File size: 3,120 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 | import OAI.MathematicalPhysics.DefocusingNLS.Profile.CompactifiedSlowDerivative
import Mathlib.Analysis.Calculus.ContDiff.Deriv
import Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas
namespace OAI
/-! # Smoothness and exact jets of the compactified slow solution -/
open Set
open scoped ContDiff
namespace DefocusingNLS
noncomputable def slowAsymptoticJet (q : ℂ) (m : ℕ) : ℕ → ℂ
| 0 => 1
| n + 1 => q * ((m : ℂ) - 1 - q) * slowAsymptoticJet (q + 1) m n
theorem contDiffOn_compactifiedSlowSolution (n : ℕ) (q : ℂ) (m : ℕ) (x : ℂ)
(hq : -1 < q.re) (hx : 0 ≤ x.re) (hx0 : x ≠ 0) :
ContDiffOn ℝ n (compactifiedSlowSolution q m x) (Ici 0) := by
induction n generalizing q with
| zero =>
exact contDiffOn_zero.mpr (continuousOn_compactifiedSlowSolution q m x hq hx hx0)
| succ n ih =>
rw [show ((n + 1 : ℕ) : ℕ∞ω) = (n : ℕ∞ω) + 1 by simp,
contDiffOn_succ_iff_derivWithin (uniqueDiffOn_Ici 0)]
refine ⟨fun t ht => (hasDerivWithinAt_compactifiedSlowSolution q m x hq hx hx0 ht).differentiableWithinAt,
by simp, ?_⟩
have hq' : -1 < (q + 1).re := by change -1 < q.re + 1; linarith
have hi : ContDiffOn ℝ n
(fun t => q * ((m : ℂ) - 1 - q) / x * compactifiedSlowSolution (q + 1) m x t)
(Ici 0) := contDiffOn_const.mul (ih (q + 1) hq')
apply hi.congr
intro t ht
exact (hasDerivWithinAt_compactifiedSlowSolution q m x hq hx hx0 ht).derivWithin
((uniqueDiffOn_Ici 0) t ht)
theorem iteratedDerivWithin_compactifiedSlowSolution (n : ℕ) (q : ℂ) (m : ℕ) (x : ℂ)
(hq : -1 < q.re) (hx : 0 ≤ x.re) (hx0 : x ≠ 0) {t : ℝ} (ht : t ∈ Icc (0 : ℝ) 1) :
iteratedDerivWithin n (compactifiedSlowSolution q m x) (Icc 0 1) t =
slowAsymptoticJet q m n / x ^ n * compactifiedSlowSolution (q + n) m x t := by
induction n generalizing q with
| zero => simp [slowAsymptoticJet]
| succ n ih =>
have hq' : -1 < (q + 1).re := by change -1 < q.re + 1; linarith
have hd : EqOn (derivWithin (compactifiedSlowSolution q m x) (Icc 0 1))
(fun u => q * ((m : ℂ) - 1 - q) / x * compactifiedSlowSolution (q + 1) m x u)
(Icc 0 1) := by
intro u hu
exact ((hasDerivWithinAt_compactifiedSlowSolution q m x hq hx hx0 hu.1).mono
(fun _ h => h.1)).derivWithin ((uniqueDiffOn_Icc (by norm_num : (0 : ℝ) < 1)) u hu)
rw [iteratedDerivWithin_succ', iteratedDerivWithin_congr hd ht,
iteratedDerivWithin_const_mul_field, ih (q + 1) hq']
have he : q + 1 + n = q + (n + 1 : ℕ) := by push_cast; ring
rw [he, slowAsymptoticJet, pow_succ]
field_simp
theorem iteratedDerivWithin_compactifiedSlowSolution_zero (n : ℕ) (q : ℂ) (m : ℕ) (x : ℂ)
(hq : -1 < q.re) (hx : 0 ≤ x.re) (hx0 : x ≠ 0) :
iteratedDerivWithin n (compactifiedSlowSolution q m x) (Icc 0 1) 0 =
slowAsymptoticJet q m n / x ^ n := by
rw [iteratedDerivWithin_compactifiedSlowSolution n q m x hq hx hx0 ⟨le_rfl, by norm_num⟩,
compactifiedSlowSolution_zero, mul_one]
end DefocusingNLS
end OAI
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