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A computational study of a class of multivaluedtronquée solutions of the third Painlevé equation
Marco Fasondini University of the Free State
Supervisors:
André Weideman Stellenbosch University
Bengt Fornberg University of Colorado at Boulder
Introducing PIII
PIII :d2u
dz2=
1
u
(du
dz
)2− 1z
du
dz+αu2 + β
z+ γu3 +
δ
u
Phase portrait
u(z) = r(z)eiθ(z)
Modulus plot
• Poles: u ∼ cz − z0
, z −→ z0
c = ±1, (red/yellow)• Zeros: u ∼ c(z − z0), z −→ z0c = ±1, (purple/green)
Introducing PIII
PIII :d2u
dz2=
1
u
(du
dz
)2− 1z
du
dz+αu2 + β
z+ γu3 +
δ
uPhase portrait
u(z) = r(z)eiθ(z)
Modulus plot
• Poles: u ∼ cz − z0
, z −→ z0
c = ±1, (red/yellow)• Zeros: u ∼ c(z − z0), z −→ z0c = ±1, (purple/green)
Tronquée PIII solutions
Tronquée solutions, Boutroux [1913]
PIII :d2u
dz2=
1
u
(du
dz
)2− 1z
du
dz+αu2 + β
z+ γu3 +
δ
u
Lin, Dai and Tibboel [2014] provedthe existence of tronquée PIIIsolutions whose pole-free sectorshave angular widths of
• π and 2π if γ = 1 and δ = −1, and
•3π
2and 3π if α = 1,γ = 0 and δ = −1.
McCoy, Tracy and Wu [1977] derived asymptotic formulaefor tronquée solutions with parameters α = −β = 2ν, γ = 1and δ = −1 and applied their results to the Ising model.
Tronquée PIII solutions
PIII :d2u
dz2=
1
u
(du
dz
)2− 1z
du
dz+αu2 + β
z+ γu3 +
δ
u
The MTW solutions: α = −β = 2ν, γ = 1, δ = −1
u(x) ∼ 1− λΓ(ν + 12
)2−2νx−ν−(1/2)e−2x, x→∞
u(x) ∼ Bxσ, x→ 0+
σ(λ) =2
πarcsin(πλ) ,
B(σ, ν) = 2−2σΓ2(12(1− σ)
)Γ(12(1 + σ) + ν
)Γ2(12(1 + σ)
)Γ(12(1− σ) + ν
)−1π< λ <
1
π
u(x; ν,−λ) = 1u(x; ν, λ)
Tronquée PIII solutions
PIII :d2u
dz2=
1
u
(du
dz
)2− 1z
du
dz+αu2 + β
z+ γu3 +
δ
u
The MTW solutions: α = −β = 2ν, γ = 1, δ = −1
u(x) ∼ 1− λΓ(ν + 12
)2−2νx−ν−(1/2)e−2x, x→∞
u(x) ∼ Bxσ, x→ 0+, −1π< λ <
1
π
Tronquée PIII solutions
PIII :d2u
dz2=
1
u
(du
dz
)2− 1z
du
dz+αu2 + β
z+ γu3 +
δ
u
The MTW solutions: α = −β = 2ν, γ = 1, δ = −1
u(x) ∼ 1− λΓ(ν + 12
)2−2νx−ν−(1/2)e−2x, x→∞
u(x) ∼ Bxσ, x→ 0+, −1π< λ <
1
π
Tronquée PIII solutions
PIII :d2u
dz2=
1
u
(du
dz
)2− 1z
du
dz+αu2 + β
z+ γu3 +
δ
u
The MTW solutions: α = −β = 2ν, γ = 1, δ = −1
u(x) ∼ 1− λΓ(ν + 12
)2−2νx−ν−(1/2)e−2x, x→∞
u(x) ∼ Bxσ, x→ 0+, −1π< λ <
1
π
Tronquée PIII solutions
PIII :d2u
dz2=
1
u
(du
dz
)2− 1z
du
dz+αu2 + β
z+ γu3 +
δ
u
The MTW solutions: α = −β = 2ν, γ = 1, δ = −1
u(x) ∼ 1− λΓ(ν + 12
)2−2νx−ν−(1/2)e−2x, x→∞
u(x) ∼ Bxσ, x→ 0+, −1π< λ <
1
π
Tronquée PIII solutions
PIII :d2u
dz2=
