A computational study of a class of multivalued tronqu ee solutions - - PowerPoint PPT Presentation

A computational study of a class of multivalued tronqu´ ee solutions of the third Painlev´ e equation

Marco Fasondini

University of the Free State

Supervisors: Andr´ e Weideman

Stellenbosch University

Bengt Fornberg

University of Colorado at Boulder

Introducing PIII

PIII : d2u dz2 = 1 u du dz 2 − 1 z du dz + αu2 + β z + γu3 + δ u Phase portrait u(z) = r(z)eiθ(z) Modulus plot

• Poles: u ∼

c z − z0 , z − → z0 c = ±1, (red/yellow)

• Zeros: u ∼ c(z − z0),

z − → z0 c = ±1, (purple/green)

Introducing PIII

PIII : d2u dz2 = 1 u du dz 2 − 1 z du dz + αu2 + β z + γu3 + δ u Phase portrait u(z) = r(z)eiθ(z) Modulus plot

• Poles: u ∼

c z − z0 , z − → z0 c = ±1, (red/yellow)

• Zeros: u ∼ c(z − z0),

z − → z0 c = ±1, (purple/green)

Tronqu´ ee PIII solutions

Tronqu´ ee solutions, Boutroux [1913] PIII : d2u dz2 = 1 u du dz 2 − 1 z du dz + αu2 + β z + γu3 + δ u Lin, Dai and Tibboel [2014] proved the existence

• f

tronqu´ ee PIII solutions whose pole-free sectors have angular widths of

• π and 2π if γ = 1 and δ = −1, and
• 3π

2 and 3π if α = 1, γ = 0 and δ = −1. McCoy, Tracy and Wu [1977] derived asymptotic formulae for tronqu´ ee solutions with parameters α = −β = 2ν, γ = 1 and δ = −1 and applied their results to the Ising model.

Tronqu´ ee PIII solutions

PIII : d2u dz2 = 1 u du dz 2 − 1 z du dz + αu2 + β z + γu3 + δ u The MTW solutions: α = −β = 2ν, γ = 1, δ = −1 u(x) ∼ 1 − λ Γ

• ν + 1

2

• 2−2νx−ν−(1/2)e−2x,

x → ∞ u(x) ∼ Bxσ, x → 0+ σ(λ) = 2 π arcsin(πλ) , B(σ, ν) = 2−2σΓ21

2(1 − σ)

• Γ

1

2(1 + σ) + ν

• Γ21

2(1 + σ)

• Γ

1

2(1 − σ) + ν

• −1

π < λ < 1 π u(x; ν, −λ) = 1 u(x; ν, λ)

Tronqu´ ee PIII solutions

PIII : d2u dz2 = 1 u du dz 2 − 1 z du dz + αu2 + β z + γu3 + δ u The MTW solutions: α = −β = 2ν, γ = 1, δ = −1 u(x) ∼ 1 − λ Γ

• ν + 1

2

• 2−2νx−ν−(1/2)e−2x,

x → ∞ u(x) ∼ Bxσ, x → 0+, −1 π < λ < 1 π

Tronqu´ ee PIII solutions

PIII : d2u dz2 = 1 u du dz 2 − 1 z du dz + αu2 + β z + γu3 + δ u The MTW solutions: α = −β = 2ν, γ = 1, δ = −1 u(x) ∼ 1 − λ Γ

• ν + 1

2

• 2−2νx−ν−(1/2)e−2x,

x → ∞ u(x) ∼ Bxσ, x → 0+, −1 π < λ < 1 π

Tronqu´ ee PIII solutions

PIII : d2u dz2 = 1 u du dz 2 − 1 z du dz + αu2 + β z + γu3 + δ u The MTW solutions: α = −β = 2ν, γ = 1, δ = −1 u(x) ∼ 1 − λ Γ

• ν + 1

2

• 2−2νx−ν−(1/2)e−2x,

x → ∞ u(x) ∼ Bxσ, x → 0+, −1 π < λ < 1 π

Tronqu´ ee PIII solutions

PIII : d2u dz2 = 1 u du dz 2 − 1 z du dz + αu2 + β z + γu3 + δ u The MTW solutions: α = −β = 2ν, γ = 1, δ = −1 u(x) ∼ 1 − λ Γ

• ν + 1

2

• 2−2νx−ν−(1/2)e−2x,

x → ∞ u(x) ∼ Bxσ, x → 0+, −1 π < λ < 1 π

Tronqu´ ee PIII solutions

PIII : d2u dz2 = 1 u du dz 2 − 1 z du dz + αu2 + β z + γu3 + δ u The MTW solutions: α = −β = 2ν, γ = 1, δ = −1 u(x) ∼ 1 − λ Γ

• ν + 1

2

• 2−2νx−ν−(1/2)e−2x,

x → ∞ u(x) ∼ Bxσ, x → 0+, −1 π < λ < 1 π

Tronqu´ ee PIII solutions

PIII : d2u dz2 = 1 u du dz 2 − 1 z du dz + αu2 + β z + γu3 + δ u The MTW solutions: α = −β = 2ν, γ = 1, δ = −1 u(x) ∼ 1 − λ Γ

