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

Transcription

1 Special solutions of the third and fifth Painlevé equations and vortex solutions of the complex Sine-Gordon equations Peter A Clarkson School of Mathematics, Statistics and Actuarial Science University of Kent, Canterbury, CT2 7NF, UK P.A.Clarkson@kent.ac.uk Symmetry, Separation, Super-integrability and Special Functions University of Minnesota, Minneapolis, September 2010

2 1. Introduction Outline 2. Rational and special function solutions of third Painlevé equation d 2 w dz = 1 ) 2 dw 1 dw 2 w dz z dz + αw2 + βz + γw 3 + δ w 3. Rational and special function solutions of the fifth Painlevé equation d 2 w 1 dz = 2 2w + 1 ) ) 2 dw 1 dw w 1)2 + αw + β ) + γw w 1 dz z dz z 2 w z 4. Vortex solutions of the complex Sine-Gordon I equation 2 ψ + ψ)2 ψ 1 ψ 2 + ψ1 ψ 2 ) = 0, ψ = ψ x, ψ y ) 5. Vortex solutions of the complex Sine-Gordon II equation 2 ψ + ψ)2 ψ 2 ψ ψ1 ψ 2 )2 ψ 2 ) = 0, ψ = ψ x, ψ y ) + δww + 1) w 1 Symmetry, Separation, Super-integrability and Special Functions, University of Minnesota, Minneapolis, September

3 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 equations Solutions related by linear recurrence relations Painlevé 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 difference equations Solutions related by nonlinear recurrence relations Symmetry, Separation, Super-integrability and Special Functions, University of Minnesota, Minneapolis, September

4 Painlevé Equations d 2 w dz = 2 6w2 + z d 2 w dz = 2 2w3 + zw + α d 2 w dz = 1 ) 2 dw 1 dw 2 w dz z dz + αw2 + β + γw 3 + δ z w d 2 w dz = 1 ) 2 dw w dz 2 w3 + 4zw 2 + 2z 2 α)w + β w d 2 w 1 dz = 2 2w + 1 ) ) 2 dw 1 dw w 1)2 + w 1 dz z dz z 2 d 2 w dz 2 = γw δww + 1) + z w 1 1 w + 1 w ww 1)w z) z 2 z 1) 2 ) ) 2 dw w { z dz α + βz w where α, β, γ and δ are arbitrary constants. αw + β w ) 1 z + 1 z w z δzz 1) + w 1) 2 w z) γz 1) ) dw } dz P I P II P III P IV P V P VI Symmetry, Separation, Super-integrability and Special Functions, University of Minnesota, Minneapolis, September

5 History of the Painlevé Equations Derived by Painlevé, 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 function of the two-dimensional Ising model is expressible in terms of solutions of P III. Ablowitz & Segur [1977] demonstrated a close connection between completely integrable PDEs solvable by inverse scattering, the so-called soliton equations, such as the Korteweg-de Vries equation and the nonlinear Schrödinger equation, and the Painlevé equations. Flaschka & Newell [1980] introduced the isomonodromy deformation method inverse scattering for ODEs), which expresses the Painlevé equation as the compatibility 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 Painlevé equations by Okamoto in 1980 s. Subsequent developments by Noumi, Umemura, Yamada,... The Painlevé equations are a chapter in the Digital Library of Mathematical Functions, which is a rewrite/update of Abramowitz & Stegun s Handbook of Mathematical Functions see Symmetry, Separation, Super-integrability and Special Functions, University of Minnesota, Minneapolis, September

6 Some Properties of the Painlevé Equations P II P VI have Bäcklund transformations which relate solutions of a given Painlevé equation to solutions of the same Painlevé equation, though with different values of the parameters with associated Affine Weyl groups that act on the parameter space. P II P VI have rational and algebraic solutions for certain values of the parameters. P II P VI have special function solutions expressed in terms of the classical special functions [P II : Airy Aiz), Biz); P III : Bessel J ν z), Y ν z), J ν z), K ν z); P IV : parabolic cylinder D ν z); P V : Whittaker M κ,µ z), W κ,µ z) [equivalently Kummer Ma, b, z), Ua, b, z) or confluent hypergeometric 1 F 1 a; c; z)]; P VI : hypergeometric 2 F 1 a, b; c; z)], for certain values of the parameters. These rational, algebraic and special function solutions of P II P VI can usually be written in determinantal form, frequently as Wronskians. P I P VI can be written as a non-autonomous) Hamiltonian system and the Hamiltonian satisfies a second-order, second-degree differential equation. P I P VI possess Lax pairs isomonodromy problems). P I P VI form a coalescence cascade, also known as a degeneration diagram P VI P V P IV P III P II P I Symmetry, Separation, Super-integrability and Special Functions, University of Minnesota, Minneapolis, September

