When is a Stokes line not a Stokes line?
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- Tracy Harmon
- 6 years ago
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1 T H E U N I V E R S I T Y O H F E D I N B U R G When is a Stokes line not a Stokes line? C.J.Howls University of Southampton, UK A.B. Olde Daalhuis University of Edinburgh, UK Work supported by EPSRC, CWI, RIMS
2 Collaboration Chris Howls, Phil Langman (Southampton) Adri Olde Daalhuis (Edinburgh) Jon Chapman, David Mortimer (Oxford) John King, Richard Tew, Giles Body (Nottingham) EPSRC-funded research.
3 So far... We ve seen what exponential asymptotics is for. Extends range of validity of expansions. Increases numerical accuracy. Provides a systematic way to calculate Stokes constants.
4 This talk... Recall some basic ideas: Stokes lines and caustics. Introduce you to an apparent asymptotic paradox. Explain why our understanding of asymptotics is missing a key tool of analytic continuation.
5 Stokes phenomenon and caustics
6 Stokes phenomenon and caustics z(λ)
7 Stokes phenomenon and caustics z(λ) 2 Main Bow Supernumerary Fringes
8 !! Stokes phenomenon and caustics Ai(z)!!!! z
9 !! Stokes phenomenon and caustics Ai(z)!!!! z Convergent Series: Airy 1838
10 !! Stokes phenomenon and caustics Ai(z)!!!! z Divergent Asymptotic Series: Stokes 1864 Convergent Series: Airy 1838
11 !! Stokes phenomenon and caustics Ai(z)!!!! z Ai(z) sin(2 z 3/2 /3 + π/4) πz 1/4 Ai(z) e 2z3/2 /3 2 πz 1/4
12 ! Stokes phenomenon and caustics Divergence of asymptotics Ai(z)!!!! z Ai(z) sin(2 z 3/2 /3 + π/4) πz 1/4 Ai(z) e 2z3/2 /3 2 πz 1/4
13 ! Stokes phenomenon and caustics Divergence of asymptotics Ai(z)!!!! z Ai(z) sin(2 z 3/2 /3 + π/4) πz 1/4 Ai(z) e 2z3/2 /3 2 πz 1/4 Coalescence of two exponents as z 0
14 ! Stokes phenomenon and caustics Divergence of asymptotics Ai(z)!!!! z Full expansion Ai(z) sin(2 z 3/2 /3 + π/4) πz 1/4 Ai(z) e 2z3/2 /3 2 πz 1/4 a r (z) r=0 Coalescence of two exponents as z 0
15 ! Stokes phenomenon and caustics a r = Divergence of every term in asymptotics Ai(z) Γ(r) (Difference in exponents) Γ(r) ( 4 3 z3/2), r!!!! z Full expansion Ai(z) sin(2 z 3/2 /3 + π/4) πz 1/4 Ai(z) e 2z3/2 /3 2 πz 1/4 a r (z) r=0 Coalescence of two exponents as z 0
16 Stokes phenomenon and caustics z C 0 Second exponential switches on in complex z- plane, where first maximally dominates it in size: arg(z)= 2π/3.
17 Stokes phenomenon and caustics z C 0 e 2z3/2 /3 2 πz 1/4 Second exponential switches on in complex z- plane, where first maximally dominates it in size: arg(z)= 2π/3.
18 Stokes phenomenon and caustics z C e 2z3/2 /3 2 πz 1/4 0 Second exponential switches on in complex z- plane, where first maximally dominates it in size: arg(z)= 2π/3.
19 Stokes phenomenon and caustics z C e 2z3/2 /3 e2z3/2/3 2 + i πz1/4 2 πz 1/4 0 Second exponential switches on in complex z- plane, where first maximally dominates it in size: arg(z)= 2π/3.
20 Stokes phenomenon and caustics z C e 2z3/2 /3 e2z3/2/3 2 + i πz1/4 2 πz 1/4 0 Stokes constant Second exponential switches on in complex z- plane, where first maximally dominates it in size: arg(z)= 2π/3.
21 Stokes phenomenon and caustics z C e 2z3/2 /3 e2z3/2/3 2 + i πz1/4 2 πz 1/4 0 Stokes lines Second exponential switches on in complex z- plane, where first maximally dominates it in size: arg(z)= 2π/3.
22 Stokes phenomenon and caustics z C anti-stokes lines 0 Anti-Stokes lines, where real parts of exponentials are identical. Vital in nonlinear problems or for numerical integration.
23 Stokes phenomenon and caustics z C turning point/ caustic 0 Stokes-lines/anti-Stokes lines sprout from turning point/caustic and extend to infinity, or to another turning point or singularity, e.g...