• ν + 1

2

• 2−2νx−ν−(1/2)e−2x,

x → ∞ u(x) ∼ Bxσ, x → 0+, −1 π < λ < 1 π

Tronqu´ ee PIII solutions

PIII : d2u dz2 = 1 u du dz 2 − 1 z du dz + αu2 + β z + γu3 + δ u The MTW solutions: α = −β = 2ν, γ = 1, δ = −1 u(x) ∼ 1 − λ Γ

• ν + 1

2

• 2−2νx−ν−(1/2)e−2x,

x → ∞ u(x) ∼ Bxσ, x → 0+, −1 π < λ < 1 π

Tronqu´ ee PIII solutions

PIII : d2u dz2 = 1 u du dz 2 − 1 z du dz + αu2 + β z + γu3 + δ u The MTW solutions: α = −β = 2ν, γ = 1, δ = −1 u(x) ∼ 1 − λ Γ

• ν + 1

2

• 2−2νx−ν−(1/2)e−2x,

x → ∞ u(x) ∼ Bxσ, x → 0+, −1 π < λ < 1 π

Tronqu´ ee PIII solutions

PIII : d2u dz2 = 1 u du dz 2 − 1 z du dz + αu2 + β z + γu3 + δ u The MTW solutions: α = −β = 2ν, γ = 1, δ = −1 u(x) ∼ 1 − λ Γ

• ν + 1

2

• 2−2νx−ν−(1/2)e−2x,

x → ∞ u(x) ∼ Bxσ, x → 0+, −1 π < λ < 1 π

Tronqu´ ee PIII solutions

PIII : d2u dz2 = 1 u du dz 2 − 1 z du dz + αu2 + β z + γu3 + δ u The MTW solutions: α = −β = 2ν, γ = 1, δ = −1 u(x) ∼ 1 − λ Γ

• ν + 1

2

• 2−2νx−ν−(1/2)e−2x,

x → ∞ u(x) ∼ Bxσ, x → 0+, −1 π < λ < 1 π

Tronqu´ ee PIII solutions

PIII : d2u dz2 = 1 u du dz 2 − 1 z du dz + αu2 + β z + γu3 + δ u The MTW solutions: α = −β = 2ν, γ = 1, δ = −1 u(x) ∼ 1 − λ Γ

• ν + 1

2

• 2−2νx−ν−(1/2)e−2x,

x → ∞ u(x) ∼ Bxσ, x → 0+, −1 π < λ < 1 π

Tronqu´ ee PIII solutions

PIII : d2u dz2 = 1 u du dz 2 − 1 z du dz + αu2 + β z + γu3 + δ u The MTW solutions: α = −β = 2ν, γ = 1, δ = −1 u(x) ∼ 1 − λ Γ

• ν + 1

2

• 2−2νx−ν−(1/2)e−2x,

x → ∞ u(x) ∼ Bxσ, x → 0+, −1 π < λ < 1 π

Tronqu´ ee PIII solutions

PIII : d2u dz2 = 1 u du dz 2 − 1 z du dz + αu2 + β z + γu3 + δ u The MTW solutions: α = −β = 2ν, γ = 1, δ = −1 u(x) ∼ 1 − λ Γ

• ν + 1

2

• 2−2νx−ν−(1/2)e−2x,

x → ∞ u(x) ∼ Bxσ, x → 0+, −1 π < λ < 1 π

Tronqu´ ee PIII solutions

PIII : d2u dz2 = 1 u du dz 2 − 1 z du dz + αu2 + β z + γu3 + δ u The MTW solutions: α = −β = 2ν, γ = 1, δ = −1 u(x) ∼ 1 − λ Γ

• ν + 1

2

• 2−2νx−ν−(1/2)e−2x,

x → ∞ u(x) ∼ Bxσ, x → 0+, −1 π < λ < 1 π

Tronqu´ ee PIII solutions

PIII : d2u dz2 = 1 u du dz 2 − 1 z du dz + αu2 + β z + γu3 + δ u The MTW solutions: α = −β = 2ν, γ = 1, δ = −1 u(x) ∼ 1 − λ Γ

• ν + 1

2

• 2−2νx−ν−(1/2)e−2x,

x → ∞ u(x) ∼ Bxσ, x → 0+, −1 π < λ < 1 π

Tronqu´ ee PIII solutions

PIII : d2u dz2 = 1 u du dz 2 − 1 z du dz + αu2 + β z + γu3 + δ u The MTW solutions: α = −β = 2ν, γ = 1, δ = −1 u(x) ∼ 1 − λ Γ

• ν + 1

2

• 2−2νx−ν−(1/2)e−2x,

x → ∞ u(x) ∼ Bxσ, x → 0+, −1 π < λ < 1 π

Tronqu´ ee PIII solutions

PIII : d2u dz2 = 1 u du dz 2 − 1 z du dz + αu2 + β z + γu3 + δ u The MTW solutions: α = −β = 2ν, γ = 1, δ = −1 u(x) ∼ 1 − λ Γ

