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

Tronqu´ ee P III solutions The MTW solutions: α = − β = 2 ν , γ = 1 , δ = − 1 . For the case λ > 1 π , let λ = cosh( πµ ) , µ > 0 , then [ u ( x ; ν, − λ ) = 1 /u ( x ; ν, λ ) ] π ν + 1 2 − 2 ν x − ν − (1 / 2) e − 2 x , � � Green u ( x ) ∼ 1 − λ Γ x → ∞ 2 � ν � u ( x ) 1 x → 0 + Red ∼ µ (1 − cos [ φ ( x, ν, µ )]) + sin [ φ ( x, ν, µ )] , x 2 µ φ ( x, ν, µ ) = 2 µ ln( x/ 4) − 4 arg [Γ( µi )] + 2 arg [Γ( ν + µi )]

Tronqu´ ee P III solutions The MTW solutions: α = − β = 2 ν , γ = 1 , δ = − 1 . For the case λ > 1 π , let λ = cosh( πµ ) , µ > 0 , then [ u ( x ; ν, − λ ) = 1 /u ( x ; ν, λ ) ] π 2 − 2 ν x − ν − (1 / 2) e − 2 x , ν + 1 � � Green u ( x ) ∼ 1 − λ Γ x → ∞ 2 � ν � u ( x ) 1 x → 0 + ∼ µ (1 − cos [ φ ( x, ν, µ )]) + sin [ φ ( x, ν, µ )] , Red x 2 µ φ ( x, ν, µ ) = 2 µ ln( x/ 4) − 4 arg [Γ( µi )] + 2 arg [Γ( ν + µi )]

Tronqu´ ee P III solutions The MTW solutions: α = − β = 2 ν , γ = 1 , δ = − 1 . For the case λ > 1 π , let λ = cosh( πµ ) , µ > 0 , then [ u ( x ; ν, − λ ) = 1 /u ( x ; ν, λ ) ] π ν + 1 2 − 2 ν x − ν − (1 / 2) e − 2 x , � � Green u ( x ) ∼ 1 − λ Γ x → ∞ 2 � ν � u ( x ) 1 x → 0 + Red ∼ µ (1 − cos [ φ ( x, ν, µ )]) + sin [ φ ( x, ν, µ )] , x 2 µ φ ( x, ν, µ ) = 2 µ ln( x/ 4) − 4 arg [Γ( µi )] + 2 arg [Γ( ν + µi )]

Tronqu´ ee P III solutions The MTW solutions: α = − β = 2 ν , γ = 1 , δ = − 1 . For the case λ > 1 π , let λ = cosh( πµ ) , µ > 0 , then [ u ( x ; ν, − λ ) = 1 /u ( x ; ν, λ ) ] π ν + 1 2 − 2 ν x − ν − (1 / 2) e − 2 x , � � Green u ( x ) ∼ 1 − λ Γ x → ∞ 2 � ν � u ( x ) 1 x → 0 + Red ∼ µ (1 − cos [ φ ( x, ν, µ )]) + sin [ φ ( x, ν, µ )] , x 2 µ φ ( x, ν, µ ) = 2 µ ln( x/ 4) − 4 arg [Γ( µi )] + 2 arg [Γ( ν + µi )]

Tronqu´ ee P III solutions The MTW solutions: α = − β = 2 ν , γ = 1 , δ = − 1 . For the case λ > 1 π , let λ = cosh( πµ ) , µ > 0 , then [ u ( x ; ν, − λ ) = 1 /u ( x ; ν, λ ) ] π ν + 1 2 − 2 ν x − ν − (1 / 2) e − 2 x , � � Green u ( x ) ∼ 1 − λ Γ x → ∞ 2 � ν � u ( x ) 1 x → 0 + Red ∼ µ (1 − cos [ φ ( x, ν, µ )]) + sin [ φ ( x, ν, µ )] , x 2 µ φ ( x, ν, µ ) = 2 µ ln( x/ 4) − 4 arg [Γ( µi )] + 2 arg [Γ( ν + µi )]

Tronqu´ ee P III solutions The MTW solutions: α = − β = 2 ν , γ = 1 , δ = − 1 . For the case λ > 1 π , let λ = cosh( πµ ) , µ > 0 , then [ u ( x ; ν, − λ ) = 1 /u ( x ; ν, λ ) ] π ν + 1 2 − 2 ν x − ν − (1 / 2) e − 2 x , � � Green u ( x ) ∼ 1 − λ Γ x → ∞ 2 � ν � u ( x ) 1 x → 0 + Red ∼ µ (1 − cos [ φ ( x, ν, µ )]) + sin [ φ ( x, ν, µ )] , x 2 µ φ ( x, ν, µ ) = 2 µ ln( x/ 4) − 4 arg [Γ( µi )] + 2 arg [Γ( ν + µi )]

