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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 | import OAI.MathematicalPhysics.DefocusingNLS.Spectrum.SpectralAirySlopeLimit
import OAI.MathematicalPhysics.DefocusingNLS.Spectrum.SpectralScalarRatio
import Mathlib.Analysis.SpecificLimits.Basic
namespace OAI
/-! The full complex logarithmic slope has the growing Airy limit.
Nonzero conserved flux rules out zeros and the decaying alternative. -/
open Set Filter Topology
namespace DefocusingNLS
theorem spectralAiry_complex_slope_tendsto
(q : ℝ → ℂ × ℂ) (T : ℝ) (hT : 1≤T)
(hq : ContinuousOn q (Ici T))
(hD : ∀ t, T≤t → HasDerivAt q (spectralScalarField (-(t : ℂ)) (q t)) t)
(hJ : spectralScalarFlux (q T)≠0) :
Tendsto (fun t => (q t).2/((Real.sqrt t : ℂ)*(q t).1)) atTop (𝓝 1) := by
obtain ⟨S,hTS,hm,hbound⟩ := spectralAiry_eventual_mass_bound q T hT hq hD hJ
let J := spectralScalarFlux (q T)
let m := spectralScalarMass (q S)
let eta := fun t => spectralScalarFlux (q t)/(Real.sqrt t*spectralScalarMass (q t))
have hflux (t : ℝ) (ht : S≤t) : spectralScalarFlux (q t)=J :=
spectralScalarFlux_const T t q (fun x => -x) (hq.mono (fun _ hx => hx.1))
(fun x hx => by simpa only [Complex.ofReal_neg] using hD x hx.1.le) t ⟨hTS.trans ht,le_rfl⟩
have hinv : Tendsto (fun t : ℝ => (Real.sqrt t)⁻¹) atTop (𝓝 0) := by
simpa only [Function.comp_def,Real.sqrt_inv,Real.sqrt_zero] using
(Real.continuous_sqrt.tendsto 0).comp
(tendsto_inv_atTop_zero : Tendsto (fun t : ℝ => t⁻¹) atTop (𝓝 0))
have hrate : Tendsto (fun t : ℝ => (|J|/m)*(Real.sqrt t)⁻¹) atTop (𝓝 0) := by
simpa only [mul_zero] using hinv.const_mul (|J|/m)
have heta : Tendsto eta atTop (𝓝 0) := by
apply squeeze_zero_norm' _ hrate
filter_upwards [eventually_ge_atTop S] with t ht
have htp : 0<t := lt_of_lt_of_le (by linarith [hT,hTS] : 0<S) ht
have hs : 0<Real.sqrt t := Real.sqrt_pos.mpr htp
have hmass : 0<spectralScalarMass (q t) := hm.trans_le (hbound t ht).2
change ‖spectralScalarFlux (q t)/(Real.sqrt t*spectralScalarMass (q t))‖≤_
rw [Real.norm_eq_abs,abs_div,abs_of_pos (mul_pos hs hmass),hflux t ht]
calc
_ ≤ |J|/(Real.sqrt t*m) :=
div_le_div_of_nonneg_left (abs_nonneg _) (mul_pos hs hm)
(mul_le_mul_of_nonneg_left (hbound t ht).2 hs.le)
_ = _ := by ring
have hz := spectralAirySlope_tendsto_one q T hT hq hD hJ
have hzc : Tendsto (fun t => (spectralAirySlope q t : ℂ)) atTop (𝓝 1) := by
exact (Complex.continuous_ofReal.tendsto 1).comp hz
have hec : Tendsto (fun t => (eta t : ℂ)) atTop (𝓝 0) := by
exact (Complex.continuous_ofReal.tendsto 0).comp heta
have hsum : Tendsto (fun t => (spectralAirySlope q t : ℂ)+Complex.I*(eta t : ℂ)) atTop (𝓝 1) := by
simpa only [mul_zero,add_zero] using hzc.add (hec.const_mul Complex.I)
apply hsum.congr'
filter_upwards [eventually_ge_atTop S] with t ht
have htp : 0<t := lt_of_lt_of_le (by linarith [hT,hTS] : 0<S) ht
have hqt : (q t).1≠0 := spectralScalarFlux_ne_zero_value (q t) (by rw [hflux t ht]; exact hJ)
exact (spectralScalarRatio_components (q t) (Real.sqrt t) (Real.sqrt_pos.mpr htp).ne' hqt).symm
end DefocusingNLS
end OAI
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