• ν + 1

2

• 2−2νx−ν−(1/2)e−2x,

x → ∞ u(x) ∼ Bxσ, x → 0+, −1 π < λ < 1 π

Tronqu´ ee PIII solutions

PIII : d2u dz2 = 1 u du dz 2 − 1 z du dz + αu2 + β z + γu3 + δ u The MTW solutions: α = −β = 2ν, γ = 1, δ = −1 u(x) ∼ 1 − λ Γ

• ν + 1

2

• 2−2νx−ν−(1/2)e−2x,

x → ∞ u(x) ∼ Bxσ, x → 0+, −1 π < λ < 1 π

Tronqu´ ee PIII solutions

PIII : d2u dz2 = 1 u du dz 2 − 1 z du dz + αu2 + β z + γu3 + δ u The MTW solutions: α = −β = 2ν, γ = 1, δ = −1 u(x) ∼ 1 − λ Γ

• ν + 1

2

• 2−2νx−ν−(1/2)e−2x,

x → ∞ u(x) ∼ Bxσ, x → 0+, −1 π < λ < 1 π

Tronqu´ ee PIII solutions

PIII : d2u dz2 = 1 u du dz 2 − 1 z du dz + αu2 + β z + γu3 + δ u The MTW solutions: α = −β = 2ν, γ = 1, δ = −1 u(x) ∼ 1 − λ Γ

• ν + 1

2

• 2−2νx−ν−(1/2)e−2x,

x → ∞ u(x) ∼ Bxσ, x → 0+, −1 π < λ < 1 π

Tronqu´ ee PIII solutions

PIII : d2u dz2 = 1 u du dz 2 − 1 z du dz + αu2 + β z + γu3 + δ u The MTW solutions: α = −β = 2ν, γ = 1, δ = −1 u(x) ∼ 1 − λ Γ

• ν + 1

2

• 2−2νx−ν−(1/2)e−2x,

x → ∞ u(x) ∼ Bxσ, x → 0+, −1 π < λ < 1 π

Tronqu´ ee PIII solutions

The MTW solutions: α = −β = 2ν, γ = 1, δ = −1. For the case λ > 1 π, let λ = cosh(πµ) π , µ > 0, then [u(x; ν, −λ) = 1/u(x; ν, λ)] Green u(x) ∼ 1 − λ Γ

• ν + 1

2

• 2−2νx−ν−(1/2)e−2x,

x → ∞ Red u(x) x ∼ 1 2µ ν µ (1 − cos [φ(x, ν, µ)]) + sin [φ(x, ν, µ)]

• ,

x → 0+ φ(x, ν, µ) = 2µ ln(x/4) − 4 arg [Γ(µi)] + 2 arg [Γ(ν + µi)]

Tronqu´ ee PIII solutions

The MTW solutions: α = −β = 2ν, γ = 1, δ = −1. For the case λ > 1 π, let λ = cosh(πµ) π , µ > 0, then [u(x; ν, −λ) = 1/u(x; ν, λ)] Green u(x) ∼ 1 − λ Γ

• ν + 1

2

• 2−2νx−ν−(1/2)e−2x,

x → ∞ Red u(x) x ∼ 1 2µ ν µ (1 − cos [φ(x, ν, µ)]) + sin [φ(x, ν, µ)]

• ,

x → 0+ φ(x, ν, µ) = 2µ ln(x/4) − 4 arg [Γ(µi)] + 2 arg [Γ(ν + µi)]

Tronqu´ ee PIII solutions

The MTW solutions: α = −β = 2ν, γ = 1, δ = −1. For the case λ > 1 π, let λ = cosh(πµ) π , µ > 0, then [u(x; ν, −λ) = 1/u(x; ν, λ)] Green u(x) ∼ 1 − λ Γ

• ν + 1

2

• 2−2νx−ν−(1/2)e−2x,

x → ∞ Red u(x) x ∼ 1 2µ ν µ (1 − cos [φ(x, ν, µ)]) + sin [φ(x, ν, µ)]

• ,

x → 0+ φ(x, ν, µ) = 2µ ln(x/4) − 4 arg [Γ(µi)] + 2 arg [Γ(ν + µi)]

Tronqu´ ee PIII solutions

The MTW solutions: α = −β = 2ν, γ = 1, δ = −1. For the case λ > 1 π, let λ = cosh(πµ) π , µ > 0, then [u(x; ν, −λ) = 1/u(x; ν, λ)] Green u(x) ∼ 1 − λ Γ

• ν + 1

2

• 2−2νx−ν−(1/2)e−2x,

x → ∞ Red u(x) x ∼ 1 2µ ν µ (1 − cos [φ(x, ν, µ)]) + sin [φ(x, ν, µ)]

• ,

x → 0+ φ(x, ν, µ) = 2µ ln(x/4) − 4 arg [Γ(µi)] + 2 arg [Γ(ν + µi)]

Tronqu´ ee PIII solutions

The MTW solutions: α = −β = 2ν, γ = 1, δ = −1. For the case λ > 1 π, let λ = cosh(πµ) π , µ > 0, then [u(x; ν, −λ) = 1/u(x; ν, λ)] Green u(x) ∼ 1 − λ Γ