24 Gravitational waves in Reissner-Nordström black hole Turning points c anti-stokes lines a b t 1 t 2 t 3 t 4 f e' r- r+ t 6 t 5 d e pole c' Complex r plane Stokes lines
25 Pearcey function ( ) = exp i t 4 + yt 2 + xt Ψ 2 x,y + ( ( )) A highly oscillatory problem...
26 Pearcey function
27 Pearcey function Stokes line Stokes line Number and type of contributing saddles caustics/turning points
28 Summary Stokes lines, anti-stokes lines, turning points/ caustics exist and play key role in asymptotic continuation of approximations. Stokes line, anti-stokes lines sprout from turning point or singularities Stokes line, anti-stokes lines run between turning points or singularities or run to infinity.
29 Summary End of talk? Until recently, pretty much yes... But... The activity of the Stokes lines themselves can be dynamically switched on and off!
30 New phenomenon Activity of Stokes line plays a significant role in wide classes of asymptotics problems involving a (non-asymptotic) parameter: Integrals, ODEs and PDE time-evolution problems (linear and nonlinear) Effect on highly oscillatory problems. We show how an exponential approach is vital to the explanation of activity of Stokes lines. Explained by introducing a new concept: Higher Order Stokes Phenomenon
31 Activity of Stokes lines Sudden change in activity of Stokes line noticed by Berk, Nevins and Roberts (1982) in third order WKB approach to: i d3 y dy + 3i dx3 dx + xy = 0 In complex x plane they noticed crossing of Stokes lines. From monodromy of solutions they noticed the need for a third Stokes line at each crossing point.
32 Activity of Stokes lines x-plane i d3 y dy + 3i dx3 dx + xy = 0 Im x Re x
33 Activity of Stokes lines x-plane i d3 y dy + 3i dx3 dx + xy = 0 y exp ( x ) i k j (ξ)dξ x T P Im x Re x
34 Activity of Stokes lines x-plane i d3 y dy + 3i dx3 dx + xy = 0 ( x ) y exp i k j (ξ)dξ x T P Im x k 3 j 3k j + x = 0 Re x
35 Activity of Stokes lines x-plane { } Turning points i d3 y dy + 3i dx3 dx + xy = 0 ( x ) y exp i k j (ξ)dξ x T P k 3 j 3k j + x = 0 k i (x T P ) = k j (x T P )
36 Activity of Stokes lines x-plane Stokes lines i d3 y dy + 3i dx3 dx + xy = 0 ( x ) y exp( i k j (ξ)dξ ) x T P k 3 j 3k j + x = 0 Im { x } i (k i k j ) dx x T P = 0
37 Activity of Stokes lines x-plane i d3 y dy + 3i dx3 dx + xy = 0 ( x ) y exp( i k j (ξ)dξ ) x T P k 3 j 3k j + x = 0 Im { x } i (k i k j ) dx x T P = 0
38 Activity of Stokes lines x-plane i d3 y dy + 3i dx3 dx + xy = 0 ( x ) y exp( i k j (ξ)dξ ) x T P k 3 j 3k j + x = 0 Im { x } i (k i k j ) dx x T P = 0
39 Activity of Stokes lines x-plane i d3 y dy + 3i dx3 dx + xy = 0 ( x ) y exp( i k j (ξ)dξ ) x T P k 3 j 3k j + x = 0 Im { x } i (k i k j ) dx x T P = 0
40 Activity of Stokes lines x-plane i d3 y dy + 3i dx3 dx + xy = 0 ( x ) y exp( i k j (ξ)dξ ) x T P k 3 j 3k j + x = 0 Im { x } i (k i k j ) dx x T P = 0 New Stokes line
41 Activity of Stokes lines x-plane i d3 y dy + 3i dx3 dx + xy = 0 ( x ) y exp( i k j (ξ)dξ ) x T P k 3 j 3k j + x = 0 Im New Stokes line j = 1, 2, 3 { x } i (k i k j ) dx x T P = 0
42 Activity of Stokes lines x-plane i d3 y dy + 3i dx3 dx + xy = 0 ( x ) y exp( i k j (ξ)dξ ) x T P k 3 j 3k j + x = 0 2 > 3 1 > 2 Im { x } i (k i k j ) dx x T P = 0 New Stokes line j = 1, 2, 3
43 Activity of Stokes lines x-plane i d3 y dy + 3i dx3 dx + xy = 0 ( x ) y exp( i k j (ξ)dξ ) x T P k 3 j 3k j + x = 0 2 > 3 1 > 3 1 > 2 Im { x } i (k i k j ) dx x T P = 0 New Stokes line j = 1, 2, 3
44 Activity of Stokes lines x-plane i d3 y dy + 3i dx3 dx + xy = 0 ( x ) y exp( i k j (ξ)dξ ) x T P k 3 j 3k j + x = 0 Im { x } i (k i k j ) dx x T P Why change in activity of Stokes line inside crossing point? = 0
45 Activity of Stokes lines x-plane i d3 y dy + 3i dx3 dx + xy = 0 ( x ) y exp( i k j (ξ)dξ ) x T P 1 > 3 k 3 j 3k j + x = 0 Im { x } i (k i k j ) dx x T P Why change in activity of Stokes line inside crossing point? = 0
46 Activity of Stokes lines x-plane i d3 y dy + 3i dx3 dx + xy = 0 ( x ) y exp( i k j (ξ)dξ ) x T P k 3 j 3k j + x = 0 Im { x } i (k i k j ) dx x T P Virtual turning point: 1 & 3 coalesce, but regular point. = 0
47 Activity of Stokes lines BNR explained in terms of somewhat complicated integral representations of the specific equation. Here we show that change in activity of a Stokes line is a general feature of asymptotic expansions involving 3 or more asymptotic contributions with parameters. Not confined to complex plane: real effects possible. Start explanation with an integral.
