Semi-classical Orthogonal Polynomials and the Painlev e Equations - PowerPoint PPT Presentation

Semi-classical Orthogonal Polynomials and the Painlev´ e Equations Peter A Clarkson School of Mathematics, Statistics and Actuarial Science University of Kent, Canterbury, CT2 7NF, UK P.A.Clarkson@kent.ac.uk South African Symposium of Numerical and Applied Mathematics University of Stellenbosch, South Africa March 2016

Alternative discrete Painlev´ e I equation x n + x n +1 = y 2 y 0 ( t ) = − Ai ′ ( t ) n − t x 0 ( t ) = 0 , x n ( y n + y n − 1 ) = n Ai( t ) Second Painlev´ e equation d 2 q d z 2 = 2 q 3 + zq + A with A a constant. References • P A Clarkson, A F Loureiro & W Van Assche , “Unique positive so- lution for the alternative discrete Painlev´ e I equation”, Journal of Differ- ence Equations and Applications , DOI: 10.1080/10652469.2015.1098635 (2016) • P A Clarkson , “On Airy Solutions of the Second Painlev´ e Equation”, Studies in Applied Mathematics , DOI: 10.1111/sapm.12123 (2016) SANUM, Stellenbosch, March 2016

Painlev´ e Equations d 2 q d z 2 = 6 q 2 + z P I d 2 q d z 2 = 2 q 3 + zq + A P II � d q � 2 d z + Aq 2 + B d 2 q d z 2 = 1 − 1 d q + Cq 3 + D P III q d z z z q � d q � 2 d 2 q d z 2 = 1 + 3 2 q 3 + 4 zq 2 + 2( z 2 − A ) q + B P IV 2 q d z q � 1 �� d q � 2 � � d 2 q d z + ( q − 1) 2 1 − 1 d q Aq + B d z 2 = 2 q + P V z 2 q − 1 d z z q + Cq z + Dq ( q + 1) q − 1 � 1 �� d q � 2 � 1 � d q d 2 q d z 2 = 1 1 1 1 1 q + q − 1 + − z + z − 1 + P VI q − z d z q − z d z 2 � � + q ( q − 1)( q − z ) A + Bz q 2 + C ( z − 1) ( q − 1) 2 + Dz ( z − 1) z 2 ( z − 1) 2 ( q − z ) 2 with A , B , C and D arbitrary constants. SANUM, Stellenbosch, March 2016

Special function solutions of Painlev´ e equations Number of Associated Special Number of (essential) orthogonal function parameters parameters polynomial P I 0 — Airy P II 1 0 — Ai( z ) , Bi( z ) Bessel 2 1 P III — J ν ( z ) , I ν ( z ) , K ν ( z ) Parabolic Hermite P IV 2 1 D ν ( z ) H n ( z ) Kummer Associated M ( a, b, z ) , U ( a, b, z ) Laguerre P V 3 2 Whittaker L ( k ) n ( z ) M κ,µ ( z ) , W κ,µ ( z ) hypergeometric Jacobi P VI 4 3 P ( α,β ) ( z ) 2 F 1 ( a, b ; c ; z ) n SANUM, Stellenbosch, March 2016

Monic Orthogonal Polynomials Let P n ( x ) , n = 0 , 1 , 2 , . . . , be the monic orthogonal polynomials of degree n in x , with respect to the positive weight ω ( x ) , such that � b P m ( x ) P n ( x ) ω ( x ) d x = h n δ m,n , h n > 0 , m, n = 0 , 1 , 2 , . . . a One of the important properties that orthogonal polynomials have is that they satisfy the three-term recurrence relation xP n ( x ) = P n +1 ( x ) + α n P n ( x ) + β n P n − 1 ( x ) where the recurrence coefficients are given by � � ∆ n +1 ∆ n β n = ∆ n +1 ∆ n − 1 α n = − , ∆ 2 ∆ n +1 ∆ n n with � � � � � � � � µ 0 µ 1 . . . µ n − 1 µ 0 µ 1 . . . µ n − 2 µ n � � � � � � � � µ 1 µ 2 . . . µ n µ 1 µ 2 . . . µ n − 1 µ n +1 � � � � � ∆ n = , ∆ n = . . . . . . . ... ... � . . . � � . . . . � . . . . . . . � � � � � � � � µ n − 1 µ n . . . µ 2 n − 2 µ n − 1 µ n . . . µ 2 n − 3 µ 2 n − 1 � b x k ω ( x ) d x are the moments of the weight ω ( x ) . and µ k = a SANUM, Stellenbosch, March 2016