• ν + 1

2

• 2−2νx−ν−(1/2)e−2x,

x → ∞ Red u(x) x ∼ 1 2µ ν µ (1 − cos [φ(x, ν, µ)]) + sin [φ(x, ν, µ)]

• ,

x → 0+ φ(x, ν, µ) = 2µ ln(x/4) − 4 arg [Γ(µi)] + 2 arg [Γ(ν + µi)]

Tronqu´ ee PIII solutions

The MTW solutions: α = −β = 2ν, γ = 1, δ = −1. For the case λ > 1 π, let λ = cosh(πµ) π , µ > 0, then [u(x; ν, −λ) = 1/u(x; ν, λ)] Green u(x) ∼ 1 − λ Γ

• ν + 1

2

• 2−2νx−ν−(1/2)e−2x,

x → ∞ Red u(x) x ∼ 1 2µ ν µ (1 − cos [φ(x, ν, µ)]) + sin [φ(x, ν, µ)]

• ,

x → 0+ φ(x, ν, µ) = 2µ ln(x/4) − 4 arg [Γ(µi)] + 2 arg [Γ(ν + µi)]

Tronqu´ ee PIII solutions

The MTW solutions: α = −β = 2ν, γ = 1, δ = −1. For the case λ > 1 π, let λ = cosh(πµ) π , µ > 0, then [u(x; ν, −λ) = 1/u(x; ν, λ)] Green u(x) ∼ 1 − λ Γ

• ν + 1

2

• 2−2νx−ν−(1/2)e−2x,

x → ∞ Red u(x) x ∼ 1 2µ ν µ (1 − cos [φ(x, ν, µ)]) + sin [φ(x, ν, µ)]

• ,

x → 0+ φ(x, ν, µ) = 2µ ln(x/4) − 4 arg [Γ(µi)] + 2 arg [Γ(ν + µi)]

Tronqu´ ee PIII solutions

The MTW solutions: α = −β = 2ν, γ = 1, δ = −1. For the case λ > 1 π, let λ = cosh(πµ) π , µ > 0, then [u(x; ν, −λ) = 1/u(x; ν, λ)] Green u(x) ∼ 1 − λ Γ

• ν + 1

2

• 2−2νx−ν−(1/2)e−2x,

x → ∞ Red u(x) x ∼ 1 2µ ν µ (1 − cos [φ(x, ν, µ)]) + sin [φ(x, ν, µ)]

• ,

x → 0+ φ(x, ν, µ) = 2µ ln(x/4) − 4 arg [Γ(µi)] + 2 arg [Γ(ν + µi)]

Tronqu´ ee PIII solutions

The MTW solutions: α = −β = 2ν, γ = 1, δ = −1. For the case λ > 1 π, let λ = cosh(πµ) π , µ > 0, then [u(x; ν, −λ) = 1/u(x; ν, λ)] Green u(x) ∼ 1 − λ Γ

• ν + 1

2

• 2−2νx−ν−(1/2)e−2x,

x → ∞ Red u(x) x ∼ 1 2µ ν µ (1 − cos [φ(x, ν, µ)]) + sin [φ(x, ν, µ)]

• ,

x → 0+ φ(x, ν, µ) = 2µ ln(x/4) − 4 arg [Γ(µi)] + 2 arg [Γ(ν + µi)]

Tronqu´ ee PIII solutions

The MTW solutions: α = −β = 2ν, γ = 1, δ = −1. For the case λ > 1 π, let λ = cosh(πµ) π , µ > 0, then [u(x; ν, −λ) = 1/u(x; ν, λ)] Green u(x) ∼ 1 − λ Γ

• ν + 1

2

• 2−2νx−ν−(1/2)e−2x,

x → ∞ Red u(x) x ∼ 1 2µ ν µ (1 − cos [φ(x, ν, µ)]) + sin [φ(x, ν, µ)]

• ,

x → 0+ φ(x, ν, µ) = 2µ ln(x/4) − 4 arg [Γ(µi)] + 2 arg [Γ(ν + µi)]

Tronqu´ ee PIII solutions

The MTW solutions: α = −β = 2ν, γ = 1, δ = −1. For the case λ > 1 π, let λ = cosh(πµ) π , µ > 0, then [u(x; ν, −λ) = 1/u(x; ν, λ)] Green u(x) ∼ 1 − λ Γ

• ν + 1

2

• 2−2νx−ν−(1/2)e−2x,

x → ∞ Red u(x) x ∼ 1 2µ ν µ (1 − cos [φ(x, ν, µ)]) + sin [φ(x, ν, µ)]

• ,

x → 0+ φ(x, ν, µ) = 2µ ln(x/4) − 4 arg [Γ(µi)] + 2 arg [Γ(ν + µi)]

Tronqu´ ee PIII solutions

The MTW solutions: α = −β = 2ν, γ = 1, δ = −1. For the case λ > 1 π, let λ = cosh(πµ) π , µ > 0, then [u(x; ν, −λ) = 1/u(x; ν, λ)] Green u(x) ∼ 1 − λ Γ

• ν + 1

2

• 2−2νx−ν−(1/2)e−2x,

x → ∞ Red u(x) x ∼ 1 2µ ν µ (1 − cos [φ(x, ν, µ)]) + sin [φ(x, ν, µ)]

