Autonomous limit of 4-dimensional Painlev-type equations and - PowerPoint PPT Presentation

Autonomous limit of 4-dimensional Painlevé-type equations and degeneration of curves of genus two Akane Nakamura 2 the University of Sydney, Australia 9th March, 2016 2 Supported by the Australian Research Council. Akane Nakamura 3 Autonomous 4-dim. Painlevé 9th March, 2016 1 / 59

Goal Geometrically understand the higher-dimensional analogues of the 2-dimensional Painlevé equations. Idea Characterize integrable systems studying the degeneration of the Liouville tori and the spectral curves. Contents 1. A review of Painlevé-type equations 2. Classification of 4-dimensional Painlevé-type equations 3. Autonomouos (isospectral) limit of Painlevé-type equations 4. Degeneration of the spectral curves and the Liouville tori Akane Nakamura 4 Autonomous 4-dim. Painlevé 9th March, 2016 2 / 59

The Painlevé equations The Painlevé equations are 8 types of nonlinear second order ordinary di ff erential equations with the Painlevé property (its general solution has no critical singularities that depend on initial values) other than linear equations, di ff erential equations satisfied by elliptic functions, equations solvable by quadratures. The Painlevé equations govern the isomonodromic deformation of certain linear equations. The Painlevé equations can be expressed as Hamiltonian systems . These 8 equations are linked by certain limiting processes ( degeneration ). Akane Nakamura 5 Autonomous 4-dim. Painlevé 9th March, 2016 3 / 59

d y 2 6 y 2 + t , P I = dt 2 d y 2 2 y 3 + t y + ↵ , P II = dt 2 ! 2 d y 2 d y 1 + 3 2 y 3 + 4 t y 2 + 2 ( t 2 � ↵ ) y + � P IV y , = dt 2 2 y dt ! 2 d 2 y dt + y 2 d y 1 � 1 d y t 2 � 1 P D 8 t , = III dt 2 dt t y ! 2 d 2 y dt � 2 y 2 d y 1 � 1 d y t 2 + � 4 t � 1 P D 7 y , = III dt 2 dt t y ! 2 d 2 y dt + ↵ y 1 4 t + � y 3 d y 1 � 1 d y 4 t 2 + � 4 t 2 + � P D 6 4 y , �� , 0 = III dt 2 dt t y ! 2 d y 2 1 ! d y dt + ( y � 1 ) 2 1 � 1 d y ( ↵ y + � y ) + � y t + � y ( y + 1 ) P V , = 2 y + dt 2 t 2 y � 1 dt t y � 1 ! 2 d y 2 ⌘ d y 1 ! d y 1 1 1 1 1 1 P VI � = y + y � 1 + t + t � 1 + dt 2 2 y � t dt y � t dt ! + y ( y � 1 )( y � t ) ↵ + � t y 2 + � ( t � 1 ) ( y � 1 ) 2 + � t ( t � 1 ) . t 2 ( t � 1 ) 2 ( y � t ) 2 Akane Nakamura 6 Autonomous 4-dim. Painlevé 9th March, 2016 4 / 59

The Painlevé equations The Painlevé equations are 8 types of nonlinear second order ordinary di ff erential equations with the Painlevé property (its general solution has no critical singularities that depend on initial values) other than linear equations, di ff erential equations satisfied by elliptic functions, equations solvable by quadratures. The Painlevé equations govern the isomonodromic deformation of certain linear equations. The Painlevé equations can be expressed as Hamiltonian systems . These 8 equations are linked by certain limiting processes ( degeneration ). Akane Nakamura 7 Autonomous 4-dim. Painlevé 9th March, 2016 5 / 59

The Painlevé equations and Isomonodromic Deformation The Painlevé equations has another important aspect; they govern isomonodromic deformation of certain linear equations. Let us consider m coupled linear ODE of first order: d dx y ( x ) = A ( x ) y ( x ) . Here y ( x ) is an m -component vector, and A ( x ) is an m ⇥ m matrix, rational in x and has poles at t ν ( ⌫ = 1 , . . ., n ) and at t 1 = 1 . Let us consider a fundamental matrix solution Y ( x ) : d dxY ( x ) = A ( x ) Y ( x ) . (1) Akane Nakamura 8 Autonomous 4-dim. Painlevé 9th March, 2016 6 / 59

t = P 1 \ { t 1 , . . ., t n , t 1 } , and ⇡ : P 1 Let P 1 t ! P 1 t the universal covering. Let � be a path in P 1 t , starting at the point x , and ending at x γ such that ⇡ ( x ) = ⇡ ( x γ ) . There exists a nonsingular constant matrix M γ such that Y ( x γ ) = Y ( x ) M γ . The mapping [ � ] 7! M γ defines a representation of the fundamental group of P 1 t , the monodromy representation associated with the di ff erential system. @ ( 1 ) @ xY ( x , t ) = A ( x , t ) Y ( x , t ) , Given a di ff erential system (1), is it possible to deform it while preserving its monodromy representation? The answer is that to ensure the isomonodromy of the deformation, Y ( x ) , as a function of deformation parameters, has to satisfy a set of linear partial di ff erential equations . @ Y ( x , t ) = B i ( x , t ) Y ( x , t ) . @ t i Akane Nakamura 9 Autonomous 4-dim. Painlevé 9th March, 2016 7 / 59