7 Hamiltonian Form of P III Jimbo & Miwa [1981], Okamoto [1987]) The Hamiltonian associated with P III is H III q, p, z; α, β) = p 2 q 2 zpq 2 β 1)pq + zp where p and q satisfy β 2 α) zq z dq dz = H III = 2pq 2 zq 2 β 1)q + z p z dp dz = H III = 2p 2 q + 2zpq + β 1)p 1 q 2 β 2 α) z Eliminating p then q = w satisfies P III whilst eliminating q then letting z pz) = 1 yx), x = z2 gives d 2 y dx 2 = 1 2y + 1 y 1 ) ) 2 dy 1 dx x dy dx + y 1)2 8x 2 Ay + B y with A = α β + 2) 2 and B = α + β 2) 2, which is P V with δ = 0. ) y 2x Symmetry, Separation, Super-integrability and Special Functions, University of Minnesota, Minneapolis, September

8 The Hamiltonian function with σz; α, β) = 1 2 H IIIq, p, z; α, β) pq β 2)2 1 4 z2 H III q, p, z; α, β) = p 2 q 2 zpq 2 β 1)pq + zp satisfies the Jimbo-Miwa-Okamoto σ-equation z d2 σ dz 2 dσ ) 2 { ) } 2 dσ + 4 z 2 z dσ ) dz dz dz 2σ β 2 α) zq zαβ 2) dσ dz { α 2 + β 2) 2} z 2 = 1 4 Conversely the solutions of the Hamiltonian system are given by q = 2zσ + 21 β)σ αz z 2 4 σ ) 2, p = σ z Symmetry, Separation, Super-integrability and Special Functions, University of Minnesota, Minneapolis, September

9 Some Applications of the Third Painlevé Equation d 2 w dz = 1 ) 2 dw 1 dw 2 w dz z dz + αw2 + β + γw 3 + δ z 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]) Symmetry, Separation, Super-integrability and Special Functions, University of Minnesota, Minneapolis, September

10 Application of P III to Orthogonal Polynomials Chen & Its [2010]) Consider the orthogonal polynomials with respect to the weight wx; z) = x α e x z/x, x [0, ), α > 0 so we seek polynomials P n x; z) which satisfy 1 0 P m x; z)p n x; z)wx; z) dx = h n z)δ m,n Consequently they satisfy the three term recurrence relation xp n x; z) = P n+1 x; z) + a n z)p n x; z) + b n z)p n 1 x; z) where a n z) and b n z) are expressible in terms of P III with α, β, γ, δ) = 22n ν), 2ν, 1, 1) Further if we define the Hankel determinant D n z) = det µ j+k z)) n 1 j,k=0 where µ k z) = with K ν z) the modified Bessel function, then 0 x ν+k e x z/x dx = 2z ν+k+1)/2 K ν+k+1 2 z) H n z) = z d dz ln D nz) satisfies the Jimbo-Miwa-Okamoto σ-equation. Symmetry, Separation, Super-integrability and Special Functions, University of Minnesota, Minneapolis, September

11 Some Applications of the Fifth Painlevé Equation d 2 w 1 dz = 2 2w + 1 ) ) 2 dw 1 dw w 1)2 + αw + β ) + γw δww + 1) + w 1 dz z dz z 2 w z 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]) Symmetry, Separation, Super-integrability and Special Functions, University of Minnesota, Minneapolis, September

12 Application of P V to Orthogonal Polynomials Chen & Dai [2010]) Consider the orthogonal polynomials with respect to the Pollaczek-Jacobi weight wx; z) = x a 1 x) b e z/x, x [0, 1], a > 0, b > 0 so we seek polynomials P n x; z) which satisfy 1 0 P m x; z)p n x; z)wx; z) dx = h n t)δ m,n Consequently they satisfy the three term recurrence relation xp n x; z) = P n+1 x; z) + a n z)p n x; z) + b n z)p n 1 x; z) where a n z) and b n z) are expressible in terms of P V with α, β, γ, δ) = 1 2 2n a + ) b)2, 1 2 b2, a, 1 2 Further if we define the Hankel determinant D n z) = det µ j+k z)) n 1 j,k=0 where µ k z) = 1 0 x k+a 1 x) b e z/x dx = Γ1 + b)u1 + b, a k, z) e z where Ua, b, z) is the Kummer function, then H n z) = z d dz ln D nz) satisfies the Jimbo-Miwa-Okamoto σ-equation. Symmetry, Separation, Super-integrability and Special Functions, University of Minnesota, Minneapolis, September

