File size: 4,951 Bytes
fac6777 | 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 | import OAI.MathematicalPhysics.DefocusingNLS.Profile.NormalizedSlowTail
namespace OAI
/-! # Strict tail cone including the removable parameter -/
open Filter Topology
namespace DefocusingNLS
theorem normalizedSlow_tail_nontrivial (q : ℂ) (M K : ℕ) (s : ℂ)
(hq : -1 < q.re) (hK : 0 < K) (hsre : s.re = 0) (hsim : s.im ≠ 0) :
∃ n, K ≤ n ∧ normalizedSlowCoefficient q (M + 1) s n ≠ 0 := by
by_contra hn
push Not at hn
have hB (j : ℕ) : normalizedSlowB q (M + 1) s (j + K) = normalizedSlowB q (M + 1) s K := by
induction j with
| zero => simp
| succ j ih =>
rw [show j + 1 + K = (j + K) + 1 by omega, normalizedSlowB_succ,
hn (j + K) (by omega), sub_zero, ih]
have hBK : normalizedSlowB q (M + 1) s K = 0 := by
have hl := (normalizedSlowB_tendsto_zero q (M + 1) s hq hsre hsim).comp
(tendsto_add_atTop_nat K)
have hc : Tendsto (fun j => normalizedSlowB q (M + 1) s (j + K)) atTop
(𝓝 (normalizedSlowB q (M + 1) s K)) := by simpa only [hB] using tendsto_const_nhds
exact tendsto_nhds_unique hc hl
have hq' : -1 < (q + 1).re := by change -1 < q.re + 1; linarith
have hq0 : q + 1 ≠ 0 := by
intro hz
have he := congrArg Complex.re hz
simp only [Complex.add_re, Complex.one_re, Complex.zero_re] at he
linarith
obtain ⟨j, hj, hgj⟩ := slowLaguerre_tail_nontrivial (q + 1) (M + 1) K s hq' hq0 hK hsre hsim
have hz := hB (j + 1 - K)
rw [Nat.sub_add_cancel (by omega : K ≤ j + 1), hBK] at hz
simp only [normalizedSlowB, Nat.add_sub_cancel] at hz
exact hgj hz
theorem normalizedSlow_cone_increment (q : ℂ) (M : ℕ) (s : ℂ) (n : ℕ)
(hn : 1 ≤ n) (hq : -1 < q.re) (hsre : s.re = 0) (hsim : s.im ≠ 0) :
coneForm M s (normalizedSlowB q (M + 1) s (n + 1))
(normalizedSlowC q (M + 1) s (n + 1)) -
coneForm M s (normalizedSlowB q (M + 1) s n) (normalizedSlowC q (M + 1) s n) =
(q.re + n - M / 2) * Complex.normSq (normalizedSlowCoefficient q (M + 1) s n) := by
have h := normalizedSlow_recurrence q M s n hn hq hsre hsim
rw [normalizedSlowB_succ q _ s n] at h
have h' := coneForm_increment (M : ℝ) s (q + n)
(normalizedSlowB q (M + 1) s n) (normalizedSlowC q (M + 1) s n)
(normalizedSlowCoefficient q (M + 1) s n) hsre (by simpa using h)
rw [normalizedSlowC_succ, normalizedSlowB_succ q _ s n]
simpa only [Complex.add_re, Complex.natCast_re] using h'
theorem normalizedSlow_strict_cone (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) :
coneForm ((ℓ : ℝ) + 5) s (normalizedSlowB q (ℓ + 6) s K)
(normalizedSlowC q (ℓ + 6) s K) < 0 := by
have hq' : -1 < q.re := by rw [hq]; linarith [Nat.cast_nonneg (α := ℝ) ℓ]
let P := fun n => coneForm ((ℓ : ℝ) + 5) s (normalizedSlowB q (ℓ + 6) s n)
(normalizedSlowC q (ℓ + 6) s n)
have hi (n : ℕ) (hn : 1 ≤ n) : P (n + 1) - P n =
(σ + n - 5 / 2) * Complex.normSq (normalizedSlowCoefficient q (ℓ + 6) s n) := by
have h := normalizedSlow_cone_increment q (ℓ + 5) s n hn hq' hsre hsim
have he : ℓ + 5 + 1 = ℓ + 6 := by omega
rw [he] at h
push_cast at h
rw [hq] at h
convert h using 1
ring
apply negative_of_increasing_tail K
(coneForm_tendsto_zero _ _ _ _
(normalizedSlowB_tendsto_zero q (ℓ + 6) s hq' hsre hsim)
(normalizedSlowC_tendsto_zero q (ℓ + 6) s hq' hsre hsim))
· intro n hn
have h := mode_increment_nonneg σ n (normalizedSlowCoefficient q (ℓ + 6) s n)
hσ (hK.trans hn)
linarith [hi n (by omega)]
· obtain ⟨n, hn, hg⟩ := normalizedSlow_tail_nontrivial q (ℓ + 5) K s hq' (by omega) hsre hsim
have he : ℓ + 5 + 1 = ℓ + 6 := by omega
rw [he] at hg
refine ⟨n, hn, ?_⟩
have hn' : (3 : ℝ) ≤ n := by exact_mod_cast hK.trans hn
have hpos : 0 < (σ + n - 5 / 2) * Complex.normSq (normalizedSlowCoefficient q (ℓ + 6) s n) :=
mul_pos (by linarith) (Complex.normSq_pos.mpr hg)
linarith [hi n (by omega)]
/-- The tail denominator is nonzero and its ratio lies in the strict disk,
including at `q = 0`. -/
theorem normalizedSlow_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) :
normalizedSlowC q (ℓ + 6) s K ≠ 0 ∧
Complex.normSq (normalizedSlowB q (ℓ + 6) s K / normalizedSlowC q (ℓ + 6) s K +
s / ((ℓ : ℝ) + 5 : ℂ)) < Complex.normSq (s / ((ℓ : ℝ) + 5 : ℂ)) := by
have hM : 0 < (ℓ : ℝ) + 5 := by positivity
have hc := normalizedSlow_strict_cone σ ℓ K q s hσ hK hq hsre hsim
have hd := tail_disk_of_negative_cone ((ℓ : ℝ) + 5) s
(normalizedSlowB q (ℓ + 6) s K) (normalizedSlowC q (ℓ + 6) s K) hM hc
push_cast at hd
exact hd
end DefocusingNLS
end OAI
|