48 An integral with 3 saddles ( ) = exp f ( z;a) ε I ε;a V 2 V 1 ( ) where V 1 = e 3πi 8, V 2 = e πi 8 f ( z) = i( 1 4 z z2 + az) ε <<1
49 An integral with 3 saddles where Not quite Pearcey, but found in uniform asymptotics ( ) = exp f ( z;a) ε I ε;a V 2 V 1 ( ) V 1 = e 3πi 8, V 2 = e πi 8 f ( z) = i( 1 4 z z2 + az) ε <<1 Control Parameter
50 An integral with 3 saddles ( ) = exp f ( z;a) ε I ε;a V 2 V 1 ( ) Saddles: z 0, z 1, z 2 z n 3 + zn + a = 0, n = 0,1, 2
51 An integral with 3 saddles ( ) = exp f ( z;a) ε I ε;a V 2 V 1 ( ) Saddles: z 0, z 1, z 2 z n 3 + zn + a = 0, n = 0,1, 2 Expansion: ( ) ~ exp f n ( a) ε I ε;a ( ) subset a n ( ) T r f n ( a) f ( z n ;a) = 1 4 iz n( z n + 3a ) r ( ) ε r
52 An integral with 3 saddles ( ) = exp f ( z;a) ε I ε;a V 2 V 1 ( ) Saddles: z 0, z 1, z 2 z n 3 + zn + a = 0, n = 0,1, 2 Expansion: ( ) ~ exp f n ( a) ε I ε;a ( ) subset a n ( ) T r f n ( a) f ( z n ;a) = 1 4 iz n( z n + 3a ) r ( ) ε r Subset changes when: ( ) f m ( a) { a : ( f n a ) ε > 0} Stokes phenomenon
53 Integration plane a-plane Borel plane
54 Stokes phenomenon Integration plane a-plane Borel plane
55 Stokes phenomenon Stokes line Integration plane a-plane Borel plane
56 Stokes phenomenon Saddles connect Integration plane a-plane Borel plane
57 Stokes phenomenon Integration plane a-plane Corresponding singularities line up horizontally Borel plane
58 Integration plane 3 a-plane Borel plane
59 Integration plane 3 a-plane Borel plane
60 Stokes phenomenon a 5 Integration plane a 5 3 a-plane Borel plane
61 a 6 Integration plane a 6 3 a-plane Borel plane
62 Stokes phenomenon a 7 Integration plane a 7 3 a-plane Borel plane
63 Stokes phenomenon a 7 Integration plane a 7 3 a-plane Borel plane
64 No Stokes phenomenon a 7 Integration plane a 7 3 a-plane Borel plane
65 Connection is between saddles that are are currently not contributing a 7 Integration plane a 7 3 a-plane Borel plane
66 Connection is between saddles that are are currently not contributing Irrelevant Stokes line a 7 Integration plane a 7 3 a-plane Borel plane
67 a-plane a 7 3
68 Stokes crossing point: a regular point a-plane
69 Stokes crossing point: a regular point Three in a row a 7 3 a-plane Borel plane
70 Stokes crossing point: a regular point a 7 3 Stokes line a-plane Borel plane
71 Stokes crossing point: a regular point a 7 3 No Stokes line Stokes line a-plane Borel plane
72 Stokes crossing point: a regular point even though singularity 2 is crossing cut from singularity 0: signature of Stokes phenomenon! a 7 3 No Stokes line Stokes line a-plane Borel plane
73 Stokes crossing point: a regular point Stokes line stops at regular point!?! a 7 3 No Stokes line Stokes line a-plane Borel plane