• ,

x → 0+ φ(x, ν, µ) = 2µ ln(x/4) − 4 arg [Γ(µi)] + 2 arg [Γ(ν + µi)]

Tronqu´ ee PIII solutions

The MTW solutions: α = −β = 2ν, γ = 1, δ = −1. For the case λ > 1 π, let λ = cosh(πµ) π , µ > 0, then [u(x; ν, −λ) = 1/u(x; ν, λ)] Green u(x) ∼ 1 − λ Γ

• ν + 1

2

• 2−2νx−ν−(1/2)e−2x,

x → ∞ Red u(x) x ∼ 1 2µ ν µ (1 − cos [φ(x, ν, µ)]) + sin [φ(x, ν, µ)]

• ,

x → 0+ φ(x, ν, µ) = 2µ ln(x/4) − 4 arg [Γ(µi)] + 2 arg [Γ(ν + µi)]

Tronqu´ ee PIII solutions

As λ is varied, multiple tronqu´ ee solutions occur on sheets

• ther than the main sheet of the MTW solutions

Lin, Dai and Tibboel [2014] proved the existence of tronqu´ ee PIII solutions whose pole-free sectors have angular widths

• f
• π and 2π if γ = 1 and δ = −1, and
• 3π

2 and 3π if α = 1, γ = 0 and δ = −1.

Tronqu´ ee PIII solutions

Tronqu´ ee PIII solutions

Tronqu´ ee PIII solutions

Tronqu´ ee PIII solutions

Tronqu´ ee PIII solutions

Tronqu´ ee PIII solutions

Tronqu´ ee PIII solutions

Tronqu´ ee PIII solutions

Tronqu´ ee PIII solutions

Tronqu´ ee PIII solutions

Tronqu´ ee PIII solutions

Tronqu´ ee PIII solutions

Tronqu´ ee PIII solutions

Tronqu´ ee PIII solutions

Tronqu´ ee PIII solutions

Tronqu´ ee PIII solutions

Tronqu´ ee PIII solutions

Tronqu´ ee PIII solutions

Tronqu´ ee PIII solutions

Tronqu´ ee PIII solutions

As λ is varied, multiple tronqu´ ee solutions occur on sheets other than the main sheet of the MTW solutions, u(z) On the region {z ∈ C | π ≤ arg z ≤ 9π} we have found

• 16 tronqu´

ee solutions if ν ∈ Z. They have the properties u(z) ∼ −1, |z| → ∞,

• k + 1

2

• π < arg z <
• k + 3

2

• π

k = 8, 6, 4, 2 k = 8, 6, 4 k = 8, 6 k = 8 u(z) ∼ 1, |z| → ∞,

• k + 1

2

• π < arg z <
• k + 3

2

• π

k = 7, 5, 3 k = 7, 5 k = 7 Conjecture: The sequences continue indefinitely

Tronqu´ ee PIII solutions

As λ is varied, multiple tronqu´ ee solutions occur on sheets other than the main sheet of the MTW solutions, u(z) On the region {z ∈ C | π ≤ arg z ≤ 9π} we have found

• 16 tronqu´

ee solutions if ν ∈ Z.

Tronqu´ ee PIII solutions

As λ is varied, multiple tronqu´ ee solutions occur on sheets other than the main sheet of the MTW solutions, u(z) On the region {z ∈ C | π ≤ arg z ≤ 9π} we have found

• 16 tronqu´

ee solutions if ν ∈ Z.

Tronqu´ ee PIII solutions

As λ is varied, multiple tronqu´ ee solutions occur on sheets other than the main sheet of the MTW solutions, u(z) On the region {z ∈ C | π ≤ arg z ≤ 9π} we have found

• 16 tronqu´

ee solutions if ν ∈ Z. They have the properties u(z) ∼ −1, |z| → ∞,

• k + 1

2

• π < arg z <
• k + 3

2

• π

k = 8, 6, 4, 2 k = 8, 6, 4 k = 8, 6 k = 8 u(z) ∼ 1, |z| → ∞,

• k + 1

2

• π < arg z <
• k + 3

2

• π

k = 7, 5, 3 k = 7, 5 k = 7 Conjecture: The sequences continue indefinitely

Tronqu´ ee PIII solutions

As λ is varied, multiple tronqu´ ee solutions occur on sheets other than the main sheet of the MTW solutions, u(z) On the region {z ∈ C | π ≤ arg z ≤ 9π} we have found

• 16 tronqu´

ee solutions if ν ∈ Z.

Tronqu´ ee PIII solutions

As λ is varied, multiple tronqu´ ee solutions occur on sheets other than the main sheet of the MTW solutions, u(z) On the region {z ∈ C | π ≤ arg z ≤ 9π} we have found

• 16 tronqu´

ee solutions if ν ∈ Z.