13 Classical Solutions of the Third Painlevé Equation d 2 w dz = 1 ) 2 dw 1 dw 2 w dz z dz + αw2 + β z + γw 3 + δ w Three Cases: 1. If γδ 0 then set γ = 1 and δ = 1, without loss of generality. 2. If γ 0 and δ = 0 or γ = 0 and δ If γ = 0 and δ = 0. Symmetry, Separation, Super-integrability and Special Functions, University of Minnesota, Minneapolis, September

14 Theorem Classical Solutions of P III d 2 w dz = 1 ) 2 dw 1 dw 2 w dz z dz + αw2 + β z P III with γ = δ = 1 has rational solutions if and only if ε 1 α + ε 2 β = 4n with n Z and ε 1 = ±1, ε 2 = ±1, independently. + w 3 1 w P III with γ = δ = 1 has solutions in terms of the solution of the Riccati equation if and only if zw = ε 1 zw 2 + αε 1 1)w + ε 2 z ε 1 α + ε 2 β = 4n + 2 with n Z and ε 1 = ±1, ε 2 = ±1, independently. The Riccati equation has solution where wz) = ε 1 ϕ z)/ϕz) ϕz) = z ν {C 1 J ν ζ) + C 2 Y ν ζ)}, ν = 1 2 αε 2, ζ = ε 1 ε 2 z with J ν ζ) and Y ν ζ) Bessel functions. P III Symmetry, Separation, Super-integrability and Special Functions, University of Minnesota, Minneapolis, September

15 P III Associated Special Polynomials Theorem PAC [2003], Kajiwara [2003]) Suppose that S n z; µ) satisfies the recursion relation [ S n+1 S n 1 = z S n S n S n) 2] S n S n + z + µ)sn 2 with S 1 z; µ) = S 0 z; µ) = 1. Then w n = wz; α n, β n, 1, 1) = 1 + d dz ln S n 1z; µ 1) S n z; µ) satisfies P III w n = w n) 2 w n w n z + α nwn 2 + β n + wn 3 1 z w n with α n = 2n + 2µ 1 and β n = 2n 2µ + 1. S nz; µ 1) S n 1 z; µ) S n z; µ) S n 1 z; µ 1) The first few polynomials, which are monic polynomials of degree 1 2nn + 1), are S 1 z; µ) = ζ S 2 z; µ) = ζ 3 µ S 3 z; µ) = ζ 6 5µζ 3 + 9µζ 5µ 2 S 4 z; µ) = ζ 10 15µζ µζ 5 225µζ µ 2 ζ 2 175µ 3 ζ + 36µ 2 with ζ = z µ. Symmetry, Separation, Super-integrability and Special Functions, University of Minnesota, Minneapolis, September

16 Determinantal form of Rational Solutions of P III Theorem Kajiwara & Masuda [1999]) Let ϕ k z; µ) be the polynomial defined by ϕ k z; µ)λ k = 1 + λ) µ expzλ) k=0 and τ n z; µ) be the n n determinant given by the Wronskian ϕ 1 ϕ 3 ϕ 2n 1 τ n z; µ) = Wϕ 1, ϕ 3,..., ϕ 2n 1 ) = ϕ 1 ϕ 3 ϕ 2n ϕ n 1) 1 ϕ n 1) 3 ϕ n 1) 2n 1 with τ 1 z; µ) = τ 0 z; µ) = 1, then satisfies P III with w n = wz; α n, β n, 1, 1) = τ nz; µ 1)τ n 1 z; µ) τ n z; µ)τ n 1 z; µ 1) α n = 2n + 2µ 1, β n = 2n 2µ + 1, γ n = 1, δ n = 1 Here ϕ k z; µ) = L µ k) k z), with L m) ζ) the associated Laguerre polynomial. k Symmetry, Separation, Super-integrability and Special Functions, University of Minnesota, Minneapolis, September

