File size: 4,840 Bytes
d5b562f | 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 | import OAI.MathematicalPhysics.DefocusingNLS.Certificates.NormalizedMatchingColumns
import OAI.MathematicalPhysics.DefocusingNLS.Profile.SlowParameterHolomorphy
namespace OAI
/-! # The analytic boundary column and its normalized tail representation -/
open Matrix
namespace DefocusingNLS
/-- The derivative-shift formula gives an explicitly analytic boundary column. -/
noncomputable def slowBoundaryColumn (q : ℂ) (m : ℕ) (s : ℂ) : Fin 2 → ℂ :=
![regularizedSlowSolution q m (-s), q * regularizedSlowSolution (q + 1) (m + 1) (-s)]
theorem slowBoundaryColumn_eq_jet (q : ℂ) (m : ℕ) (s : ℂ)
(hq : -1 < q.re) (hsre : s.re = 0) (hsim : s.im ≠ 0) :
slowBoundaryColumn q m s =
![regularizedSlowSolution q m (-s), -deriv (regularizedSlowSolution q m) (-s)] := by
have hs0 : -s ≠ 0 := neg_ne_zero.mpr (fun h => hsim (by simp [h]))
have hd := (hasDerivAt_regularizedSlowSolution_shift_closed q m (-s) hq
(by simp [hsre]) hs0).deriv
simp [slowBoundaryColumn, hd]
theorem differentiableAt_slowBoundaryColumn (q : ℂ) (m : ℕ) (s : ℂ)
(hq : -1 < q.re) (hsre : s.re = 0) (hsim : s.im ≠ 0) :
DifferentiableAt ℂ (fun z => slowBoundaryColumn z m s) q := by
have hs0 : -s ≠ 0 := neg_ne_zero.mpr (fun h => hsim (by simp [h]))
have hq' : -1 < (q + 1).re := by change -1 < q.re + 1; linarith
have h₀ := differentiableAt_regularizedSlowSolution_parameter_closed q m (-s) hq
(by simp [hsre]) hs0
have h₁ := differentiableAt_regularizedSlowSolution_parameter_closed (q + 1) (m + 1)
(-s) hq' (by simp [hsre]) hs0
apply differentiableAt_pi.mpr
intro i
fin_cases i
· change DifferentiableAt ℂ (fun z => regularizedSlowSolution z m (-s)) q
exact h₀
· change DifferentiableAt ℂ (fun z => z * regularizedSlowSolution (z + 1) (m + 1) (-s)) q
convert! differentiableAt_id.mul (h₁.comp q (differentiableAt_id.add_const 1)) using 1
theorem analyticOnNhd_slowBoundaryColumn (m : ℕ) (s : ℂ)
(hsre : s.re = 0) (hsim : s.im ≠ 0) :
AnalyticOnNhd ℂ (fun q => slowBoundaryColumn q m s) {q : ℂ | -1 < q.re} := by
apply DifferentiableOn.analyticOnNhd _ (isOpen_lt continuous_const Complex.continuous_re)
intro q hq
exact (differentiableAt_slowBoundaryColumn q m s hq hsre hsim).differentiableWithinAt
theorem slowBoundaryColumn_ne_zero (q : ℂ) (m : ℕ) (s : ℂ)
(hq : -1 < q.re) (hsre : s.re = 0) (hsim : s.im ≠ 0) :
slowBoundaryColumn q m s ≠ 0 := by
rw [slowBoundaryColumn_eq_jet q m s hq hsre hsim]
intro hz
have h₀ := congrFun hz 0
have h₁ := congrFun hz 1
simp only [Matrix.cons_val_zero, Matrix.cons_val_one, Pi.zero_apply] at h₀ h₁
have hs0 : -s ≠ 0 := neg_ne_zero.mpr (fun h => hsim (by simp [h]))
exact (regularizedSlowSolution_jet_nonzero q m (-s) hq (by simp [hsre]) hs0).elim
(fun h => h h₀) (fun h => h (neg_eq_zero.mp h₁))
theorem normalizedMatching_tail_disk (σ : ℝ) (ℓ K : ℕ) (q s : ℂ)
(hσ : -(1 / 32 : ℝ) ≤ σ) (hK : 3 ≤ K)
(hq : q.re = σ + (ℓ : ℝ) / 2) (hsre : s.re = 0) (hsim : s.im ≠ 0) :
normalizedMatchingC q (ℓ + 6) s K ≠ 0 ∧
Complex.normSq (normalizedMatchingB q (ℓ + 6) s K /
normalizedMatchingC q (ℓ + 6) s K + s / ((ℓ : ℝ) + 5 : ℂ)) <
Complex.normSq (s / ((ℓ : ℝ) + 5 : ℂ)) := by
have hq' : -1 < q.re := by rw [hq]; linarith [Nat.cast_nonneg (α := ℝ) ℓ]
have hp := shiftedPochhammer_ne_zero q (K - 1) hq'
obtain ⟨hC, hd⟩ := normalizedSlow_tail_disk σ ℓ K q s hσ hK hq hsre hsim
refine ⟨div_ne_zero hC hp, ?_⟩
have he : normalizedMatchingB q (ℓ + 6) s K / normalizedMatchingC q (ℓ + 6) s K =
normalizedSlowB q (ℓ + 6) s K / normalizedSlowC q (ℓ + 6) s K := by
unfold normalizedMatchingB normalizedMatchingC
field_simp
rwa [he]
/-- The backward image of the ratio column differs from the true analytic
boundary column by the explicit nonzero tail scalar. -/
theorem normalizedMatching_ratio_column (q : ℂ) (M K : ℕ) (s : ℂ)
(hK : 0 < K) (hq : -1 < q.re) (hsre : s.re = 0) (hsim : s.im ≠ 0)
(hC : normalizedMatchingC q (M + 1) s K ≠ 0) :
backwardProduct M s q K *ᵥ
![normalizedMatchingB q (M + 1) s K / normalizedMatchingC q (M + 1) s K, 1] =
(normalizedMatchingC q (M + 1) s K)⁻¹ • slowBoundaryColumn q (M + 1) s := by
have hv : ![normalizedMatchingB q (M + 1) s K / normalizedMatchingC q (M + 1) s K, 1] =
(normalizedMatchingC q (M + 1) s K)⁻¹ •
![normalizedMatchingB q (M + 1) s K, normalizedMatchingC q (M + 1) s K] := by
ext i
fin_cases i <;> simp [div_eq_mul_inv, mul_comm, hC]
rw [hv, Matrix.mulVec_smul, normalizedMatching_backwardProduct q M K s hK hq hsre hsim,
slowBoundaryColumn_eq_jet q (M + 1) s hq hsre hsim]
end DefocusingNLS
end OAI
|