Semi-classical Orthogonal Polynomials Consider the Pearson equation satisfied by the weight ω ( x ) d d x [ σ ( x ) ω ( x )] = τ ( x ) ω ( x ) • Classical orthogonal polynomials : σ ( x ) and τ ( x ) are polynomials with deg ( σ ) ≤ 2 and deg ( τ ) = 1 ω ( x ) σ ( x ) τ ( x ) exp( − x 2 ) 1 − 2 x Hermite x ν exp( − x ) Laguerre x 1 + ν − x (1 − x ) α (1 + x ) β 1 − x 2 Jacobi β − α − (2 + α + β ) x • Semi-classical orthogonal polynomials : σ ( x ) and τ ( x ) are polynomi- als with either deg ( σ ) > 2 or deg ( τ ) > 1 ω ( x ) σ ( x ) τ ( x ) 3 x 3 + tx ) exp( − 1 t − x 2 1 Airy | x | ν exp( − x 2 + tx ) 1 + ν + tx − 2 x 2 x semi-classical Hermite | x | 2 ν +1 exp( − x 4 + tx 2 ) 2 ν + 2 + 2 tx 2 − 4 x 4 x Generalized Freud SANUM, Stellenbosch, March 2016

If the weight has the form ω ( x ; t ) = ω 0 ( x ) exp( tx ) � ∞ x k ω 0 ( x ) exp( tx ) d x exist for all k ≥ 0 . where the integrals −∞ • The recurrence coefficients α n ( t ) and β n ( t ) satisfy the Toda system d α n d β n d t = β n − β n +1 , d t = β n ( α n − α n − 1 ) • The k th moment is given by �� ∞ � � ∞ x k ω 0 ( x ) exp( tx ) d x = d k = d k µ 0 µ k ( t ) = ω 0 ( x ) exp( tx ) d x d t k d t k −∞ −∞ • Since µ k ( t ) = d k µ 0 d t k , then ∆ n ( t ) and � ∆ n ( t ) can be expressed as Wronskians � � � d j + k µ 0 � n − 1 d t , . . . , d n − 1 µ 0 µ 0 , d µ 0 ∆ n ( t ) = W = det d t n − 1 d t j + k j,k =0 � � d t , . . . , d n − 2 µ 0 d t n − 2 , d n µ 0 µ 0 , d µ 0 = d � ∆ n ( t ) = W d t ∆ n ( t ) d t n SANUM, Stellenbosch, March 2016

An Alternative Discrete Painlev´ e I Equation x n + x n +1 = y 2 y 0 ( t ) = − Ai ′ ( t ) n − t x 0 ( t ) = 0 , x n ( y n + y n − 1 ) = n Ai( t ) • PAC, A Loureiro & W Van Assche , “Unique positive solution for the alternative discrete Painlev´ e I equation”, Journal of Difference Equations and Applications , DOI: 10.1080/10652469.2015.1098635 (2016) SANUM, Stellenbosch, March 2016

x n + x n +1 = y 2 y 0 ( t ) = − Ai ′ ( t ) n − t x 0 ( t ) = 0 , x n ( y n + y n − 1 ) = n Ai( t ) The system is highly sensitive to the initial conditions [50 digits] Ai(0) = 3 1 / 3 Γ( 2 y 0 (0) = − Ai ′ (0) 3 ) x 0 (0) = 0 Γ( 1 3 ) SANUM, Stellenbosch, March 2016

y 0 (0) = 0 . 7290111 ... y 0 (0) = 0 . 729 y 0 (0) = 0 . 72902 x 0 (0) = 0 x 0 (0) = 0 x 0 (0) = 0 SANUM, Stellenbosch, March 2016

