=

(1

2vn+

1

vn − 1

)(dvndz

)2

− 1

z

dvndz− n2(vn − 1)2

2z2vn+vn2z

which is PV with α = 0, β = −12n

2, γ = 12 and δ = 0, and so is equivalent to PIII.


Suppose that ϕn(r) satisfies

d2ϕndr2

+1

r

dϕndr

+ϕn

1− ϕ2n

{(dϕndr

)2

− n2

r2

}+ ϕn(1− ϕ2

n) = 0 (2)

Then ϕn(r) also satisfies the differential-difference equationsdϕndr

+n

rϕn − (1− ϕ2

n)ϕn−1 = 0 (3a)

dϕn−1

dr− n− 1

rϕn−1 + (1− ϕ2

n−1)ϕn = 0 (3b)

and the difference equation

ϕn+1 + ϕn−1 =2n

r

ϕn1− ϕ2

n

(4)

which is discrete Painleve II (Nijhoff & Papageorgiou [1991]).

• Solving (3a) for ϕn−1(r) and substituting in (3b) yields equation (2). Also eliminat-ing the derivatives in (3), after letting n→ n + 1 in (3b), yields equation (4).

• If n = 1 then equations (3) have the solution

ϕ0(r) = 1, ϕ1(r) =C1I1(r)− C2K1(r)

C1I0(r) + C2K0(r)

with I0(r), K0(r), I1(r) and K1(r) the imaginary Bessel functions and C1 and C2

arbitrary constants. Then one can use (4) to determine ϕn(r), n = 2, 3, . . . .Symmetry, Separation, Super-integrability and Special Functions, University of Minnesota, Minneapolis, September 2010 29

• For bounded solutions at r = 0 then C2 = 0 and so

ϕ0(r) = 1, ϕ1(r) =d

drln I0(r) =

I1(r)

I0(r)

since I ′0(r) = I1(r). Hence

ϕ2(r) = −rϕ21(r) + 2ϕ1(r)− rr [ϕ2

1(r)− 1]

ϕ3(r) =ϕ3

1(r)− rϕ21(r)− 2ϕ1(r) + r

ϕ1(r) [rϕ21(r) + ϕ1(r)− r]

ϕ4(r) =r(r2 + 5)ϕ4

1(r) + 4ϕ31(r)− 2r(r2 + 3)ϕ2

1(r) + r3

r [(r2 − 1)ϕ41(r) + 4rϕ3

1(r)− 2(r2 + 2)ϕ21(r)− 4rϕ1(r) + r2]

and so on.

• The asymptotic behaviour of the vortex solution ϕn(r) is given by

ϕn(r) =rn

2n n!

{1− r2

4(n + 1)+O

(r4)}

, as r → 0

ϕn(r) = 1− n

2r− n2

8r2− n(n2 + 1)

16r3+O(r−4), as r →∞


Theorem (Tracy & Widom (1998])Suppose that ϕn(r) satisfies

dϕndr

+n

rϕn − (1− ϕ2

n)ϕn−1 = 0

dϕn−1

dr− n− 1

rϕn−1 + (1− ϕ2

n−1)ϕn = 0

then wn(r) =ϕn(r)

ϕn−1(r)satisfies

d2wndr2

=1

wn

(dwndr

)2

− 1

r

dwndr− 2(n− 1)

rw2n +

2n

r+ w3

n −1

wn

i.e. PIII with the parameters α3 = −2(n − 1), β3 = 2n, γ3 = 1 and δ3 = −1. Since−α3 + β3 = 4n − 2, with n ∈ Z+, then this equation has solutions in terms of theimaginary Bessel functions I0(r), K0(r), I1(r) and K1(r).