1
u
(du
dz
)2− 1z
du
dz+αu2 + β
z+ γu3 +
δ
u
The MTW solutions: α = −β = 2ν, γ = 1, δ = −1
u(x) ∼ 1− λΓ(ν + 12
)2−2νx−ν−(1/2)e−2x, x→∞
u(x) ∼ Bxσ, x→ 0+, −1π< λ <
1
π
Tronquée PIII solutions
PIII :d2u
dz2=
1
u
(du
dz
)2− 1z
du
dz+αu2 + β
z+ γu3 +
δ
u
The MTW solutions: α = −β = 2ν, γ = 1, δ = −1
u(x) ∼ 1− λΓ(ν + 12
)2−2νx−ν−(1/2)e−2x, x→∞
u(x) ∼ Bxσ, x→ 0+, −1π< λ <
1
π
Tronquée PIII solutions
PIII :d2u
dz2=
1
u
(du
dz
)2− 1z
du
dz+αu2 + β
z+ γu3 +
δ
u
The MTW solutions: α = −β = 2ν, γ = 1, δ = −1
u(x) ∼ 1− λΓ(ν + 12
)2−2νx−ν−(1/2)e−2x, x→∞
u(x) ∼ Bxσ, x→ 0+, −1π< λ <
1
π
Tronquée PIII solutions
PIII :d2u
dz2=
1
u
(du
dz
)2− 1z
du
dz+αu2 + β
z+ γu3 +
δ
u
The MTW solutions: α = −β = 2ν, γ = 1, δ = −1
u(x) ∼ 1− λΓ(ν + 12
)2−2νx−ν−(1/2)e−2x, x→∞
u(x) ∼ Bxσ, x→ 0+, −1π< λ <
1
π
Tronquée PIII solutions
PIII :d2u
dz2=
1
u
(du
dz
)2− 1z
du
dz+αu2 + β
z+ γu3 +
δ
u
The MTW solutions: α = −β = 2ν, γ = 1, δ = −1
u(x) ∼ 1− λΓ(ν + 12
)2−2νx−ν−(1/2)e−2x, x→∞
u(x) ∼ Bxσ, x→ 0+, −1π< λ <
1
π
Tronquée PIII solutions
PIII :d2u
dz2=
1
u
(du
dz
)2− 1z
du
dz+αu2 + β
z+ γu3 +
δ
u
The MTW solutions: α = −β = 2ν, γ = 1, δ = −1
u(x) ∼ 1− λΓ(ν + 12
)2−2νx−ν−(1/2)e−2x, x→∞
u(x) ∼ Bxσ, x→ 0+, −1π< λ <
1
π
Tronquée PIII solutions
PIII :d2u
dz2=
1
u
(du
dz
)2− 1z
du
dz+αu2 + β
z+ γu3 +
δ
u
The MTW solutions: α = −β = 2ν, γ = 1, δ = −1
u(x) ∼ 1− λΓ(ν + 12
)2−2νx−ν−(1/2)e−2x, x→∞
u(x) ∼ Bxσ, x→ 0+, −1π< λ <
1
π
Tronquée PIII solutions
PIII :d2u
dz2=
1
u
(du
dz
)2− 1z
du
dz+αu2 + β
z+ γu3 +
δ
u
The MTW solutions: α = −β = 2ν, γ = 1, δ = −1
u(x) ∼ 1− λΓ(ν + 12
)2−2νx−ν−(1/2)e−2x, x→∞
u(x) ∼ Bxσ, x→ 0+, −1π< λ <
1
π
Tronquée PIII solutions
PIII :d2u
dz2=
1
u
(du
dz
)2− 1z
du
dz+αu2 + β
z+ γu3 +
δ
u
The MTW solutions: α = −β = 2ν, γ = 1, δ = −1
u(x) ∼ 1− λΓ(ν + 12
)2−2νx−ν−(1/2)e−2x, x→∞
u(x) ∼ Bxσ, x→ 0+, −1π< λ <
1
π
Tronquée PIII solutions
PIII :d2u
dz2=
1
u
(du
dz
)2− 1z
du
dz+αu2 + β
z+ γu3 +
δ
u
The MTW solutions: α = −β = 2ν, γ = 1, δ = −1
u(x) ∼ 1− λΓ(ν + 12
)2−2νx−ν−(1/2)e−2x, x→∞
u(x) ∼ Bxσ, x→ 0+, −1π< λ <
1
π
Tronquée PIII solutions
PIII :d2u
dz2=
1
u
(du
dz
)2− 1z
du
dz+αu2 + β
z+ γu3 +
δ
u
The MTW solutions: α = −β = 2ν, γ = 1, δ = −1
u(x) ∼ 1− λΓ(ν + 12
)2−2νx−ν−(1/2)e−2x, x→∞
u(x) ∼ Bxσ, x→ 0+, −1π< λ <
1
π
Tronquée PIII solutions
PIII :d2u
dz2=
1
u
(du
dz
)2− 1z
du
dz+αu2 + β
z+ γu3 +
δ
u
The MTW solutions: α = −β = 2ν, γ = 1, δ = −1
u(x) ∼ 1− λΓ(ν + 12
)2−2νx−ν−(1/2)e−2x, x→∞
u(x) ∼ Bxσ, x→ 0+, −1π< λ <