Tronqu´ ee PIII solutions

As λ is varied, multiple tronqu´ ee solutions occur on sheets other than the main sheet of the MTW solutions, u(z) On the region {z ∈ C | π ≤ arg z ≤ 9π} we have found

• 16 tronqu´

ee solutions if ν ∈ Z. They have the properties u(z) ∼ −1, |z| → ∞,

• k + 1

2

• π < arg z <
• k + 3

2

• π

k = 8, 6, 4, 2 k = 8, 6, 4 k = 8, 6 k = 8 u(z) ∼ 1, |z| → ∞,

• k + 1

2

• π < arg z <
• k + 3

2

• π

k = 7, 5, 3 k = 7, 5 k = 7 Conjecture: The sequences continue indefinitely

Tronqu´ ee PIII solutions

As λ is varied, multiple tronqu´ ee solutions occur on sheets other than the main sheet of the MTW solutions, u(z) On the region {z ∈ C | π ≤ arg z ≤ 9π} we have found

• 16 tronqu´

ee solutions if ν ∈ Z.

Tronqu´ ee PIII solutions

As λ is varied, multiple tronqu´ ee solutions occur on sheets other than the main sheet of the MTW solutions, u(z) On the region {z ∈ C | π ≤ arg z ≤ 9π} we have found

• 16 tronqu´

ee solutions if ν ∈ Z.

Tronqu´ ee PIII solutions

As λ is varied, multiple tronqu´ ee solutions occur on sheets other than the main sheet of the MTW solutions, u(z) On the region {z ∈ C | π ≤ arg z ≤ 9π} we have found

• 16 tronqu´

ee solutions if ν ∈ Z. They have the properties u(z) ∼ −1, |z| → ∞,

• k + 1

2

• π < arg z <
• k + 3

2

• π

k = 8, 6, 4, 2 k = 8, 6, 4 k = 8, 6 k = 8 u(z) ∼ 1, |z| → ∞,

• k + 1

2

• π < arg z <
• k + 3

2

• π

k = 7, 5, 3 k = 7, 5 k = 7 Conjecture: The sequences continue indefinitely

Tronqu´ ee PIII solutions

As λ is varied, multiple tronqu´ ee solutions occur on sheets other than the main sheet of the MTW solutions, u(z) On the region {z ∈ C | π ≤ arg z ≤ 9π} we have found

• 16 tronqu´

ee solutions if ν ∈ Z.

Tronqu´ ee PIII solutions

As λ is varied, multiple tronqu´ ee solutions occur on sheets other than the main sheet of the MTW solutions, u(z) On the region {z ∈ C | π ≤ arg z ≤ 9π} we have found

• 16 tronqu´

ee solutions if ν ∈ Z.

Tronqu´ ee PIII solutions

As λ is varied, multiple tronqu´ ee solutions occur on sheets other than the main sheet of the MTW solutions, u(z) On the region {z ∈ C | π ≤ arg z ≤ 9π} we have found

• 16 tronqu´

ee solutions if ν ∈ Z. They have the properties u(z) ∼ −1, |z| → ∞,

• k + 1

2

• π < arg z <
• k + 3

2

• π

k = 8, 6, 4, 2 k = 8, 6, 4 k = 8, 6 k = 8 u(z) ∼ 1, |z| → ∞,

• k + 1

2

• π < arg z <
• k + 3

2

• π

k = 7, 5, 3 k = 7, 5 k = 7 Conjecture: The sequences continue indefinitely

Tronqu´ ee PIII solutions

As λ is varied, multiple tronqu´ ee solutions occur on sheets other than the main sheet of the MTW solutions, u(z) On the region {z ∈ C | π ≤ arg z ≤ 9π} we have found

• 16 tronqu´

ee solutions if ν ∈ Z.

Tronqu´ ee PIII solutions

As λ is varied, multiple tronqu´ ee solutions occur on sheets other than the main sheet of the MTW solutions, u(z) On the region {z ∈ C | π ≤ arg z ≤ 9π} we have found

• 16 tronqu´

ee solutions if ν ∈ Z.

Tronqu´ ee PIII solutions

As λ is varied, multiple tronqu´ ee solutions occur on sheets other than the main sheet of the MTW solutions, u(z) On the region {z ∈ C | π ≤ arg z ≤ 9π} we have found

• 16 tronqu´

ee solutions if ν ∈ Z. They have the properties u(z) ∼ −1, |z| → ∞,

• k + 1

2

• π < arg z <
• k + 3

2

• π

k = 8, 6, 4, 2 k = 8, 6, 4 k = 8, 6 k = 8 u(z) ∼ 1, |z| → ∞,

• k + 1

2

• π < arg z <
• k + 3

2

• π

k = 7, 5, 3 k = 7, 5 k = 7 Conjecture: The sequences continue indefinitely

Tronqu´ ee PIII solutions

As λ is varied, multiple tronqu´ ee solutions occur on sheets other than the main sheet of the MTW solutions, u(z) On the region {z ∈ C | π ≤ arg z ≤ 9π} we have found

• 16 tronqu´

ee solutions if ν ∈ Z.

Tronqu´ ee PIII solutions

As λ is varied, multiple tronqu´ ee solutions occur on sheets other than the main sheet of the MTW solutions, u(z) On the region {z ∈ C | π ≤ arg z ≤ 9π} we have found

• 16 tronqu´

ee solutions if ν ∈ Z.