17 Determinantal form of Bessel Function Solutions of P III Theorem Okamoto [1987], Masuda [2004]) Let τ n z; ν) by the n n determinant ψ ν ψ ν 1)... ψ n 1) ν ψ ν 1) ψ ν 2)... ψ n) ν τ n z; ν) =, ψ ν..... k) = z d ) k ψ. ν, dz ψ n 1)... ψ 2n 2) where ν ψ ν z) = ψ n) ν ν { C 1 J ν z) + C 2 Y ν z), if ε = 1, C 1 I ν z) + C 2 K ν z), if ε = 1, with J ν z), Y ν z), I ν z) and K ν z) Bessel functions and C 1 and C 2 arbitrary constants, then w n z; ν) = ε τ n+1z; ν)τ n z; ν + 1) τ n+1 z; ν + 1)τ n z; ν), for n 1, satisfies P III with α n = 2εn + ν 1), β n = 2n ν), γ n = 1, δ n = 1 Symmetry, Separation, Super-integrability and Special Functions, University of Minnesota, Minneapolis, September

18 Classical Solutions of the Fifth Painlevé Equation d 2 w dz 2 = 1 2w + 1 ) ) 2 dw 1 w 1 dz z dw dz + w 1)2 z 2 αw + β ) w + γw z + δww + 1) w 1) Two Cases: 1. If δ 0 then set δ = 1 2, without loss of generality. 2. If δ = 0 and γ 0, when it is equivalent to P III. Symmetry, Separation, Super-integrability and Special Functions, University of Minnesota, Minneapolis, September

19 Classical Solutions of P V d 2 w dz 2 = Theorem 1 2w + 1 ) ) 2 dw 1 w 1 dz z dw dz + w 1)2 αw 2 + β) z 2 w + γw z ww + 1) 2w 1) P V P V with δ = 1 2, has solutions in terms of Whittaker functions M κ,µz), W κ,µ z), or equivalently Kummer functions Ma, b, z), Ua, b, z), or confluent hypergeometric functions 1 F 1 a; c; z), if and only if ε 1 2α + ε2 2β + ε3 γ = 2n + 1 where n Z, with ε j = ±1, j = 1, 2, 3, independently. P V with δ = 1 2, has a rational solution if and only if one of the following holds with m, n Z and µ an arbitrary constant. i), α = 1 2 m + n µ)2, β = 1 2 m n)2 and γ = µ, ii), α = 1 2 m n)2, β = 1 2 m + n µ)2 and γ = µ, iii), α = 1 8 µ2, β = 1 8 µ 2m + 2n)2 and γ = m n, iv), α = 1 8 µ 2m + 2n)2, β = 1 8 µ2 and γ = m + n, v), α = 1 8 2m + 1)2, β = 1 8 2n + 1)2 and γ = m n µ, with µ 0. Symmetry, Separation, Super-integrability and Special Functions, University of Minnesota, Minneapolis, September

20 Determinantal form of Whittaker Function Solutions of P V Theorem Okamoto [1987], Masuda [2004]) where Let τ i,j) n z) = ϕ 0) i,j z) ϕ1) i,j z)... ϕn 1) i,j z) ϕ 1) i,j z) ϕ2) i,j z)... ϕn) i,j z) z) ϕ n) z)... ϕ2n 2) z) ϕ n 1) i,j i,j i,j, ϕ k) i,j z) = z d dz ) k ϕ i,j z) ϕ i,j z) = A 1 F 1 a + i; c + j; z) + Bz 1 c j 1F 1 a c i j; 2 c j; z) with 1 F 1 a; c; z) the confluent hypergeometric function and A, B arbitrary constants, so ϕ i,j z) satisfies Then is a solution of P V for z d2 ϕ i,j dz 2 wz; α, β, γ, δ) = + c + j z) dϕ i,j dz a + i)ϕ i,j = 0 c a ) n τ n 0,0) z) τ 1,1) n+1 z) c a 1 τ n 1,0) z) τ 0,1) n+1 z) α, β, γ, δ) = 1 2 c a)2, 1 2 a + ) n)2, n + 1 c, 1 2 Symmetry, Separation, Super-integrability and Special Functions, University of Minnesota, Minneapolis, September