74 Eh?
75 Eh? How can we explain this?
76 Eh? How can we explain this? Higher order Stokes phenomenon!
77 Higher order Stokes line Integration plane a-plane Borel plane
78 Higher order Stokes line Integration plane a-plane Collinearity of singularities Borel plane
79 Higher order Stokes line Nothing happening to saddles! Integration plane a-plane Collinearity of singularities Borel plane
80 What does happen at a higher order Stokes phenomenon? Ordinary Stokes phenomenon: Exponentially small contribution is switched on or off Stokes curves: Higher order Stokes phenomenon ( ) > 0 { a :ε 1 f 1 ( a) f 0 ( a) } Possibility of a Stokes phenomenon is switched on/off A Stokes multipler changes value Higher order Stokes curves: ( f 1 ( a) f 0 ( a) ) a : f 2 ( a) f 1 ( a) ( ) > 0
81 What does happen at a higher order Stokes phenomenon? Ordinary Stokes phenomenon: Exponentially small contribution is switched on or off Stokes curves: Higher order Stokes phenomenon ( ) > 0 { a :ε 1 f 1 ( a) f 0 ( a) } Possibility of a Stokes phenomenon is switched on/off A Stokes multipler changes value Higher order Stokes curves: ( f 1 ( a) f 0 ( a) ) a : f 2 ( a) f 1 ( a) ( ) > 0
82 What does happen at a higher order Stokes phenomenon? Ordinary Stokes phenomenon: Exponentially small contribution is switched on or off Stokes curves: Higher order Stokes phenomenon ( ) > 0 { a :ε 1 f 1 ( a) f 0 ( a) } Possibility of a Stokes phenomenon is switched on/off A Stokes multipler itself changes value Higher order Stokes curves: ( f 1 ( a) f 0 ( a) ) a : f 2 ( a) f 1 ( a) Condition for collinearity ( ) > 0
83 What does happen at a Stokes phenomenon? Subdominant singularity crosses integration contour from dominant singularity.
84 What does happen at a Stokes phenomenon? Subdominant singularity crosses integration contour from dominant singularity. Integration contour loops around subdominant singularity.
85 What does happen at a Stokes phenomenon? Subdominant singularity crosses integration contour from dominant singularity. Integration contour loops around subdominant singularity. SUBDOMINANT CONTRIBUTION ACQUIRED.
86 What does happen at a higher order Stokes phenomenon?
87 What does happen at a higher order Stokes phenomenon? 0 can t see 2: Stokes constant =0
88 What does happen at a higher order Stokes phenomenon? 0 can t see 2: Stokes constant =0 0, 1 & 2 collinear
89 What does happen at a higher order Stokes phenomenon? 0 can t see 2: Stokes constant =0 0, 1 & 2 collinear 0 sees 2: Stokes constant 0
90 What does happen at a higher order Stokes phenomenon? 0 can t see 2: Stokes constant =0 0, 1 & 2 colinear 0 sees 2: Stokes constant 0 NB: NO CHANGE IN FORM OF ORDINARY ASYMPTOTICS!
91 What does happen at a higher order Stokes phenomenon? Singularity generating dominant contribution
92 What does happen at a higher order Stokes phenomenon? Singularity generating dominant contribution is on different sheet to subdominant singularity NB: NO CHANGE IN FORM OF ORDINARY ASYMPTOTICS!
93 Frequency of Higher Order Stokes Stokes line: 2 exponents line up horizontally in Borel plane. f 1 f 2 f 1 F 12 >0 Higher order Stokes line: 3 exponents line up in any direction in Borel plane. Same number of constraints (1): f 2 f 3 F 12 /F 23 >0 Same codimension Just as common
94 Higher Order Stokes Phenomenon Higher order Stokes phenomenon can occur whenever 3 or more different asymptotic contributions/independent Borel plane singularities exist. Inhomogeneous second order linear odes Higher order linear homogeneous odes Second order nonlinear odes Single/multiple integrals PDEs...
95 Ordinary vs. Higher order Stokes ε real { a :ε 1 ( f 1 ( a) f 0 ( a) ) > 0} ε complex
96 Ordinary vs. Higher order Stokes ε real { a :ε 1 ( f 1 ( a) f 0 ( a) ) > 0} ε complex
97 Ordinary vs. Higher order Stokes ε real ε complex Ordinary Stokes dependent on asymptotic parameter.
98 Ordinary vs. Higher order Stokes a : ( f 1 ( a) f 0 ( a) ) ε real f 2 ( a) f 1 ( a) ε complex Higher order Stokes independent of asymptotic parameter! ( ) > 0
99 Link to Hyperasymptotics Hyperasymptotics previously used to find Stokes constants. If Stokes constants are changing... Hyper and higher order Stokes phenomenon intimately linked. Examples shows how HoSP can be crucial for deducing larger time behaviour in evolution equations.