Orthogonal Polynomials on Complex Contours Consider the semi-classical Airy weight � � 3 x 3 + tx − 1 ω ( x ; t ) = exp , t > 0 on the curve C from e 2 π i / 3 ∞ to e − 2 π i / 3 ∞ . The moments are � � � 3 x 3 + tx − 1 µ 0 ( t ) = exp d x = Ai( t ) C � d x = d k � � x k exp 3 x 3 + tx − 1 d t k Ai( t ) = Ai ( k ) ( t ) µ k ( t ) = C where Ai( t ) is the Airy function , the Hankel determinant is � d j + k � � � Ai( t ) , Ai ′ ( t ) , . . . , Ai ( n − 1) ( t ) ∆ n ( t ) = W = det d t j + k Ai( t ) j,k =0 with ∆ 0 ( t ) = 1 , and the recursion coefficients are β n ( t ) = d 2 α n ( t ) = d d t ln ∆ n +1 ( t ) ∆ n ( t ) , d t 2 ln ∆ n ( t ) with d t ln Ai( t ) = Ai ′ ( t ) α 0 ( t ) = d Ai( t ) , β 0 ( t ) = 0 SANUM, Stellenbosch, March 2016

The recurrence coefficients α n ( t ) and β n ( t ) satisfy the discrete system ( α n + α n − 1 ) β n − n = 0 (1) α 2 n + β n + β n +1 − t = 0 and the differential system (Toda) d α n d β n d t = β n +1 − β n , d t = β n ( α n − α n − 1 ) (2) Letting x n = − β n and y n = − α n in (1) and (2) yields x n + x n +1 = y 2 n − t (3) x n ( y n + y n − 1 ) = n which is the discrete system we’re interested in, and d x n d y n (4) d t = x n ( y n − 1 − y n ) , d t = x n +1 − x n Then eliminating x n +1 and y n − 1 between (3) and (4) yields d y n d x n d t = y 2 (5) n − 2 x n − t, d t = − 2 x n y n + n SANUM, Stellenbosch, March 2016

Consider the system d y n d x n d t = y 2 n − 2 x n − t, d t = − 2 x n y n + n • Eliminating x n yields d 2 y n d t 2 = 2 y 3 n − 2 ty n − 2 n − 1 which is equivalent to d 2 q d z 2 = 2 q 3 + zq + n + 1 2 i.e. P II with A = n + 1 2 . • Eliminating y n yields � d x n � 2 d 2 x n n + 2 tx n − n 2 1 + 4 x 2 d t 2 = 2 x n d t 2 x n which is equivalent to � d v � 2 d 2 v − 2 v 2 − zv − n 2 d z 2 = 1 2 v d z 2 v an equation known as P 34 . SANUM, Stellenbosch, March 2016

x n + x n +1 = y 2 y 0 ( t ) = − Ai ′ ( t ) n − t x 0 ( t ) = 0 , x n ( y n + y n − 1 ) = n Ai( t ) Solving for x n yields n + 1 n = y 2 + n − t y n + y n +1 y n + y n − 1 which is known as alt-d P I ( Fokas, Grammaticos & Ramani [1993] ). We have seen that y n and x n satisfy d 2 y n d t 2 = 2 y 3 n − 2 ty n − 2 n − 1 � d x n � 2 d 2 x n n + 2 tx n − n 2 1 + 4 x 2 d t 2 = 2 x n d t 2 x n which have “Airy-type” solutions x n ( t ) = − d 2 y n ( t ) = d d t ln τ n ( t ) τ n +1 ( t ) , d t 2 ln τ n ( t ) where � d j + k � τ n ( t ) = det d t j + k Ai( t ) , n ≥ 1 j,k =0 and τ 0 ( t ) = 1 . SANUM, Stellenbosch, March 2016

Theorem ( PAC, Loureiro & Van Assche [2016] ) For positive values of t , there exists a unique solution of x n + x n +1 = y 2 n − t x n ( y n + y n − 1 ) = n with x 0 ( t ) = 0 for which x n +1 ( t ) > 0 and y n ( t ) > 0 for all n ≥ 0 . This solution corresponds to the initial value y 0 ( t ) = − Ai ′ ( t ) Ai( t ) . Theorem ( PAC, Loureiro & Van Assche [2016] ) For positive values of t , there exists a unique solution of n + 1 n = y 2 + n − t y n + y n +1 y n + y n − 1 for which y n ( t ) ≥ 0 for all n ≥ 0 . This solution corresponds to the initial values y 0 ( t ) = − Ai ′ ( t ) 1 Ai( t ) , y 1 ( t ) = − y 0 ( t ) + y 2 0 ( t ) − t SANUM, Stellenbosch, March 2016