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ef96466 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 | import OAI.MathematicalPhysics.DefocusingNLS.Spectrum.SpectralAllOrders
import OAI.MathematicalPhysics.DefocusingNLS.Spectrum.SpectralHolomorphicAllOrders
import OAI.MathematicalPhysics.DefocusingNLS.Certificates.ContinuousHalfLineBound
import OAI.MathematicalPhysics.DefocusingNLS.Profile.RadialExteriorAllOrders
namespace OAI
/-! Applying the all-order slow-column construction to the canonical nonlinear profile. -/
open Polynomial Filter Set
open scoped BoundedContinuousFunction
namespace DefocusingNLS
local notation "E₄" => (ℂ × ℂ) × (ℂ × ℂ)
theorem canonical_circular_expansion_data (ν b : ℂ) (n : ℕ) (L : ℝ)
(hX : HasRadialExterior ν n b L) (hb : b ≠ 0) :
∃ q : ℝ →ᵇ ℂ,
(∀ t, 0 ≤ t → q t=(radialExteriorCanonical ν n b L t).1) ∧
∀ J : ℕ, ∃ j : ℕ, J ≤ j ∧ ∃ e : ℝ →ᵇ ℂ, ∀ t, 0 ≤ t →
q t-radialExteriorPolynomialFunction (radialExteriorExpansion ν n b j) t=
(Real.exp (-(2*(j : ℝ))*t) : ℂ)*e t := by
obtain ⟨hqcont,hqexp,_⟩ := radialExteriorCanonical_spec hX
obtain ⟨q,hq⟩ := continuous_tendsto_halfLine_bounded
(fun t => (radialExteriorCanonical ν n b L t).1) hqcont.fst b hqexp.tendsto.fst_nhds
refine ⟨q,hq,?_⟩
intro J
obtain ⟨j,hJj,v,T,hv⟩ := radialExteriorCanonical_allOrders ν b n L hX hb J
let P := radialExteriorExpansion ν n b j
let κ : ℝ := 2*(j : ℝ)
let f : ℝ → ℂ := fun t => (Real.exp (κ*t) : ℂ)*
(q t-radialExteriorPolynomialFunction P t)
have hf : Continuous f := by
dsimp [f,radialExteriorPolynomialFunction]
fun_prop
have hef : ∀ᶠ t in atTop, ‖f t‖ ≤ ‖v‖ := by
filter_upwards [eventually_ge_atTop (max T 0)] with t ht
have htT : T ≤ t := (le_max_left _ _).trans ht
have ht0 : 0 ≤ t := (le_max_right _ _).trans ht
have he := congrArg Prod.fst (hv t htT)
simp only [radialPolynomialJet,Prod.fst_add,radialExteriorUnweight,
Prod.smul_fst,Complex.real_smul] at he
have hexp : (Real.exp (κ*t) : ℂ)*(Real.exp (-κ*t) : ℂ)=1 := by
rw [← Complex.ofReal_mul,← Real.exp_add]
simp
have heq : f t=(v t).1 := by
dsimp [f]
rw [hq t ht0,he]
change (Real.exp (κ*t) : ℂ)*
((radialExteriorPolynomialFunction P t+(Real.exp (-κ*t) : ℂ)*(v t).1)-
radialExteriorPolynomialFunction P t)=(v t).1
rw [add_sub_cancel_left,← mul_assoc,hexp,one_mul]
rw [heq]
exact (norm_fst_le (v t)).trans (v.norm_coe_le_norm t)
obtain ⟨e,he⟩ := exists_bounded_extension_nonneg f hf ⟨‖v‖,hef⟩
refine ⟨j,hJj,e,?_⟩
intro t ht
rw [he t ht]