1
π
Tronquée PIII solutions
PIII :d2u
dz2=
1
u
(du
dz
)2− 1z
du
dz+αu2 + β
z+ γu3 +
δ
u
The MTW solutions: α = −β = 2ν, γ = 1, δ = −1
u(x) ∼ 1− λΓ(ν + 12
)2−2νx−ν−(1/2)e−2x, x→∞
u(x) ∼ Bxσ, x→ 0+, −1π< λ <
1
π
Tronquée PIII solutions
PIII :d2u
dz2=
1
u
(du
dz
)2− 1z
du
dz+αu2 + β
z+ γu3 +
δ
u
The MTW solutions: α = −β = 2ν, γ = 1, δ = −1
u(x) ∼ 1− λΓ(ν + 12
)2−2νx−ν−(1/2)e−2x, x→∞
u(x) ∼ Bxσ, x→ 0+, −1π< λ <
1
π
Tronquée PIII solutions
PIII :d2u
dz2=
1
u
(du
dz
)2− 1z
du
dz+αu2 + β
z+ γu3 +
δ
u
The MTW solutions: α = −β = 2ν, γ = 1, δ = −1
u(x) ∼ 1− λΓ(ν + 12
)2−2νx−ν−(1/2)e−2x, x→∞
u(x) ∼ Bxσ, x→ 0+, −1π< λ <
1
π
Tronquée PIII solutions
PIII :d2u
dz2=
1
u
(du
dz
)2− 1z
du
dz+αu2 + β
z+ γu3 +
δ
u
The MTW solutions: α = −β = 2ν, γ = 1, δ = −1
u(x) ∼ 1− λΓ(ν + 12
)2−2νx−ν−(1/2)e−2x, x→∞
u(x) ∼ Bxσ, x→ 0+, −1π< λ <
1
π
Tronquée PIII solutions
PIII :d2u
dz2=
1
u
(du
dz
)2− 1z
du
dz+αu2 + β
z+ γu3 +
δ
u
The MTW solutions: α = −β = 2ν, γ = 1, δ = −1
u(x) ∼ 1− λΓ(ν + 12
)2−2νx−ν−(1/2)e−2x, x→∞
u(x) ∼ Bxσ, x→ 0+, −1π< λ <
1
π
Tronquée PIII solutionsThe 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− λΓ(ν + 12
)2−2νx−ν−(1/2)e−2x, x→∞
Redu(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)]
Tronquée PIII solutionsThe 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− λΓ(ν + 12
)2−2νx−ν−(1/2)e−2x, x→∞
Redu(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)]
Tronquée PIII solutionsThe 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− λΓ(ν + 12
)2−2νx−ν−(1/2)e−2x, x→∞
Redu(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)]
Tronquée PIII solutionsThe 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− λΓ(ν + 12
)2−2νx−ν−(1/2)e−2x, x→∞
Redu(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)]
Tronquée PIII solutionsThe 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− λΓ(ν + 12
)2−2νx−ν−(1/2)e−2x, x→∞
Redu(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)]
Tronquée PIII solutionsThe 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− λΓ(ν + 12
)2−2νx−ν−(1/2)e−2x, x→∞
Redu(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)]
Tronquée PIII solutionsThe 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− λΓ(ν + 12
)2−2νx−ν−(1/2)e−2x, x→∞
Redu(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)]
Tronquée PIII solutionsThe 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− λΓ(ν + 12
)2−2νx−ν−(1/2)e−2x, x→∞