100 PDE illustrative example Derive large time behaviour of: ( ) θ t θ x = ε 2 θ xxx x 2 x (, + ), t 0, ( ) = arctan x [ ) θ x,0 θ xx,θ x,θ bounded as x
101 PDE illustrative example Dominant balances θ x ~1 ( 1 + x 2 ) θ( x) ~ arctan x
102 PDE illustrative example Dominant balances θ x ~1 ( 1 + x 2 ) θ( x) ~ arctan x Satisfies initial condition
103 PDE illustrative example Dominant balances θ x ~1 ( 1 + x 2 ) θ( x) ~ arctan x Satisfies initial condition θ x ~ ε 2 θ xxx θ ~ Asin ( x ε) + Bcos( x ε ) + C
104 PDE illustrative example Dominant balances θ x ~1 ( 1 + x 2 ) θ( x) ~ arctan x Satisfies initial condition θ x ~ ε 2 θ xxx θ ~ Asin ( x ε) + Bcos( x ε ) + C Doesn t satisfy initial condition, not obviously present in solution.
105 PDE illustrative example Can solve PDE by Fourier integral (cheating!) u(x, t; ε) = I 1 (x, t; ε) + I 2 (x, t; ε) + I 3 (x; ε) + I 4 (x; ε) I 1 (x, t; ε) = I 3 (x; ε) = 0 0 iπp 2 1 p 2 e (p(1+ix)+ip(1 p )t)/ε dp iπ p(1 p 2 ) e p(1+ix)/ε dp Endpoint, saddle point, pole asymptotics. BUT can derive same results directly from expansion coefficients.
106 PDE illustrative example Base (hyper)asymptotics on
107 PDE illustrative example Base (hyper)asymptotics on ( ) ~ arctan x + a n x,t θ x,t r=1 n ( )ε n
108 PDE illustrative example Base (hyper)asymptotics on ( ) ~ arctan x + a n x,t θ x,t r=1 ( )ε n Large time behaviour evolves from sum, via hyperasymptotics: 3 different types of contribution. n
109 PDE illustrative example θ s θ e ( x,t) = arctan x + 4πε 2 x + t 1 + x + t Always contributing ( x,t) = 2 Re ( 3t )1 4 π 3 2 i + t + x ( i + 2t x) θ p Sometimes contributing ( x,t ) = 2π 2 e 1 ε cos x ε ( ) ( ( ) 2 ) 2 x ( ) 1 4 exp ( 1 + x 2 ) 2 + O ε 3 2i ( i + t + x ) tε ( ) i π 4 Sub-sub dominant near t=0, but dominates large t behaviour!
110 PDE illustrative example θ s θ e ( x,t) = arctan x + 4πε 2 x + t 1 + x + t Always contributing ( x,t) = 2 Re ( 3t )1 4 π 3 2 i + t + x ( i + 2t x) θ p Sometimes contributing ( x,t ) = 2π 2 e 1 ε cos x ε ( ) ( ( ) 2 ) 2 x ( ) 1 4 exp ( 1 + x 2 ) 2 + O ε 3 2i ( i + t + x ) tε ( ) i π 4 Sub-sub dominant near t=0, but dominates large t behaviour!
111 PDE illustrative example θ s θ e ( x,t) = arctan x + 4πε 2 x + t 1 + x + t Always contributing ( x,t) = 2 Re ( 3t )1 4 π 3 2 i + t + x ( i + 2t x) θ p Sometimes contributing ( x,t ) = 2π 2 e 1 ε cos x ε ( ) ( ( ) 2 ) 2 x ( ) 1 4 exp ( 1 + x 2 ) 2 + O ε 3 2i ( i + t + x ) tε ( ) i π 4 Sub-sub dominant near t=0, but dominates large t behaviour!
112 Higher Order Stokes Phenomenon e+p+s e+s? e+p+s Contradiction: no active Stokes line on x = 0 for t > A e+p e+p A e e+s
113 Higher Order Stokes Phenomenon Switches off Stokes line ep Higher order Stokes line
114 Higher Order Stokes Phenomenon: Hyper explanation Ordinary Stokes phenomenon
115 Higher Order Stokes Phenomenon: Hyper explanation Ordinary Stokes phenomenon Birth of exponentially small terms across curve (surface) in (x,t) plane: switches on/off terms. Change in number of contributions to full asymptotic expansion.