21 Determinantal form of Rational Solutions of P V Theorem Define W n f) = W f, df ) dz,..., dn 1 f dz n 1 where W f 1, f 2,..., f n ) is the Wronskian, then w m,nz) i) = W µ) n L m z) ) µ+1) W n+1 L m 1 z)) µ) W n L m 1 z)) µ+1) W n+1 L m z) ) w m,nz) ii) = W µ) n L m 1 z)) µ+1) W n+1 L m z) ) µ) W n L m z) ) µ+1) W n+1 L m 1 z)) with L m α) z) the associated Laguerre polynomial L µ) m z) = z µ e z d m z m! dz m m+µ e z), m 0 are rational solution of P V respectively for ) α m,n, i) β m,n, i) γ m,n, i) δ m,n i) α ii) m,n, β ii) m,n, γ ii) m,n, δ ii) m,n Cases i), ii) Thomas & PAC [2010]) = 1 2 m + n µ)2, 1 2 m ) n)2, µ, 1 2 ) = 1 2 m n)2, 1 2 m + n ) µ)2, µ, 1 2 Symmetry, Separation, Super-integrability and Special Functions, University of Minnesota, Minneapolis, September

22 P V Generalized Umemura Polynomials Theorem Masuda, Ohta & Kajiwara [2001]) Suppose that U m,n z; µ) satisfies the recursion relations U m+1,n U m 1,n = 8z [ U m,n U m,n U m,n) 2 ] + 8Um,n U m,n + z + 2µ 2 6m + 2n)U 2 m,n U m,n+1 U m,n 1 = 8z [ U m,n U m,n U m,n) 2 ] + 8Um,n U m,n + z 2µ 2 + 2m 6n)U 2 m,n with Then w iii) m,nz; µ) = w U 1, 1 z; µ) = U 1,0 z; µ) = U 0, 1 z; µ) = U 0,0 z; µ) = 1 ) z; α m,n, iii) β m,n, iii) γ m,n, iii) δ m,n iii) is a rational solution of P V for ) α m,n, iii) β m,n, iii) γ m,n, iii) δ m,n iii) and w v) m,nz; µ) = w ) z; α m,n, v) β m,n, v) γ m,n, v) δ m,n v) is a rational solution of P V for ) α m,n, v) β m,n, v) γ m,n, v) δ m,n v) U m,n 1 z; µ)u m 1,n z; µ) = U m 1,n z; µ 2)U m,n 1 z; µ + 2) = 1 8 µ2, 1 8 µ 2m + ) 2n)2, m n, 1 2 = U m,n 1z; µ + 1)U m,n+1 z; µ 1) U m 1,n z; µ 1)U m+1,n z; µ + 1) = 1 8 2m + 1)2, 1 8 2n + ) 1)2, m n µ, 1 2 Symmetry, Separation, Super-integrability and Special Functions, University of Minnesota, Minneapolis, September

23 Theorem Masuda, Ohta & Kajiwara [2001]) Let ϕ k z; µ) = L µ) k 1 2 z) and ψ kz; µ) = L µ) k 1 2 z), with ϕ kz; µ) = ψ k z; µ) = 0 for k < 0, and where L µ) k x) is the associated Laguerre polynomial. Also define ψ 1 z; µ) ψ m+2 z; µ) ψ m+1 z; µ) ψ m n+2 z; µ) S m,n z; µ) = ψ 2m 1 z; µ) ψ m z; µ) ψ m 1 z; µ) ψ m n z; µ) ϕ n m z; µ) ϕ n+1 z; µ) ϕ n z; µ) ϕ 2n 1 z; µ) ϕ n m+2 z; µ) ϕ n+1 z; µ) ϕ n+2 z; µ) ϕ 1 z; µ) then ) w m,nz; iii) µ) = w z; α m,n, iii) β m,n, iii) γ m,n, iii) δ m,n iii) S m,n 1 z; µ) S m 1,n z; µ) = S m 1,n z; µ 2) S m,n 1 z; µ + 2) is a rational solution of P V for α m,n, iii) β m,n, iii) γ m,n, iii) δ m,n iii) and w v) m,nz; µ) = w ) ) z; α m,n, v) β m,n, v) γ m,n, v) δ m,n v) is a rational solution of P V for ) α m,n, v) β m,n, v) γ m,n, v) δ m,n v) = 1 8 µ2, 1 8 µ 2m + ) 2n)2, m n, 1 2 = 2n + 1 2m + 1 S m,n 1 z; µ + 1) S m,n+1 z; µ 1) S m 1,n z; µ 1) S m+1,n z; µ + 1) = 1 8 2m + 1)2, 1 8 2n + ) 1)2, m n µ, 1 2 Symmetry, Separation, Super-integrability and Special Functions, University of Minnesota, Minneapolis, September