Redu(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)]
Tronquée PIII solutionsThe 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− λΓ(ν + 12
)2−2νx−ν−(1/2)e−2x, x→∞
Redu(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)]
Tronquée PIII solutionsThe 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− λΓ(ν + 12
)2−2νx−ν−(1/2)e−2x, x→∞
Redu(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)]
Tronquée PIII solutionsThe 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− λΓ(ν + 12
)2−2νx−ν−(1/2)e−2x, x→∞
Redu(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)]
Tronquée PIII solutionsThe 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− λΓ(ν + 12
)2−2νx−ν−(1/2)e−2x, x→∞
Redu(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)]
Tronquée PIII solutionsThe 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− λΓ(ν + 12
)2−2νx−ν−(1/2)e−2x, x→∞
Redu(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)]
Tronquée PIII solutions
As λ is varied, multiple tronquée solutions occur on sheetsother than the main sheet of the MTW solutions
Lin, Dai and Tibboel [2014] proved the existence of tronquéePIII solutions whose pole-free sectors have angular widthsof
• π and 2π if γ = 1 and δ = −1, and
•3π
2and 3π if α = 1,γ = 0 and δ = −1.
Tronquée PIII solutions
Tronquée PIII solutions
Tronquée PIII solutions
Tronquée PIII solutions
Tronquée PIII solutions
Tronquée PIII solutions
Tronquée PIII solutions
Tronquée PIII solutions
Tronquée PIII solutions
Tronquée PIII solutions
Tronquée PIII solutions
Tronquée PIII solutions
Tronquée PIII solutions
Tronquée PIII solutions
Tronquée PIII solutions
Tronquée PIII solutions
Tronquée PIII solutions
Tronquée PIII solutions
Tronquée PIII solutions
Tronquée PIII solutionsAs λ is varied, multiple tronquée solutions occur on sheets other thanthe main sheet of the MTW solutions, u(z)
On the region {z ∈ C | π ≤ arg z ≤ 9π} we have found
• 16 tronquée solutions if ν 6∈ 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
Tronquée PIII solutionsAs λ is varied, multiple tronquée solutions occur on sheets other thanthe main sheet of the MTW solutions, u(z)
On the region {z ∈ C | π ≤ arg z ≤ 9π} we have found
• 16 tronquée solutions if ν 6∈ Z.
Tronquée PIII solutionsAs λ is varied, multiple tronquée solutions occur on sheets other thanthe main sheet of the MTW solutions, u(z)
On the region {z ∈ C | π ≤ arg z ≤ 9π} we have found
• 16 tronquée solutions if ν 6∈ Z.