Linear equation @ @ xY ( x , t ) = A ( x , t ) Y ( x , t ) , admit isomonodromic deformation. ? Y ( x , t ) satisfies 8 > > @ > > @ xY ( x , t ) = A ( x , t ) Y ( x , t ) < > > > > @ : Y ( x , t ) = B i ( x , t ) Y ( x , t ) , @ t i ? Frobenius integrability @ A ( x , t ) � @ B i ( x , t ) + [ A ( x , t ) , B i ( x , t ) ] (2) @ t i @ x The Painlevé equations are the solutions of (2) for certain A ( x , t ) ’s. Akane Nakamura 10 Autonomous 4-dim. Painlevé 9th March, 2016 8 / 59

The Painlevé equations The Painlevé equations are 8 types of nonlinear second order ordinary di ff erential equations with the Painlevé property (its general solution has no critical singularities that depend on initial values) other than linear equations, di ff erential equations satisfied by elliptic functions, equations solvable by quadratures. The Painlevé equations govern the isomonodromic deformation of certain linear equations. The Painlevé equations can be expressed as Hamiltonian systems . These 8 equations are linked by certain limiting processes ( degeneration ). Akane Nakamura 11 Autonomous 4-dim. Painlevé 9th March, 2016 9 / 59

Hamiltonians of the Painlevé equations dq dt = @ H dp dt = � @ H @ p , @ q � t ; q , p � = p 2 � q 3 � tq , H I � ↵ ; t ; q , p � = p 2 � ( q 2 + t ) p � ↵ q , H II � ↵ , � ; t ; q , p � = pq ( p � q � t ) + � p + ↵ q , H IV tH III ( D 8 ) � t ; q , p � = p 2 q 2 + qp � q � t q , tH III ( D 7 ) � ↵ ; t ; q , p � = p 2 q 2 + ↵ qp + tp + q , tH III ( D 6 ) � ↵ , � ; t ; q , p � = p 2 q 2 � ( q 2 � � q � t ) p � ↵ q , � ↵ , � , � ; t ; q , p � = p ( p + t ) q ( q � 1 ) + � pq + � p � ( ↵ + � ) tq , tH V � ↵ , � , � , ✏ ; t ; q , p � = q ( q � 1 )( q � t ) p 2 t ( t � 1 ) H VI + { ✏ q ( q � 1 ) � ( 2 ↵ + � + � + ✏ ) q ( q � t ) + � ( q � 1 )( q � t ) } p + ↵ ( ↵ + � )( q � t ) . Akane Nakamura 12 Autonomous 4-dim. Painlevé 9th March, 2016 10 / 59

The Painlevé equations The Painlevé equations are 8 types of nonlinear second order ordinary di ff erential equations with the Painlevé property (its general solution has no critical singularities that depend on initial values) other than linear equations, di ff erential equations satisfied by elliptic functions, equations solvable by quadratures. The Painlevé equations govern the isomonodromic deformation of certain linear equations. The Painlevé equations can be expressed as Hamiltonian systems . These 8 equations are linked by certain limiting processes ( degeneration ). Akane Nakamura 13 Autonomous 4-dim. Painlevé 9th March, 2016 11 / 59

The first to fifth Painlevé equations can e derived from the sisth Painlevé equation by degeneration process. 2+2 - 2+3/2 - 3/2+3/2 ⌦ � ⌦ P III ( D 6 ) @ P III ( D 7 ) @ P III ( D 8 ) � ✓ ⌦ � @ @ @ R @ R 1+1+1+1 - 2+1+1 ⌦ - 3/2+1+1 � 4 - 7/2 P VI P V J P III ( D 6 ) @ P II P I ✓ � � ✓ J � @ � R @ 3+1 5/2+1 J J ^ � - � P IV P II These degeneration processes correspond to the confluence of the singularities of corresponding linear equations. (cf. confluence of hypergeometric functions:Gauss to Kummer, Bessel, Hermite, Airy) Confluence of Singular Points of linear equations 2 + 2 - q c c q S ⌦ � ⌦ 1 + 1 + 1 + 1 - 2 + 1 + 1 4 S S w ⌦ - - - q q q q c q q q c q k g J 7 ◆ ◆ J ^ J ◆ 3 + 1 - g c q q Akane Nakamura 14 Autonomous 4-dim. Painlevé 9th March, 2016 12 / 59

Various extensions and analogues of the 2-dimensional Painlevé equations are known. The relations among the equations from di ff erent origins are not obvious clear and the overall picture is not clear Recently, (roughly speaking,) a “classification" of the 4-dimensional Painlevé-type equations was accomplished (Sakai [9], Kawakami-N.-Sakai [3], Kawakami [4]). The classification is based on the classification of the Fuchsian linear equations with 4 accessory parameters up to Katz’ operations and the degeneration processes. There are 40 types of 4-dimensional Painlevé-type equations in their list. Akane Nakamura 15 Autonomous 4-dim. Painlevé 9th March, 2016 13 / 59