24 Determinantal form of Rational Solutions of P V Theorem Cases iii) v) Thomas & PAC [2010]) Let ϕ k z; a) = e z/2 L a) k 1 2 z) and ψ kz; a) = L a) k 1 2z), with La) k x) the associated Laguerre polynomial. Also define the double Wronskian V m,n z; θ) = exp 1 2 mz) Wϕ 1, ϕ 3,..., ϕ 2m 1, ψ 1, ψ 3,..., ψ 2n 1 ) with a = θ m n. Then w iii) m,nz; θ) = w z; α iii) m,n, β iii) m,n, γ iii) m,n, δ iii) m,n ) = V m,n 1z; θ + 1) V m 1,n z; θ 1) V m 1,n z; θ + 1) V m,n 1 z; θ 1) is a rational solution of P V for ) α m,n, iii) β m,n, iii) γ m,n, iii) δ m,n iii) = 1 8 θ m + n)2, 1 8 θ + m ) n)2, m n, 1 2 and ) w m,nz; v) µ) = w z; α m,n, v) β m,n, v) γ m,n, v) δ m,n v) is a rational solution of P V for ) α m,n, v) β m,n, v) γ m,n, v) δ m,n v) = 2n + 1 2m + 1 V m,n 1 z; θ) V m,n+1 z; θ) V m 1,n z; θ) V m+1,n z; θ) = 1 8 2m + 1)2, 1 8 2n + ) 1)2, θ, 1 2 Symmetry, Separation, Super-integrability and Special Functions, University of Minnesota, Minneapolis, September

25 Vortex Solutions of the Complex Sine-Gordon I Equation 2 ψ + ψ)2 ψ 1 ψ 2 + ψ1 ψ 2 ) = 0 Symmetry, Separation, Super-integrability and Special Functions, University of Minnesota, Minneapolis, September

26 Complex Sine-Gordon I equation The 2-dimensional complex Sine-Gordon I equation 2 ψ + ψ)2 ψ 1 ψ 2 + ψ1 ψ 2 ) = 0 where ψ = ψ x, ψ y ), is associated with the Lagrangian { } ψ 2 E SG1 = 1 ψ ψ 2 dx dy Making the transformation R 2 ψx, y) = cosϕx, y)) e iηx,y), ψx, y) = cosϕx, y)) e iηx,y) yields the Pohlmeyer-Regge-Lund model Pohlmeyer [1976], Lund & Regge [1976]) 2 ϕ + cos ϕ sin 3 ϕ η)2 = 1 2 sin2ϕ) sin2ϕ) 2 η = 4 ϕ η Note that setting η = 0 and rescaling ϕ yields the Sine-Gordon equation 2 ϕ + sin ϕ = 0 Symmetry, Separation, Super-integrability and Special Functions, University of Minnesota, Minneapolis, September

27 Complex Sine-Gordon I equation The 2-dimensional complex Sine-Gordon I equation 2 ψ + ψ)2 ψ 1 ψ 2 + ψ1 ψ 2 ) = 0 1) where ψ = ψ x, ψ y ), has a separable solution in polar coordinates where ϕ n r) satisfies d 2 ϕ n dr r dϕ n dr + ψr, θ) = ϕ n r) e inθ { ϕ dϕn ) } 2 n n2 + ϕ 1 ϕ 2 n dr r 2 n 1 ϕ 2 n) = 0 2) This equation also arises in the study of the theory of entanglement in extended quantum systems Casini, Fosco & Huerta [2005], Casini & Huerta [2005, 2008]). Symmetry, Separation, Super-integrability and Special Functions, University of Minnesota, Minneapolis, September

28 { d 2 ϕ n dr + 1 dϕ n 2 r dr + ϕ dϕn ) } 2 n n2 + ϕ 1 ϕ 2 n dr r 2 n 1 ϕ 2 n) = 0 2) This can be transformed into P V in two different ways Making the transformation in 2) yields d 2 u n dz 2 = ϕ n r) = 1 + u nz) 1 u n z), with r = 1 2 z ) ) 2 dun 1 2u n u n 1 dz z which is P V with α = 1 8 n2, β = 1 8 n2, γ = 0 and δ = 1 2. Making the transformation in 2) yields d 2 v n dz 2 = ϕ n r) = du n dz + n2 u n 1) 2 u 2 n 1) u nu n + 1) 8z 2 u n 2u n 1) 1 1 vn z), with r = z ) ) 2 dvn 1 2v n v n 1 dz z dv n dz n2 v n 1) 2 2z 2 v n + v n 2z which is P V with α = 0, β = 1 2 n2, γ = 1 2 and δ = 0, and so is equivalent to P III. Symmetry, Separation, Super-integrability and Special Functions, University of Minnesota, Minneapolis, September