Tronquée PIII solutionsAs λ is varied, multiple tronquée solutions occur on sheets other thanthe main sheet of the MTW solutions, u(z)
On the region {z ∈ C | π ≤ arg z ≤ 9π} we have found
• 16 tronquée solutions if ν 6∈ 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
Tronquée PIII solutionsAs λ is varied, multiple tronquée solutions occur on sheets other thanthe main sheet of the MTW solutions, u(z)
On the region {z ∈ C | π ≤ arg z ≤ 9π} we have found
• 16 tronquée solutions if ν 6∈ Z.
Tronquée PIII solutionsAs λ is varied, multiple tronquée solutions occur on sheets other thanthe main sheet of the MTW solutions, u(z)
On the region {z ∈ C | π ≤ arg z ≤ 9π} we have found
• 16 tronquée solutions if ν 6∈ Z.
Tronquée PIII solutionsAs λ is varied, multiple tronquée solutions occur on sheets other thanthe main sheet of the MTW solutions, u(z)
On the region {z ∈ C | π ≤ arg z ≤ 9π} we have found
• 16 tronquée solutions if ν 6∈ 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
Tronquée PIII solutionsAs λ is varied, multiple tronquée solutions occur on sheets other thanthe main sheet of the MTW solutions, u(z)
On the region {z ∈ C | π ≤ arg z ≤ 9π} we have found
• 16 tronquée solutions if ν 6∈ Z.
Tronquée PIII solutionsAs λ is varied, multiple tronquée solutions occur on sheets other thanthe main sheet of the MTW solutions, u(z)
On the region {z ∈ C | π ≤ arg z ≤ 9π} we have found
• 16 tronquée solutions if ν 6∈ Z.
Tronquée PIII solutionsAs λ is varied, multiple tronquée solutions occur on sheets other thanthe main sheet of the MTW solutions, u(z)
On the region {z ∈ C | π ≤ arg z ≤ 9π} we have found
• 16 tronquée solutions if ν 6∈ 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
Tronquée PIII solutionsAs λ is varied, multiple tronquée solutions occur on sheets other thanthe main sheet of the MTW solutions, u(z)
On the region {z ∈ C | π ≤ arg z ≤ 9π} we have found
• 16 tronquée solutions if ν 6∈ Z.
Tronquée PIII solutionsAs λ is varied, multiple tronquée solutions occur on sheets other thanthe main sheet of the MTW solutions, u(z)
On the region {z ∈ C | π ≤ arg z ≤ 9π} we have found
• 16 tronquée solutions if ν 6∈ Z.
Tronquée PIII solutionsAs λ is varied, multiple tronquée solutions occur on sheets other thanthe main sheet of the MTW solutions, u(z)
On the region {z ∈ C | π ≤ arg z ≤ 9π} we have found
• 16 tronquée solutions if ν 6∈ 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
Tronquée PIII solutionsAs λ is varied, multiple tronquée solutions occur on sheets other thanthe main sheet of the MTW solutions, u(z)
On the region {z ∈ C | π ≤ arg z ≤ 9π} we have found
• 16 tronquée solutions if ν 6∈ Z.
Tronquée PIII solutionsAs λ is varied, multiple tronquée solutions occur on sheets other thanthe main sheet of the MTW solutions, u(z)
On the region {z ∈ C | π ≤ arg z ≤ 9π} we have found
• 16 tronquée solutions if ν 6∈ Z.
Tronquée PIII solutionsAs λ is varied, multiple tronquée solutions occur on sheets other thanthe main sheet of the MTW solutions, u(z)
On the region {z ∈ C | π ≤ arg z ≤ 9π} we have found
• 16 tronquée solutions if ν 6∈ 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
Tronquée PIII solutionsAs λ is varied, multiple tronquée solutions occur on sheets other thanthe main sheet of the MTW solutions, u(z)
On the region {z ∈ C | π ≤ arg z ≤ 9π} we have found
• 16 tronquée solutions if ν 6∈ Z.