Tronqu´ ee PIII solutions

As λ is varied, multiple tronqu´ ee solutions occur on sheets other than the main sheet of the MTW solutions, u(z) On the region {z ∈ C | π ≤ arg z ≤ 9π} we have found

• 16 tronqu´

ee solutions if ν ∈ Z.

Tronqu´ ee PIII solutions

As λ is varied, multiple tronqu´ ee solutions occur on sheets other than the main sheet of the MTW solutions, u(z) On the region {z ∈ C | π ≤ arg z ≤ 9π} we have found

• 16 tronqu´

ee solutions if ν ∈ Z.

Tronqu´ ee PIII solutions

As λ is varied, multiple tronqu´ ee solutions occur on sheets other than the main sheet of the MTW solutions, u(z) On the region {z ∈ C | π ≤ arg z ≤ 9π} we have found

• 16 tronqu´

ee solutions if ν ∈ Z.

Tronqu´ ee PIII solutions

As λ is varied, multiple tronqu´ ee solutions occur on sheets other than the main sheet of the MTW solutions, u(z) On the region {z ∈ C | π ≤ arg z ≤ 9π} we have found

• 16 tronqu´

ee solutions if ν ∈ Z.

Tronqu´ ee PIII solutions

As λ is varied, multiple tronqu´ ee solutions occur on sheets other than the main sheet of the MTW solutions, u(z) On the region {z ∈ C | π ≤ arg z ≤ 9π} we have found

• 16 tronqu´

ee solutions if ν ∈ Z.

Tronqu´ ee PIII solutions

As λ is varied, multiple tronqu´ ee solutions occur on sheets other than the main sheet of the MTW solutions, u(z) On the region {z ∈ C | π ≤ arg z ≤ 9π} we have found

• 16 tronqu´

ee solutions if ν ∈ Z.

Tronqu´ ee PIII solutions

As λ is varied, multiple tronqu´ ee solutions occur on sheets other than the main sheet of the MTW solutions, u(z) On the region {z ∈ C | π ≤ arg z ≤ 9π} we have found

• 16 tronqu´

ee solutions if ν ∈ Z.

Tronqu´ ee PIII solutions

As λ is varied, multiple tronqu´ ee solutions occur on sheets other than the main sheet of the MTW solutions, u(z) On the region {z ∈ C | π ≤ arg z ≤ 9π} we have found

• 16 tronqu´

ee solutions if ν ∈ Z.

Tronqu´ ee PIII solutions

As λ is varied, multiple tronqu´ ee solutions occur on sheets other than the main sheet of the MTW solutions, u(z) On the region {z ∈ C | π ≤ arg z ≤ 9π} we have found

• 16 tronqu´

ee solutions if ν ∈ Z.

Tronqu´ ee PIII solutions

As λ is varied, multiple tronqu´ ee solutions occur on sheets other than the main sheet of the MTW solutions, u(z) On the region {z ∈ C | π ≤ arg z ≤ 9π} we have found

• 16 tronqu´

ee solutions if ν ∈ Z.

Tronqu´ ee PIII solutions

As λ is varied, multiple tronqu´ ee solutions occur on sheets other than the main sheet of the MTW solutions, u(z) On the region {z ∈ C | π ≤ arg z ≤ 9π} we have found

• 16 tronqu´

ee solutions if ν ∈ Z.

Tronqu´ ee PIII solutions

As λ is varied, multiple tronqu´ ee solutions occur on sheets other than the main sheet of the MTW solutions, u(z) On the region {z ∈ C | π ≤ arg z ≤ 9π} we have found

• 16 tronqu´

ee solutions if ν ∈ Z.

Tronqu´ ee PIII solutions

As λ is varied, multiple tronqu´ ee solutions occur on sheets other than the main sheet of the MTW solutions, u(z) On the region {z ∈ C | π ≤ arg z ≤ 9π} we have found

• 16 tronqu´

ee solutions if ν ∈ Z.

Tronqu´ ee PIII solutions

As λ is varied, multiple tronqu´ ee solutions occur on sheets other than the main sheet of the MTW solutions, u(z) On the region {z ∈ C | π ≤ arg z ≤ 9π} we have found

• 16 tronqu´

ee solutions if ν ∈ Z.

Tronqu´ ee PIII solutions

As λ is varied, multiple tronqu´ ee solutions occur on sheets other than the main sheet of the MTW solutions, u(z) On the region {z ∈ C | π ≤ arg z ≤ 9π} we have found

• 16 tronqu´

ee solutions if ν ∈ Z. They have the properties u(z) ∼ −1, |z| → ∞,

• k + 1

2

• π < arg z <
• k + 3

2

• π

k = 8, 6, 4, 2 k = 8, 6, 4 k = 8, 6 k = 8 u(z) ∼ 1, |z| → ∞,

• k + 1

2

• π < arg z <
• k + 3

2

• π

k = 7, 5, 3 k = 7, 5 k = 7 Conjecture: The sequences continue indefinitely

Tronqu´ ee PIII solutions

As λ is varied, multiple tronqu´ ee solutions occur on sheets other than the main sheet of the MTW solutions, u(z) On the region {z ∈ C | π ≤ arg z ≤ 9π} we have found