Tronqu´ ee P III solutions The MTW solutions: α = − β = 2 ν , γ = 1 , δ = − 1 . For the case λ > 1 π , let λ = cosh( πµ ) , µ > 0 , then [ u ( x ; ν, − λ ) = 1 /u ( x ; ν, λ ) ] π ν + 1 2 − 2 ν x − ν − (1 / 2) e − 2 x , � � Green u ( x ) ∼ 1 − λ Γ x → ∞ 2 � ν � u ( x ) 1 x → 0 + Red ∼ µ (1 − cos [ φ ( x, ν, µ )]) + sin [ φ ( x, ν, µ )] , x 2 µ φ ( x, ν, µ ) = 2 µ ln( x/ 4) − 4 arg [Γ( µi )] + 2 arg [Γ( ν + µi )]

Tronqu´ ee P III solutions The MTW solutions: α = − β = 2 ν , γ = 1 , δ = − 1 . For the case λ > 1 π , let λ = cosh( πµ ) , µ > 0 , then [ u ( x ; ν, − λ ) = 1 /u ( x ; ν, λ ) ] π ν + 1 2 − 2 ν x − ν − (1 / 2) e − 2 x , � � Green u ( x ) ∼ 1 − λ Γ x → ∞ 2 � ν � u ( x ) 1 x → 0 + Red ∼ µ (1 − cos [ φ ( x, ν, µ )]) + sin [ φ ( x, ν, µ )] , x 2 µ φ ( x, ν, µ ) = 2 µ ln( x/ 4) − 4 arg [Γ( µi )] + 2 arg [Γ( ν + µi )]

Tronqu´ ee P III solutions The MTW solutions: α = − β = 2 ν , γ = 1 , δ = − 1 . For the case λ > 1 π , let λ = cosh( πµ ) , µ > 0 , then [ u ( x ; ν, − λ ) = 1 /u ( x ; ν, λ ) ] π ν + 1 2 − 2 ν x − ν − (1 / 2) e − 2 x , � � Green u ( x ) ∼ 1 − λ Γ x → ∞ 2 � ν � u ( x ) 1 x → 0 + Red ∼ µ (1 − cos [ φ ( x, ν, µ )]) + sin [ φ ( x, ν, µ )] , x 2 µ φ ( x, ν, µ ) = 2 µ ln( x/ 4) − 4 arg [Γ( µi )] + 2 arg [Γ( ν + µi )]

Tronqu´ ee P III solutions The MTW solutions: α = − β = 2 ν , γ = 1 , δ = − 1 . For the case λ > 1 π , let λ = cosh( πµ ) , µ > 0 , then [ u ( x ; ν, − λ ) = 1 /u ( x ; ν, λ ) ] π ν + 1 2 − 2 ν x − ν − (1 / 2) e − 2 x , � � Green u ( x ) ∼ 1 − λ Γ x → ∞ 2 � ν � u ( x ) 1 x → 0 + Red ∼ µ (1 − cos [ φ ( x, ν, µ )]) + sin [ φ ( x, ν, µ )] , x 2 µ φ ( x, ν, µ ) = 2 µ ln( x/ 4) − 4 arg [Γ( µi )] + 2 arg [Γ( ν + µi )]

Tronqu´ ee P III solutions The MTW solutions: α = − β = 2 ν , γ = 1 , δ = − 1 . For the case λ > 1 π , let λ = cosh( πµ ) , µ > 0 , then [ u ( x ; ν, − λ ) = 1 /u ( x ; ν, λ ) ] π ν + 1 2 − 2 ν x − ν − (1 / 2) e − 2 x , � � Green u ( x ) ∼ 1 − λ Γ x → ∞ 2 � ν � u ( x ) 1 x → 0 + Red ∼ µ (1 − cos [ φ ( x, ν, µ )]) + sin [ φ ( x, ν, µ )] , x 2 µ φ ( x, ν, µ ) = 2 µ ln( x/ 4) − 4 arg [Γ( µi )] + 2 arg [Γ( ν + µi )]

