File size: 3,288 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 | import OAI.MathematicalPhysics.DefocusingNLS.Profile.CompactifiedSlowContinuity
import Mathlib.Analysis.Calculus.FDeriv.Extend
namespace OAI
/-! # A regular derivative recurrence at inverse radius zero -/
open Filter Topology Set
namespace DefocusingNLS
attribute [local irreducible] normalizedSlowSolution
theorem hasDerivAt_compactifiedSlowSolution_pos (q : ℂ) (m : ℕ) (x : ℂ)
(hq : -1 < q.re) (hx : 0 ≤ x.re) (hx0 : x ≠ 0) {t : ℝ} (ht : 0 < t) :
HasDerivAt (compactifiedSlowSolution q m x)
(q * ((m : ℂ) - 1 - q) / x * compactifiedSlowSolution (q + 1) m x t) t := by
have ht0 : (t : ℂ) ≠ 0 := Complex.ofReal_ne_zero.mpr ht.ne'
have hy0 : x / (t : ℂ) ≠ 0 := div_ne_zero hx0 ht0
have ha : HasDerivAt (fun u : ℝ => x / (u : ℂ)) (-x / (t : ℂ) ^ 2) t := by
convert! (((hasDerivAt_const (t : ℂ) x).div (hasDerivAt_id (t : ℂ)) ht0).comp_ofReal) using 1
simp only [id_eq]
ring
have hn := hasDerivAt_normalizedSlowSolution q m (x / (t : ℂ)) hq
(slow_div_real_re_nonneg hx ht) hy0
have he : compactifiedSlowSolution q m x =ᶠ[𝓝 t]
(fun u : ℝ => normalizedSlowSolution q m (x / (u : ℂ))) := by
filter_upwards [eventually_gt_nhds ht] with u hu
exact compactifiedSlowSolution_pos q m x hu
convert! (hn.comp t ha).congr_of_eventuallyEq he using 1
rw [compactifiedSlowSolution_pos (q + 1) m x ht]
field_simp
/-- The derivative at the endpoint is obtained from the actual normalized
integral and its limit, not imposed as a formal asymptotic coefficient. -/
theorem hasDerivWithinAt_compactifiedSlowSolution_zero (q : ℂ) (m : ℕ) (x : ℂ)
(hq : -1 < q.re) (hx : 0 ≤ x.re) (hx0 : x ≠ 0) :
HasDerivWithinAt (compactifiedSlowSolution q m x)
(q * ((m : ℂ) - 1 - q) / x) (Ici 0) 0 := by
have hq' : -1 < (q + 1).re := by change -1 < q.re + 1; linarith
apply hasDerivWithinAt_Ici_of_tendsto_deriv
(s := Ioi 0)
(fun t ht => (hasDerivAt_compactifiedSlowSolution_pos q m x hq hx hx0 ht).differentiableAt.differentiableWithinAt)
(continuousWithinAt_compactifiedSlowSolution_zero q m x hq hx hx0) self_mem_nhdsWithin
have hc := (continuousWithinAt_compactifiedSlowSolution_zero (q + 1) m x hq' hx hx0).const_mul
(q * ((m : ℂ) - 1 - q) / x)
have he : (fun t : ℝ => deriv (compactifiedSlowSolution q m x) t) =ᶠ[𝓝[>] 0]
(fun t => q * ((m : ℂ) - 1 - q) / x * compactifiedSlowSolution (q + 1) m x t) := by
filter_upwards [self_mem_nhdsWithin] with t ht
exact (hasDerivAt_compactifiedSlowSolution_pos q m x hq hx hx0 ht).deriv
simpa only [compactifiedSlowSolution_zero, mul_one] using hc.congr' he.symm
theorem hasDerivWithinAt_compactifiedSlowSolution (q : ℂ) (m : ℕ) (x : ℂ)
(hq : -1 < q.re) (hx : 0 ≤ x.re) (hx0 : x ≠ 0) {t : ℝ} (ht : t ∈ Ici 0) :
HasDerivWithinAt (compactifiedSlowSolution q m x)
(q * ((m : ℂ) - 1 - q) / x * compactifiedSlowSolution (q + 1) m x t) (Ici 0) t := by
change 0 ≤ t at ht
rcases ht.eq_or_lt with he | ht
· subst t
simpa only [compactifiedSlowSolution_zero, mul_one] using
hasDerivWithinAt_compactifiedSlowSolution_zero q m x hq hx hx0
· exact (hasDerivAt_compactifiedSlowSolution_pos q m x hq hx hx0 ht).hasDerivWithinAt
end DefocusingNLS
end OAI
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