Plots of ϕ1(r), ϕ2(r), ϕ3(r), ϕ4(r)


Vortex Solutions of the Complex Sine-Gordon II Equation

∇2ψ +(∇ψ)2ψ

2− |ψ|2+ 1

2ψ(1− |ψ|2)(2− |ψ|2) = 0


Complex Sine-Gordon II equationThe 2-dimensional complex Sine-Gordon II equation

∇2ψ +(∇ψ)2ψ

2− |ψ|2+ 1

2ψ(1− |ψ|2)(2− |ψ|2) = 0

where ∇ψ = (ψx, ψy), is associated with the Lagrangian

ESG2 =

∫∫R2

{|∇ψ|2

2− |ψ|2+ (1− |ψ|2)2

}dx dy

Making the transformation

ψ(x, y) =√

2 cos(ϕ(x, y)) eiη(x,y), ψ(x, y) =√

2 cos(ϕ(x, y)) e−iη(x,y)

yields

∇2ϕ +cosϕ

sin3ϕ(∇η)2 + 1

4 sin(4ϕ) = 0

sin(2ϕ)∇2η = 4∇ϕ •∇ηSetting η = 0 and rescaling ϕ yields the Sine-Gordon equation

∇2ϕ + sinϕ = 0


The 2-dimensional complex Sine-Gordon II equation

∇2ψ +(∇ψ)2ψ

2− |ψ|2+ 1

2ψ(1− |ψ|2)(2− |ψ|2) = 0 (1)

where ∇ψ = (ψx, ψy), has a separable solution in polar coordinates

ψ(r, θ) = Q1/2n (r) einθ

an n-vortex configuration, where Qn(r) satisfies

d2Qn

dr2 =Qn − 1

Qn(Qn − 2)

(dQn

dr

)2

− 1

r

dQn

dr−Qn(Qn − 1)(Qn − 2)− 4n2Qn

r2(Qn − 2)(2)

which is solvable in terms of PV. Setting

Qn(r) =2

1−Wn(z), z = 2ir

in (2) yields

d2Wn

dz2 =

(1

2Wn+

1

Wn − 1

)(dWn

dz

)2

− 1

z

dWn

dz− 2n2(Wn − 1)2

z2Wn− Wn(Wn + 1)

2(Wn − 1)

which is PV with α = 0, β = −2n2, γ = 0 and δ = −12.


Theorem (Thomas & PAC [2010])Let ϕk(r; b, n) = e−ir L

(−b−2n)k (ir) and ψk(r; b, n) = L

(−b−2n)k (−ir), where L(a)

k (x)is the associated Laguerre polynomial and define the double Wronskian

Θn(r; b) = exp(inr)W(ϕ1, ϕ3, . . . , ϕ2n−1, ψ1, ψ3, . . . , ψ2n−1)

which is a polynomial of degree n(n + 1). Then

Qn(r) =2Θn−1(r; 2n + 1) Θn(r; 2n− 1)

Θn−1(r; 2n + 1) Θn(r; 2n− 1) + Θn−1(r; 2n− 1) Θn(r; 2n + 1)

satisfies

d2Qn

dr2 =Qn − 1

Qn(Qn − 2)

(dQn

dr

)2

− 1

r

dQn

dr−Qn(Qn − 1)(Qn − 2)− 4n2Qn

r2(Qn − 2)

Previously Barashenkov & Pelinovsky [1998] and N. Olver & Barashenkov [2005]derived a sequence of 4 Schlesinger maps to obtain Qn+1 from Qn.

Q0 = 1→ Q1 → Q2 → Q3 → Q4 → . . .


The first few functions Qn(r) are given by

Q1(r) =r2

r2 + 4

Q2(r) =r4(r2 + 24)2

r8 + 64r6 + 1152r4 + 9216r2 + 36864

Q3(r) =r6(r6 + 144r4 + 5760r2 + 92160)2

D3(r)

where

D3(r) = r18 + 324r16 + 41472r14 + 2820096r12 + 114130944r10 + 2919628800r8

+ 50960793600r6 + 611529523200r4 + 4892236185600r2 + 19568944742400

As r → 0

Qn(r) =r2n

22n(n!)2− r2n+2

22n+2 n! (n + 1)!+O(r2n+4)

and as r →∞Qn(r) = 1− 4n2

r2+

16n2(2n2 − 1)

r4+O(r−6)


Plots of Q1(r), Q2(r), Q3(r), Q4(r)


Q1(r) Q2(r)

Q3(r) Q4(r)Symmetry, Separation, Super-integrability and Special Functions, University of Minnesota, Minneapolis, September 2010 39

Roots of Qn

Q4(r) Q6(r)

Q8(r) Q10(r)


special solutions of the third and ﬁfth painleve equations ... · solutions satisfy linear...

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