Tronqu´ ee P III solutions The MTW solutions: α = − β = 2 ν , γ = 1 , δ = − 1 . For the case λ > 1 π , let λ = cosh( πµ ) , µ > 0 , then [ u ( x ; ν, − λ ) = 1 /u ( x ; ν, λ ) ] π ν + 1 2 − 2 ν x − ν − (1 / 2) e − 2 x , � � Green u ( x ) ∼ 1 − λ Γ x → ∞ 2 � ν � u ( x ) 1 x → 0 + Red ∼ µ (1 − cos [ φ ( x, ν, µ )]) + sin [ φ ( x, ν, µ )] , x 2 µ φ ( x, ν, µ ) = 2 µ ln( x/ 4) − 4 arg [Γ( µi )] + 2 arg [Γ( ν + µi )]

Tronqu´ ee P III solutions The MTW solutions: α = − β = 2 ν , γ = 1 , δ = − 1 . For the case λ > 1 π , let λ = cosh( πµ ) , µ > 0 , then [ u ( x ; ν, − λ ) = 1 /u ( x ; ν, λ ) ] π ν + 1 2 − 2 ν x − ν − (1 / 2) e − 2 x , � � Green u ( x ) ∼ 1 − λ Γ x → ∞ 2 � ν � u ( x ) 1 x → 0 + Red ∼ µ (1 − cos [ φ ( x, ν, µ )]) + sin [ φ ( x, ν, µ )] , x 2 µ φ ( x, ν, µ ) = 2 µ ln( x/ 4) − 4 arg [Γ( µi )] + 2 arg [Γ( ν + µi )]

Tronqu´ ee P III solutions As λ is varied, multiple tronqu´ ee solutions occur on sheets other than the main sheet of the MTW solutions Lin, Dai and Tibboel [2014] proved the existence of tronqu´ ee P III solutions whose pole-free sectors have angular widths of • π and 2 π if γ = 1 and δ = − 1 , and • 3 π 2 and 3 π if α = 1 , γ = 0 and δ = − 1 .

Tronqu´ ee P III solutions

Tronqu´ ee P III solutions

Tronqu´ ee P III solutions

Tronqu´ ee P III solutions

Tronqu´ ee P III solutions

Tronqu´ ee P III solutions

Tronqu´ ee P III solutions

Tronqu´ ee P III solutions

Tronqu´ ee P III solutions

Tronqu´ ee P III solutions

Tronqu´ ee P III solutions

Tronqu´ ee P III solutions

Tronqu´ ee P III solutions

Tronqu´ ee P III solutions

Tronqu´ ee P III solutions

Tronqu´ ee P III solutions

Tronqu´ ee P III solutions

Tronqu´ ee P III solutions

Tronqu´ ee P III solutions

Tronqu´ ee P III 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 ≤ 9 π } we have found ee solutions if ν �∈ Z . They have the properties • 16 tronqu´ � k + 1 � � k + 3 � u ( z ) ∼ − 1 , | z | → ∞ , π < arg z < π 2 2 k = 8 , 6 , 4 , 2 k = 8 , 6 , 4 k = 8 , 6 k = 8 � � � � k + 1 k + 3 u ( z ) ∼ 1 , | z | → ∞ , π < arg z < π 2 2 k = 7 , 5 , 3 k = 7 , 5 k = 7 Conjecture: The sequences continue indefinitely

Tronqu´ ee P III 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 ≤ 9 π } we have found ee solutions if ν �∈ Z . • 16 tronqu´

Tronqu´ ee P III 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 ≤ 9 π } we have found ee solutions if ν �∈ Z . • 16 tronqu´

Tronqu´ ee P III 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 ≤ 9 π } we have found ee solutions if ν �∈ Z . They have the properties • 16 tronqu´ � k + 1 � � k + 3 � u ( z ) ∼ − 1 , | z | → ∞ , π < arg z < π 2 2 k = 8 , 6 , 4 , 2 k = 8 , 6 , 4 k = 8 , 6 k = 8 � � � � k + 1 k + 3 u ( z ) ∼ 1 , | z | → ∞ , π < arg z < π 2 2 k = 7 , 5 , 3 k = 7 , 5 k = 7 Conjecture: The sequences continue indefinitely

Tronqu´ ee P III 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 ≤ 9 π } we have found ee solutions if ν �∈ Z . • 16 tronqu´

Tronqu´ ee P III 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 ≤ 9 π } we have found ee solutions if ν �∈ Z . • 16 tronqu´

Tronqu´ ee P III 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 ≤ 9 π } we have found ee solutions if ν �∈ Z . They have the properties • 16 tronqu´ � k + 1 � � k + 3 � u ( z ) ∼ − 1 , | z | → ∞ , π < arg z < π 2 2 k = 8 , 6 , 4 , 2 k = 8 , 6 , 4 k = 8 , 6 k = 8 � � � � k + 1 k + 3 u ( z ) ∼ 1 , | z | → ∞ , π < arg z < π 2 2 k = 7 , 5 , 3 k = 7 , 5 k = 7 Conjecture: The sequences continue indefinitely