116 Higher Order Stokes Phenomenon: Hyper explanation Ordinary Stokes phenomenon Birth of exponentially small terms across curve (surface) in (x,t) plane: switches on/off terms. Change in number of contributions to full asymptotic expansion. Higher order Stokes phenomenon
117 Higher Order Stokes Phenomenon: Hyper explanation Ordinary Stokes phenomenon Birth of exponentially small terms across curve (surface) in (x,t) plane: switches on/off terms. Change in number of contributions to full asymptotic expansion. Higher order Stokes phenomenon Change in value of Stokes constants across curve (surface) in (x,t) plane: switches on/off Stokes lines. Change in number of remainder terms.
118 Higher Order Stokes Phenomenon: Hyper explanation ( ( N y (1) (ɛ) e f1/ɛ ɛ µ (0) 1 n +ɛ N (0) 1 r=0 K 12 2πi K 13 2πi K 14 2πi a (1) r ɛ r N (1) 2 1 a (2) s=0 N (1) 3 1 a (3) s=0 N (1) 4 1 a (4) s=0 ( s F (1) ɛ; ) + K 21 2πi K 23 2πi K 24 2πi ( s F (1) ɛ; ; ) { + ( s F (1) ɛ; ; ) { + N (2) 1 1 a (1) s=0 N (2) 3 1 a (3) s=0 N (2) 4 1 a (4) s=0 ( s F (2) ɛ; ; ) + ( s F (2) ɛ; ; ) + ( s F (2) ɛ; ; ) + Tree structure: any function possessing a Borel transform
119 Higher Order Stokes Phenomenon: Hyper explanation θ e N 1 e ( x,t) = a r r=0 ( ) ε r p= p 1,p 2 Level 1 Hyper re-expansion ε N g p F ep N +1 du e u u N uε F ep 0 ( ) m=s 1,s 2 p= p 1, p 2 dv e v v N M 1 2 ( 1 vε F em ) 0 ε N g p 2πiF em N M+1 2 F mp M+1 2 du e u u M 1/2 ( 1 uf em vf mp ) +third order hyperterms 0 F ep Responsible For Stokes line ep, when F ep >0 ( x,t), F mp ( x,t), F em ( x,t)
120 θ e Higher Order Stokes Phenomenon: Hyper explanation N 1 e ( x,t) = a r r=0 m=s 1,s 2 ( ) ε r p= p 1, p 2 p= p 1,p 2 dv e v v N M 1 2 ( 1 vε F em ) 0 ε N g p F ep N +1 du e u u N uε F ep 0 ( ) ε N g p 2πiF em N M+1 2 F mp M+1 2 du e u u M 1/2 ( 1 uf em vf mp ) +third order hyperterms 0 Level 1 Hyper re-expansion Residue cancels F ep term exactly Level 2 Hyper re-expansion Pole when F em /F mp > 0
121 Higher Order Stokes Phenomenon: Hyper explanation θ e N 1 e ( x,t) = a r r=0 ( ) ε r m=s 1,s 2 p= p 1, p 2 dv e v v N M 1 2 ( 1 vε F em ) 0 ε N g p 2πiF em N M+1 2 F mp M+1 2 du e u u M 1/2 ( 1 uf em vf mp ) +third order hyperterms 0 Residue cancels F ep term exactly Pole when F em /F mp > 0 Condition for colinearity of singularities in the Borel plane
122 Higher Order Stokes Phenomenon: Hyper explanation θ e N 1 e ( x,t) = a r r=0 m=s 1,s 2 ( ) ε r p= p 1, p 2 dv e v v N M 1 2 ( 1 vε F em ) 0 ε N g p 2πiF em N M+1 2 F mp M+1 2 du e u u M 1/2 ( 1 uf em vf mp ) 0 No further possibility of Stokes line between e and p: Higher order Stokes Phenomenon +third order hyperterms
123 PDE illustrative example t x Exact ε=0.5
124 PDE illustrative example NB t ε=0.5 x Asymptotics neglecting initial sub-subdominant term