Tronquée PIII solutionsAs λ is varied, multiple tronquée solutions occur on sheets other thanthe main sheet of the MTW solutions, u(z)
On the region {z ∈ C | π ≤ arg z ≤ 9π} we have found
• 16 tronquée solutions if ν 6∈ Z.
Tronquée PIII solutionsAs λ is varied, multiple tronquée solutions occur on sheets other thanthe main sheet of the MTW solutions, u(z)
On the region {z ∈ C | π ≤ arg z ≤ 9π} we have found
• 16 tronquée solutions if ν 6∈ Z.
Tronquée PIII solutionsAs λ is varied, multiple tronquée solutions occur on sheets other thanthe main sheet of the MTW solutions, u(z)
On the region {z ∈ C | π ≤ arg z ≤ 9π} we have found
• 16 tronquée solutions if ν 6∈ Z.
Tronquée PIII solutionsAs λ is varied, multiple tronquée solutions occur on sheets other thanthe main sheet of the MTW solutions, u(z)
On the region {z ∈ C | π ≤ arg z ≤ 9π} we have found
• 16 tronquée solutions if ν 6∈ Z.
Tronquée PIII solutionsAs λ is varied, multiple tronquée solutions occur on sheets other thanthe main sheet of the MTW solutions, u(z)
On the region {z ∈ C | π ≤ arg z ≤ 9π} we have found
• 16 tronquée solutions if ν 6∈ Z.
Tronquée PIII solutionsAs λ is varied, multiple tronquée solutions occur on sheets other thanthe main sheet of the MTW solutions, u(z)
On the region {z ∈ C | π ≤ arg z ≤ 9π} we have found
• 16 tronquée solutions if ν 6∈ Z.
Tronquée PIII solutionsAs λ is varied, multiple tronquée solutions occur on sheets other thanthe main sheet of the MTW solutions, u(z)
On the region {z ∈ C | π ≤ arg z ≤ 9π} we have found
• 16 tronquée solutions if ν 6∈ Z.
Tronquée PIII solutionsAs λ is varied, multiple tronquée solutions occur on sheets other thanthe main sheet of the MTW solutions, u(z)
On the region {z ∈ C | π ≤ arg z ≤ 9π} we have found
• 16 tronquée solutions if ν 6∈ Z.
Tronquée PIII solutionsAs λ is varied, multiple tronquée solutions occur on sheets other thanthe main sheet of the MTW solutions, u(z)
On the region {z ∈ C | π ≤ arg z ≤ 9π} we have found
• 16 tronquée solutions if ν 6∈ Z.
Tronquée PIII solutionsAs λ is varied, multiple tronquée solutions occur on sheets other thanthe main sheet of the MTW solutions, u(z)
On the region {z ∈ C | π ≤ arg z ≤ 9π} we have found
• 16 tronquée solutions if ν 6∈ Z.
Tronquée PIII solutionsAs λ is varied, multiple tronquée solutions occur on sheets other thanthe main sheet of the MTW solutions, u(z)
On the region {z ∈ C | π ≤ arg z ≤ 9π} we have found
• 16 tronquée solutions if ν 6∈ Z.
Tronquée PIII solutionsAs λ is varied, multiple tronquée solutions occur on sheets other thanthe main sheet of the MTW solutions, u(z)
On the region {z ∈ C | π ≤ arg z ≤ 9π} we have found
• 16 tronquée solutions if ν 6∈ Z.
Tronquée PIII solutionsAs λ is varied, multiple tronquée solutions occur on sheets other thanthe main sheet of the MTW solutions, u(z)
On the region {z ∈ C | π ≤ arg z ≤ 9π} we have found
• 16 tronquée solutions if ν 6∈ Z.