Tronqu´ ee P III 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 ≤ 9 π } we have found ee solutions if ν �∈ Z . • 16 tronqu´

Tronqu´ ee P III 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 ≤ 9 π } we have found ee solutions if ν �∈ Z . • 16 tronqu´

Tronqu´ ee P III 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 ≤ 9 π } we have found ee solutions if ν �∈ Z . They have the properties • 16 tronqu´ � k + 1 � � k + 3 � u ( z ) ∼ − 1 , | z | → ∞ , π < arg z < π 2 2 k = 8 , 6 , 4 , 2 k = 8 , 6 , 4 k = 8 , 6 k = 8 � � � � k + 1 k + 3 u ( z ) ∼ 1 , | z | → ∞ , π < arg z < π 2 2 k = 7 , 5 , 3 k = 7 , 5 k = 7 Conjecture: The sequences continue indefinitely

Tronqu´ ee P III 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 ≤ 9 π } we have found ee solutions if ν �∈ Z . • 16 tronqu´

Tronqu´ ee P III 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 ≤ 9 π } we have found ee solutions if ν �∈ Z . • 16 tronqu´

Tronqu´ ee P III 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 ≤ 9 π } we have found ee solutions if ν �∈ Z . They have the properties • 16 tronqu´ � k + 1 � � k + 3 � u ( z ) ∼ − 1 , | z | → ∞ , π < arg z < π 2 2 k = 8 , 6 , 4 , 2 k = 8 , 6 , 4 k = 8 , 6 k = 8 � � � � k + 1 k + 3 u ( z ) ∼ 1 , | z | → ∞ , π < arg z < π 2 2 k = 7 , 5 , 3 k = 7 , 5 k = 7 Conjecture: The sequences continue indefinitely

Tronqu´ ee P III 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 ≤ 9 π } we have found ee solutions if ν �∈ Z . • 16 tronqu´

Tronqu´ ee P III 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 ≤ 9 π } we have found ee solutions if ν �∈ Z . • 16 tronqu´

Tronqu´ ee P III 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 ≤ 9 π } we have found ee solutions if ν �∈ Z . They have the properties • 16 tronqu´ � k + 1 � � k + 3 � u ( z ) ∼ − 1 , | z | → ∞ , π < arg z < π 2 2 k = 8 , 6 , 4 , 2 k = 8 , 6 , 4 k = 8 , 6 k = 8 � � � � k + 1 k + 3 u ( z ) ∼ 1 , | z | → ∞ , π < arg z < π 2 2 k = 7 , 5 , 3 k = 7 , 5 k = 7 Conjecture: The sequences continue indefinitely

Tronqu´ ee P III 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 ≤ 9 π } we have found ee solutions if ν �∈ Z . • 16 tronqu´

Tronqu´ ee P III 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 ≤ 9 π } we have found ee solutions if ν �∈ Z . • 16 tronqu´

Tronqu´ ee P III 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 ≤ 9 π } we have found ee solutions if ν �∈ Z . • 16 tronqu´

Tronqu´ ee P III 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 ≤ 9 π } we have found ee solutions if ν �∈ Z . • 16 tronqu´

Tronqu´ ee P III 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 ≤ 9 π } we have found ee solutions if ν �∈ Z . • 16 tronqu´

Tronqu´ ee P III 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 ≤ 9 π } we have found ee solutions if ν �∈ Z . • 16 tronqu´

Tronqu´ ee P III 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 ≤ 9 π } we have found ee solutions if ν �∈ Z . • 16 tronqu´

Tronqu´ ee P III 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 ≤ 9 π } we have found ee solutions if ν �∈ Z . • 16 tronqu´

Tronqu´ ee P III 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 ≤ 9 π } we have found ee solutions if ν �∈ Z . • 16 tronqu´

Tronqu´ ee P III 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 ≤ 9 π } we have found ee solutions if ν �∈ Z . • 16 tronqu´

Tronqu´ ee P III 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 ≤ 9 π } we have found ee solutions if ν �∈ Z . • 16 tronqu´

Tronqu´ ee P III 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 ≤ 9 π } we have found ee solutions if ν �∈ Z . • 16 tronqu´

Tronqu´ ee P III 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 ≤ 9 π } we have found ee solutions if ν �∈ Z . • 16 tronqu´