125 PDE illustrative example NB t x Exact ε=0.5
126 PDE illustrative example NB t ε=0.5 x Asymptotics including initially sub-sub dominant term and taking account of higher order Stokes phenomenon
127 PDE illustrative example Exact θ x
128 PDE illustrative example Exact With HoSP θ x
129 PDE illustrative example Exact With HoSP HoSP ignored θ x
130 PDE illustrative example Exact With HoSP HoSP ignored Exact vs HoSP θ x
131 PDE illustrative example Exact With HoSP HoSP ignored Exact vs HoSP θ x Neighborhood of complex caustics
132 Hyper-tree and higher order Stokes ( y (1) (ɛ) e f 1/ɛ ɛ µ n +ɛ N (0) 1 ( N (0) 1 r=0 K 12 2πi K 13 2πi K 14 2πi a (1) r ɛ r N (1) 2 1 a (2) s=0 N (1) 3 1 a (3) s=0 N (1) 4 1 a (4) s=0 ( s F (1) ɛ; ) + K 21 2πi K 23 2πi K 24 2πi ( s F (1) ɛ; ; ) { + ( s F (1) ɛ; ; ) { + N (2) 1 1 a (1) s=0 N (2) 3 1 a (3) s=0 N (2) 4 1 a (4) s=0 ( s F (2) ɛ; ; ) + ( s F (2) ɛ; ; ) + ( s F (2) ɛ; ; ) + Tree structure: any function possessing a Borel transform
133 Hyper-tree and higher order Stokes ( y (1) (ɛ) e f 1/ɛ ɛ µ n +ɛ N (0) 1 ( N (0) 1 r=0 K 12 2πi K 13 2πi K 14 2πi a (1) r ɛ r N (1) 2 1 a (2) s=0 N (1) 3 1 a (3) s=0 N (1) 4 1 a (4) s=0 ( s F (1) ɛ; ) + K 21 2πi K 23 2πi K 24 2πi ( s F (1) ɛ; ; ) { + ( s F (1) ɛ; ; ) { + N (2) 1 1 a (1) s=0 N (2) 3 1 a (3) s=0 N (2) 4 1 a (4) s=0 ( s F (2) ɛ; ; ) + ( s F (2) ɛ; ; ) + ( s F (2) ɛ; ; ) + Higher order twig on one branch...
134 Hyper-tree and higher order Stokes ( y (1) (ɛ) e f 1/ɛ ɛ µ n +ɛ N (0) 1 ( N (0) 1 r=0 K 12 2πi K 13 2πi K 14 2πi a (1) r ɛ r N (1) 2 1 a (2) s=0 N (1) 3 1 a (3) s=0 N (1) 4 1 a (4) s=0 ( s F (1) ɛ; ) + K 21 2πi K 23 2πi K 24 2πi ( s F (1) ɛ; ; ) { + ( s F (1) ɛ; ; ) { + N (2) 1 1 a (1) s=0 N (2) 3 1 a (3) s=0 N (2) 4 1 a (4) s=0 ( s F (2) ɛ; ; ) + ( s F (2) ɛ; ; ) + ( s F (2) ɛ; ; ) +...can switch off other branches
135 Hyper-tree and higher order Stokes ( y (1) (ɛ) e f 1/ɛ ɛ µ n +ɛ N (0) 1 ( N (0) 1 r=0 K 12 2πi K 13 2πi K 14 2πi a (1) r ɛ r N (1) 2 1 a (2) s=0 N (1) 3 1 a (3) s=0 N (1) 4 1 a (4) s=0 ( s F (1) ɛ; ) + K 21 2πi K 23 2πi K 24 2πi ( s F (1) ɛ; ; ) { + ( s F (1) ɛ; ; ) { + N (2) 1 1 a (1) s=0 N (2) 3 1 a (3) s=0 N (2) 4 1 a (4) s=0 ( s F (2) ɛ; ; ) + ( s F (2) ɛ; ; ) + ( s F (2) ɛ; ; ) +...can switch off other branches
136 Hyper-tree and higher order Stokes ( y (1) (ɛ) e f 1/ɛ ɛ µ n +ɛ N (0) 1 ( N (0) 1 r=0 K 12 2πi K 13 2πi K 14 2πi a (1) r ɛ r N (1) 2 1 a (2) s=0 N (1) 3 1 a (3) s=0 N (1) 4 1 a (4) s=0 ( s F (1) ɛ; ) + K 21 2πi K 23 2πi K 24 2πi ( s F (1) ɛ; ; ) { + ( s F (1) ɛ; ; ) { + N (2) 1 1 a (1) s=0 N (2) 3 1 a (3) s=0 N (2) 4 1 a (4) s=0 ( s F (2) ɛ; ; ) + ( s F (2) ɛ; ; ) + ( s F (2) ɛ; ; ) + Equivalent to Stokes constant changing, e.g., vanishing.