• 72 tronqu´

ee solutions if ν ∈ Z. They have the properties u(z) ∼ i, |z| → ∞, kπ < arg z < (k + 1) π k = 8, 7, 6, 5, 4, 3, 2, 1 k = 8, 7, 6, 5, 4, 3 k = 8, 7, 6, 5 k = 8, 7 u(z) ∼ −i, |z| → ∞, kπ < arg z < (k + 1) π k = 8, 7, 6, 5, 4, 3, 2 k = 8, 7, 6, 5, 4 k = 8, 7, 6 k = 8 Conjecture: The sequences continue indefinitely

Tronqu´ ee PIII solutions

As λ is varied, multiple tronqu´ ee solutions occur on sheets other than the main sheet of the MTW solutions, u(z) On the region {z ∈ C | π ≤ arg z ≤ 9π} we have found

• 72 tronqu´

ee solutions if ν ∈ Z. They have the properties u(z) ∼ −1, |z| → ∞,

• k + 1

2

• π < arg z <
• k + 3

2

• π

k = 8, 7, 6, 5, 4, 3, 2, 1 k = 8, 7, 6, 5, 4, 3 k = 8, 7, 6, 5 k = 8, 7 u(z) ∼ 1, |z| → ∞,

• k + 1

2

• π < arg z <
• k + 3

2

• π

k = 8, 7, 6, 5, 4, 3, 2 k = 8, 7, 6, 5, 4 k = 8, 7, 6 k = 8 Conjecture: The sequences continue indefinitely

Future plans

• Survey tronqu´

ee PIII solutions with α = 1, γ = 0 and δ = −1

• Survey single-valued PIII solutions
• Survey triply branched PIII solutions
• Extend computational method for PIII to

PV : d2u dz2 = 1 2u + 1 u − 1 du dz 2 − 1 z du dz + (u − 1)2 z2

• αu + β

u

• + γu

z + δu(u + 1) u − 1 and PV I : d2u dz2 = 1 2 1 u + 1 u − 1 + 1 u − z du dz 2 − 1 z + 1 z − 1 + 1 u − z du dz

• +u(u − 1)(u − z)

z2(z − 1)2

• α + β z

u2 + γ z − 1 (u − 1)2 + δz(z − 1) (u − z)2

How to compute PIII?

Challenges:

• Moveable poles: the ‘Pole Field Solver’ (PFS), Fornberg & Weideman [2011]

u(z), u′(z) − → Taylor coefficients − → Pad´ e approximation at z + heiθ Stage 1: pole avoidance on a coarse grid Stage 2: compute the solution on a fine grid Single-valued Painlev´ e transcendents: PI, F & W [2011] ; PII, F & W [2014, 2015] ; PIV , Reeger & F [2013, 2014]

• Multivaluedness

How to compute PIII?

Challenges:

• Moveable poles: the ‘Pole Field Solver’ (PFS), Fornberg & Weideman [2011]

u(z), u′(z) − → Taylor coefficients − → Pad´ e approximation at z + heiθ Stage 1: pole avoidance on a coarse grid Stage 2: compute the solution on a fine grid Single-valued Painlev´ e transcendents: PI, F & W [2011] ; PII, F & W [2014, 2015] ; PIV , Reeger & F [2013, 2014]

• Multivaluedness

How to compute PIII?

Second challenge: Multivaluedness Make the paths run in the right direction

−10 −5 5 10 −4 −2 2 4

1st sheet

How to compute PIII?

Second challenge: Multivaluedness Make the paths run in the right direction

−10 −5 5 10 −4 −2 2 4

1st sheet

−10 −8 −6 −4 −2 2 −4 −2 2 4

counterclockwise sheet

How to compute PIII?

Second challenge: Multivaluedness PIII : d2u dz2 = 1 u du dz 2 − 1 z du dz + αu2 + β z + γu3 + δ u Let z = eζ/2, u(z) = e−ζ/2w(ζ) in PIII, then

• PIII :

d2w dζ2 = 1 w dw dζ 2 + 1 4

• αw2 + γw3 + βeζ + δe2ζ

w

• ,

and all solutions of PIII are single-valued Hinkkanen & Laine [2001]. (z = 0 ⇒ ζ = −∞)

How to compute PIII?

z = eζ/2, u(z) = e−ζ/2w(ζ)

Tronqu´ ee PIII solutions

u(x) ∼ 1 − λ Γ

• ν + 1

2

• 2−2νx−ν−(1/2)e−2x,

x → ∞ u(x) ∼ Bxσ, x → 0+, −1 π < λ < 1 π

Tronqu´ ee PIII solutions

MTW solutions: u(x) ∼ 1 − λ Γ

• ν + 1

2

• 2−2νx−ν−(1/2)e−2x, x → ∞

Limiting solution: u(x) ∼ 1 − kx−ν−(1/2)e−2x, x → ∞

Tronqu´ ee PIII solutions

MTW solutions: u(x) ∼ 1 − λ Γ

• ν + 1

2

• 2−2νx−ν−(1/2)e−2x, x → ∞

Limiting solution: u(x) ∼ 1 − kx−ν−(1/2)e−2x, x → ∞