29 Suppose that ϕ n r) satisfies { d 2 ϕ n dr + 1 dϕ n 2 r dr + ϕ dϕn ) } 2 n n2 + ϕ 1 ϕ 2 n dr r 2 n 1 ϕ 2 n) = 0 2) Then ϕ n r) also satisfies the differential-difference equations dϕ n dr + n r ϕ n 1 ϕ 2 n)ϕ n 1 = 0 dϕ n 1 dr and the difference equation n 1 ϕ n ϕ 2 r n 1)ϕ n = 0 ϕ n+1 + ϕ n 1 = 2n r 1 ϕ 2 n which is discrete Painlevé II Nijhoff & Papageorgiou [1991]). Solving 3a) for ϕ n 1 r) and substituting in 3b) yields equation 2). Also eliminating the derivatives in 3), after letting n n + 1 in 3b), yields equation 4). If n = 1 then equations 3) have the solution ϕ 0 r) = 1, ϕ n ϕ 1 r) = C 1I 1 r) C 2 K 1 r) C 1 I 0 r) + C 2 K 0 r) with I 0 r), K 0 r), I 1 r) and K 1 r) the imaginary Bessel functions and C 1 and C 2 arbitrary constants. Then one can use 4) to determine ϕ n r), n = 2, 3,.... Symmetry, Separation, Super-integrability and Special Functions, University of Minnesota, Minneapolis, September a) 3b) 4)

30 For bounded solutions at r = 0 then C 2 = 0 and so ϕ 0 r) = 1, since I 0r) = I 1 r). Hence and so on. ϕ 1 r) = d dr ln I 0r) = I 1r) I 0 r) ϕ 2 r) = rϕ2 1r) + 2ϕ 1 r) r r [ϕ 2 1 r) 1] ϕ 3 r) = ϕ3 1r) rϕ 2 1r) 2ϕ 1 r) + r ϕ 1 r) [rϕ 2 1 r) + ϕ 1r) r] rr 2 + 5)ϕ 4 ϕ 4 r) = 1r) + 4ϕ 3 1r) 2rr 2 + 3)ϕ 2 1r) + r 3 r [r 2 1)ϕ 4 1 r) + 4rϕ3 1 r) 2r2 + 2)ϕ 2 1 r) 4rϕ 1r) + r 2 ] The asymptotic behaviour of the vortex solution ϕ n r) is given by ϕ n r) = {1 rn r2 2 n n! 4n + 1) + O r 4) }, as r 0 ϕ n r) = 1 n 2r n2 8r nn2 + 1) + Or 4 ), as r 2 16r 3 Symmetry, Separation, Super-integrability and Special Functions, University of Minnesota, Minneapolis, September

31 Theorem Suppose that ϕ n r) satisfies Tracy & Widom 1998]) then w n r) = dϕ n dr + n r ϕ n 1 ϕ 2 n)ϕ n 1 = 0 dϕ n 1 dr ϕ nr) ϕ n 1 r) satisfies = 1 dwn w n dr d 2 w n dr 2 n 1 ϕ n ϕ 2 r n 1)ϕ n = 0 ) 2 1 r dw n dr 2n 1) wn 2 + 2n r r + w3 n 1 w n i.e. P III with the parameters α 3 = 2n 1), β 3 = 2n, γ 3 = 1 and δ 3 = 1. Since α 3 + β 3 = 4n 2, with n Z +, then this equation has solutions in terms of the imaginary Bessel functions I 0 r), K 0 r), I 1 r) and K 1 r). Symmetry, Separation, Super-integrability and Special Functions, University of Minnesota, Minneapolis, September

32 Plots of ϕ 1 r), ϕ 2 r), ϕ 3 r), ϕ 4 r) Symmetry, Separation, Super-integrability and Special Functions, University of Minnesota, Minneapolis, September

33 Vortex Solutions of the Complex Sine-Gordon II Equation 2 ψ + ψ)2 ψ 2 ψ ψ1 ψ 2 )2 ψ 2 ) = 0 Symmetry, Separation, Super-integrability and Special Functions, University of Minnesota, Minneapolis, September