Tronquée PIII solutionsAs λ is varied, multiple tronquée solutions occur on sheets other thanthe main sheet of the MTW solutions, u(z)
On the region {z ∈ C | π ≤ arg z ≤ 9π} we have found
• 16 tronquée solutions if ν 6∈ Z.
Tronquée PIII solutionsAs λ is varied, multiple tronquée solutions occur on sheets other thanthe main sheet of the MTW solutions, u(z)
On the region {z ∈ C | π ≤ arg z ≤ 9π} we have found
• 16 tronquée solutions if ν 6∈ Z.
Tronquée PIII solutionsAs λ is varied, multiple tronquée solutions occur on sheets other thanthe main sheet of the MTW solutions, u(z)
On the region {z ∈ C | π ≤ arg z ≤ 9π} we have found
• 16 tronquée solutions if ν 6∈ 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
Tronquée PIII solutionsAs λ is varied, multiple tronquée solutions occur on sheets other thanthe main sheet of the MTW solutions, u(z)
On the region {z ∈ C | π ≤ arg z ≤ 9π} we have found
• 72 tronquée 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
Tronquée PIII solutionsAs λ is varied, multiple tronquée solutions occur on sheets other thanthe main sheet of the MTW solutions, u(z)
On the region {z ∈ C | π ≤ arg z ≤ 9π} we have found
• 72 tronquée 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 tronquée PIII solutions with α = 1,γ = 0 andδ = −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− 1z
du
dz+
(u− 1)2
z2
(αu +
β
u
)+ γ
u
z+ δ
u(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 −→ Padé approximation at z + heiθ
Stage 1: pole avoidance on a coarse grid
Stage 2: compute the solution on a fine grid
Single-valued Painlevé 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 −→ Padé approximation at z + heiθ
Stage 1: pole avoidance on a coarse grid
Stage 2: compute the solution on a fine grid
Single-valued Painlevé transcendents: PI, F & W [2011] ;PII, F & W [2014, 2015] ; PIV , Reeger & F [2013, 2014]• Multivaluedness
How to compute PIII?Second challenge: MultivaluednessMake the paths run in the right direction
−10 −5 0 5 10−4
−2
0
2
4
1st sheet
How to compute PIII?Second challenge: MultivaluednessMake the paths run in the right direction
−10 −5 0 5 10−4
−2
0
2
4
1st sheet
−10 −8 −6 −4 −2 0 2−4
−2
0
2
4
counterclockwise sheet
How to compute PIII?
Second challenge: Multivaluedness
PIII :d2u
dz2=
1
u
(du
dz
)2− 1z
du
dz+αu2 + β
z+ γu3 +
δ
u
Let z = eζ/2, u(z) = e−ζ/2w(ζ) in PIII , then
P̃III :d2w
dζ2=
1
w
(dw
dζ
)2+
1
4
(αw2 + γw3 + βeζ +
δe2ζ
w
),
and all solutions of P̃III are single-valued
Hinkkanen & Laine [2001]. (z = 0⇒ ζ = −∞)
How to compute PIII?
z = eζ/2, u(z) = e−ζ/2w(ζ)
Tronquée PIII solutions
u(x) ∼ 1− λΓ(ν + 12
)2−2νx−ν−(1/2)e−2x, x→∞
u(x) ∼ Bxσ, x→ 0+, −1π< λ <
1
π
Tronquée PIII solutions
MTW solutions: u(x) ∼ 1− λΓ(ν + 12
)2−2νx−ν−(1/2)e−2x, x→∞
Limiting solution: u(x) ∼ 1− kx−ν−(1/2)e−2x, x→∞
Tronquée PIII solutions
MTW solutions: u(x) ∼ 1− λΓ(ν + 12
)2−2νx−ν−(1/2)e−2x, x→∞
Limiting solution: u(x) ∼ 1− kx−ν−(1/2)e−2x, x→∞