change q t-radialExteriorPolynomialFunction P t=
(Real.exp (-κ*t) : ℂ)*((Real.exp (κ*t) : ℂ)*
(q t-radialExteriorPolynomialFunction P t))
have hexp : (Real.exp (-κ*t) : ℂ)*(Real.exp (κ*t) : ℂ)=1 := by
rw [← Complex.ofReal_mul,← Real.exp_add]
simp
rw [← mul_assoc,hexp,one_mul]
theorem canonical_circular_allOrders (ν νp νm η b : ℂ) (n : ℕ) (hn : 1 ≤ n)
(L : ℝ) (hX : HasRadialExterior ν n b L) (hb : b ≠ 0) (c : ℂ × ℂ) :
∃ Y : ℝ → E₄,
(∀ t, 0 ≤ t → HasDerivAt Y
(circularLeadingField t (Y t)+
circularBoundedField νp νm η n (radialExteriorCanonical ν n b L t).1 (Y t)) t) ∧
Tendsto Y atTop (nhds ((c.1,0),(c.2,0))) ∧
∀ J : ℕ, ∃ j : ℕ, J ≤ j ∧ ∃ v : CircularTailSpace, ∀ t, 0 ≤ t →
Y t=circularPolynomialJet (spectralOutgoingPolynomial νp νm η n
(radialExteriorExpansion ν n b j) c j) t+circularUnweight (2*(j : ℝ)) v t := by
obtain ⟨q,hq,hP⟩ := canonical_circular_expansion_data ν b n L hX hb
obtain ⟨Y,hY,hlim,he⟩ := exists_circular_allOrders ν νp νm η b n hn c q hP
refine ⟨Y,?_,hlim,he⟩
intro t ht
simpa only [hq t ht] using hY t ht
theorem canonical_circular_outgoing (ν νp νm η b : ℂ) (n : ℕ) (hn : 1 ≤ n)
(L : ℝ) (hX : HasRadialExterior ν n b L) (hb : b ≠ 0) (c : ℂ × ℂ) :
∃ Y : ℝ → E₄,
(∀ t, 0 ≤ t → HasDerivAt Y
(circularLeadingField t (Y t)+
circularBoundedField νp νm η n (radialExteriorCanonical ν n b L t).1 (Y t)) t) ∧
Tendsto Y atTop (nhds ((c.1,0),(c.2,0))) := by
obtain ⟨Y,hY,hlim,_⟩ := canonical_circular_allOrders ν νp νm η b n hn L hX hb c
exact ⟨Y,hY,hlim⟩
theorem canonical_holomorphic_circular_allOrders (ν νp νm η b : ℂ)
(n : ℕ) (hn : 1 ≤ n) (L : ℝ) (hX : HasRadialExterior ν n b L) (hb : b ≠ 0)
(c : ℂ × ℂ) :
∃ Y : ℂ → ℝ → E₄,
(∀ lam t, 0 ≤ t → HasDerivAt (Y lam)
(circularLeadingField t (Y lam t)+circularBoundedField (νp-2*lam) (νm-2*lam) η n
(radialExteriorCanonical ν n b L t).1 (Y lam t)) t) ∧
(∀ lam, Tendsto (Y lam) atTop (nhds ((c.1,0),(c.2,0)))) ∧
(∀ z t, 0 ≤ t → AnalyticAt ℂ (fun lam => Y lam t) z) ∧
(∀ lam J, ∃ j : ℕ, J ≤ j ∧ ∃ v : CircularTailSpace, ∀ t, 0 ≤ t →
Y lam t=circularPolynomialJet
(spectralOutgoingPolynomial (νp-2*lam) (νm-2*lam) η n
(radialExteriorExpansion ν n b j) c j) t+circularUnweight (2*(j : ℝ)) v t) := by
obtain ⟨q,hq,hP⟩ := canonical_circular_expansion_data ν b n L hX hb
obtain ⟨Y,hY,hlim,ha,he⟩ := exists_holomorphic_circular_allOrders ν νp νm η b n hn c q hP
refine ⟨Y,?_,hlim,ha,he⟩
intro lam t ht
simpa only [hq t ht] using hY lam t ht
end DefocusingNLS
end OAI
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