Tronqu´ ee P III 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 ≤ 9 π } we have found ee solutions if ν �∈ Z . • 16 tronqu´

Tronqu´ ee P III 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 ≤ 9 π } we have found ee solutions if ν �∈ Z . • 16 tronqu´

Tronqu´ ee P III 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 ≤ 9 π } we have found ee solutions if ν �∈ Z . • 16 tronqu´

Tronqu´ ee P III 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 ≤ 9 π } we have found ee solutions if ν �∈ Z . They have the properties • 16 tronqu´ � k + 1 � � k + 3 � u ( z ) ∼ − 1 , | z | → ∞ , π < arg z < π 2 2 k = 8 , 6 , 4 , 2 k = 8 , 6 , 4 k = 8 , 6 k = 8 � � � � k + 1 k + 3 u ( z ) ∼ 1 , | z | → ∞ , π < arg z < π 2 2 k = 7 , 5 , 3 k = 7 , 5 k = 7 Conjecture: The sequences continue indefinitely

Tronqu´ ee P III 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 ≤ 9 π } we have found ee solutions if ν ∈ Z . They have the properties • 72 tronqu´ 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 P III 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 ≤ 9 π } we have found ee solutions if ν ∈ Z . They have the properties • 72 tronqu´ � � � � k + 1 k + 3 u ( z ) ∼ − 1 , | z | → ∞ , π < arg z < π 2 2 k = 8 , 7 , 6 , 5 , 4 , 3 , 2 , 1 k = 8 , 7 , 6 , 5 , 4 , 3 k = 8 , 7 , 6 , 5 k = 8 , 7 � � � � k + 1 k + 3 u ( z ) ∼ 1 , | z | → ∞ , π < arg z < π 2 2 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 P III solutions with α = 1 , γ = 0 and δ = − 1 • Survey single-valued P III solutions • Survey triply branched P III solutions • Extend computational method for P III to � 1 � 2 : d 2 u dz + ( u − 1) 2 1 � � du − 1 du � α u + β � + γ u z + δ u ( u + 1) P V dz 2 = 2 u + z 2 u − 1 dz z u u − 1 and � 2 : d 2 u � 1 � � du � 1 � � du � dz 2 = 1 1 1 1 1 P V I u + u − 1 + − z + z − 1 + 2 u − z dz u − z dz + u ( u − 1)( u − z ) � α + β z u 2 + γ z − 1 ( u − 1) 2 + δ z ( z − 1) � z 2 ( z − 1) 2 ( u − z ) 2

How to compute P III ? Challenges: • Moveable poles : the ‘Pole Field Solver’ (PFS), Fornberg & Weideman [2011] u ( z ) , u ′ ( z ) − e approximation at z + he iθ → Pad´ → Taylor coefficients − Stage 1: pole avoidance on a coarse grid Stage 2: compute the solution on a fine grid Single-valued Painlev´ e transcendents: P I , F & W [2011] ; P II , F & W [2014, 2015] ; P IV , Reeger & F [2013, 2014] • Multivaluedness

How to compute P III ? Challenges: • Moveable poles : the ‘Pole Field Solver’ (PFS), Fornberg & Weideman [2011] u ( z ) , u ′ ( z ) − e approximation at z + he iθ → Taylor coefficients − → Pad´ Stage 1: pole avoidance on a coarse grid Stage 2: compute the solution on a fine grid Single-valued Painlev´ e transcendents: P I , F & W [2011] ; P II , F & W [2014, 2015] ; P IV , Reeger & F [2013, 2014] • Multivaluedness

How to compute P III ? Second challenge: Multivaluedness Make the paths run in the right direction 4 2 0 −2 −4 −10 −5 0 5 10 1st sheet

How to compute P III ? Second challenge: Multivaluedness Make the paths run in the right direction 4 2 0 −2 −4 −10 −5 0 5 10 1st sheet 4 2 0 −2 −4 −10 −8 −6 −4 −2 0 2 counterclockwise sheet

How to compute P III ? Second challenge: Multivaluedness dz + α u 2 + β � 2 d 2 u � du dz 2 = 1 − 1 du + γ u 3 + δ P III : u dz z z u Let z = e ζ/ 2 , u ( z ) = e − ζ/ 2 w ( ζ ) in P III , then � 2 � � d 2 w α w 2 + γ w 3 + β e ζ + δ e 2 ζ � dw dζ 2 = 1 + 1 � P III : , w dζ 4 w and all solutions of � P III are single-valued Hinkkanen & Laine [2001] . ( z = 0 ⇒ ζ = −∞ )