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TRANSCRIPT
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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
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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→∞