34 Complex Sine-Gordon II equation The 2-dimensional complex Sine-Gordon II equation 2 ψ + ψ)2 ψ 2 ψ ψ1 ψ 2 )2 ψ 2 ) = 0 where ψ = ψ x, ψ y ), is associated with the Lagrangian { } ψ 2 E SG2 = 2 ψ ψ 2 ) 2 dx dy Making the transformation yields R 2 ψx, y) = 2 cosϕx, y)) e iηx,y), 2 ϕ + cos ϕ sin 3 ϕ η) sin4ϕ) = 0 sin2ϕ) 2 η = 4 ϕ η ψx, y) = 2 cosϕx, y)) e iηx,y) Setting η = 0 and rescaling ϕ yields the Sine-Gordon equation 2 ϕ + sin ϕ = 0 Symmetry, Separation, Super-integrability and Special Functions, University of Minnesota, Minneapolis, September

35 The 2-dimensional complex Sine-Gordon II equation 2 ψ + ψ)2 ψ 2 ψ ψ1 ψ 2 )2 ψ 2 ) = 0 1) where ψ = ψ x, ψ y ), has a separable solution in polar coordinates ψr, θ) = Q 1/2 n r) e inθ an n-vortex configuration, where Q n r) satisfies d 2 Q n dr 2 = Q n 1 dqn Q n Q n 2) dr which is solvable in terms of P V. Setting in 2) yields d 2 W n dz 2 = 1 2W n + 1 W n 1 ) 2 1 dq n r dr Q nq n 1)Q n 2) Q n r) = ) dwn 2 1 W n z), z = 2ir dz ) 2 1 z which is P V with α = 0, β = 2n 2, γ = 0 and δ = n2 Q n r 2 Q n 2) dw n dz 2n2 W n 1) 2 W nw n + 1) z 2 W n 2W n 1) 2) Symmetry, Separation, Super-integrability and Special Functions, University of Minnesota, Minneapolis, September

36 Theorem Thomas & PAC [2010]) Let ϕ k r; b, n) = e ir L b 2n) k ir) and ψ k r; b, n) = L b 2n) k ir), where L a) k x) is the associated Laguerre polynomial and define the double Wronskian Θ n r; b) = expinr) Wϕ 1, ϕ 3,..., ϕ 2n 1, ψ 1, ψ 3,..., ψ 2n 1 ) which is a polynomial of degree nn + 1). Then satisfies Q n r) = d 2 Q n dr 2 = Q n 1 Q n Q n 2) 2Θ n 1 r; 2n + 1) Θ n r; 2n 1) Θ n 1 r; 2n + 1) Θ n r; 2n 1) + Θ n 1 r; 2n 1) Θ n r; 2n + 1) dqn dr ) 2 1 dq n r dr Q nq n 1)Q n 2) 4n2 Q n r 2 Q n 2) Previously Barashenkov & Pelinovsky [1998] and N. Olver & Barashenkov [2005] derived a sequence of 4 Schlesinger maps to obtain Q n+1 from Q n. Q 0 = 1 Q 1 Q 2 Q 3 Q 4... Symmetry, Separation, Super-integrability and Special Functions, University of Minnesota, Minneapolis, September

37 The first few functions Q n r) are given by Q 1 r) = Q 2 r) = r2 r r 4 r ) 2 r r r r where Q 3 r) = r6 r r r ) 2 D 3 r) D 3 r) = r r r r r r r r r As r 0 and as r Q n r) = r2n 2 2n n!) r 2n n+2 n! n + 1)! + Or2n+4 ) Q n r) = 1 4n2 r n2 2n 2 1) r 4 + Or 6 ) Symmetry, Separation, Super-integrability and Special Functions, University of Minnesota, Minneapolis, September

38 Plots of Q 1 r), Q 2 r), Q 3 r), Q 4 r) Symmetry, Separation, Super-integrability and Special Functions, University of Minnesota, Minneapolis, September

39 Q1r) Q2r) Q3r) Q4r) Symmetry, Separation, Super-integrability and Special Functions, University of Minnesota, Minneapolis, September

40 Roots of Q n Q 4 r) Q 6 r) Q 8 r) Q 10 r) Symmetry, Separation, Super-integrability and Special Functions, University of Minnesota, Minneapolis, September