137 Extensions to nonlinear PDEs Higher order Stokes phenomenon plays a significant role in smoothed-shock formation. Without it the smoothed-shock would be a caustic! Example: Burgers
138 Burgers Equation u t + uu x = εu xx, x C, t 0, ε 0 +. u(x, 0) = 1, and u 0 as x. 1 + x2 tion of the form
139 Burgers Equation ( ) x = x j + a 0 (x j )t, j = 0, 1, 2, rves are shown in Figure Inviscid multivaluedness
140 Burgers Equation x = x j + a 0 (x j )t, j = 0, 1, 2, 10 Inviscid shock
141 Burgers Equation u t + uu x = εu xx, x C, t 0, ε 0 +. u(x, 0) = 1, and u 0 as x. 1 + x2 tion of the form u(x, t; ε) u (0) ( ) (x, t; ε) + K( 01u n (n,1) (x, t; ε), ) Transseries n=1 ( ( ) ( ) ( ) a (n,1) 0 = a (1,1) n 0 2 f ) 1 n 1. x a (1,1) 0 (x 0, x 1 ) = (a 0 (x 1 ) a 0 (x 0 )) we find = = 0, so that u(x, t; ε) = a 0 (x, t) + 2K 01a (1,1) 0 (x, t) f 1 a 0 (x 1 ) a 0 (x 0 ) a 0 (x 0)(x 1 x 0 ) a 0 (x 1 ) a 0 (x 0 ) a 0 (x 1)(x 1 x 0 ). x e f 1/ε 2 f 1 x + K 01a (1,1) 0 (x, t)e f 1/ε + O(ε), result (38) is valid everywhere in the region where Re 0 and
142 Complex x-plane at t = 0.25, =0.05 ɛ 2 AS 01 Pre-shock S 0>1' C 01' 10 1 S 0>1 8 6 C 01 S 0> S 0>2' Active Stokes Anti-Stokes Higher-Order-Stokes Inactive Stokes Branch Cut Irrelevant Stokes curves not shown S 0>2 C 02 S 0>2-2 C 02' AS
143 Complex x-plane at t = 0.25, =0.05 ɛ 2 AS 01 Pre-shock S 0>1' C 01' 1 S 0>1 0-1 S 0>2' Active Stokes Anti-Stokes Higher-Order-Stokes Inactive Stokes Branch Cut Irrelevant Stokes curves not shown S 0>2 C 01 C 02 S 0>1 S 0>2 Inactive Stokes line: -2 C 02' AS
144 Complex x-plane at t = 0.25, =0.05 ɛ 2 AS 01 Pre-shock S 0>1' C 01' 1 S 0>1 C 01 S 0>1 0 Active Stokes Anti-Stokes Higher-Order-Stokes Inactive Stokes Branch Cut Irrelevant Stokes curves not shown C 02 S 0>2 Inactive Stokes line: switched off by Higher order Stokes line -1 S 0>2' S 0>2 C 02' -2 AS
145 2 Complex x-plane at t = 0.25, ɛ=0.05 AS 01 Pre-shock 1 S 0>1' C 01' S 0> C 01 S 0> C 02 S 0>2-1 S 0>2' S 0>2-2 C 02' AS
146 2 Complex x-plane at t = 0.25, ɛ=0.05 AS 01 Pre-shock S 0>1' C 01' 1 S 0>1 0 Without HOS a Stokes phenomenon would take place across real space even before shock formation. C 01 C 02 S 0>1 S 0>2-1 S 0>2' S 0>2-2 C 02' AS
147 Complex x-plane at t = 5, ɛ= C 01' AS 01 Post-shock t S 0>1 S C R C R x s C R S 0>1 S 0>2 x S 0> C 02' AS 02 Poles given by summed transseries
148 Complex x-plane at t = 5, ɛ=0.05 AS C 01' S 0>1 S 021 Shock=virtual turning point, not a caustic 0 C R x s C R S 0>1 S 0>2 S 0> C 02' AS 02
149 Complex x-plane at t = 5, ɛ=0.05 AS C 01' S 0>1 S 021 Shock=virtual turning point, not a caustic C 02' S 0>2 C R x s C R S 0>1 S 0>2 AS 02 HOSP forces asymptotic contributions onto different Riemann sheets, preventing coalescence and hence a caustic/ turning point
150 Generalisations u t + u 2 u x = εu xx, u t + F (u)u x = εu xx, etc...
151 Hokusai
152 Higher order Stokes curve
153 Virtual turning point
154 Conclusions Stokes lines switch off at regular points! Higher order Stokes phenomenon can explain activity of Stokes lines. Has crucial role role to play in asymptotic evolution of system, especially PDEs. Should include sub-sub(-sub-sub...) dominant terms! Simple hyperasymptotics can cope with this
155 KdV u t + uu x + ɛ 2 u xxx = 0 u(x, 0) = x 2 ɛ = 0.02
156 KdV t =5,ε =0.02 "kdv4fout.dat" u x Work in progress...
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