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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 62 63 64 65 66 67 68 69 | import OAI.MathematicalPhysics.DefocusingNLS.Spectrum.SpectralNormalizedResidual
import OAI.MathematicalPhysics.DefocusingNLS.Spectrum.SpectralCoefficientForcing
namespace OAI
/-! The entire bounded forcing family for the actual-profile slow-column construction. -/
open Polynomial
open scoped BoundedContinuousFunction
namespace DefocusingNLS
noncomputable def spectralAnalyticTailData (νp νm η : ℂ) (m : ℕ)
(P : ℂ[X]) (c : ℂ × ℂ) (j : ℕ) (A B : ℝ →ᵇ ℂ) (lam : ℂ) : CircularTailSpace :=
let U := spectralOutgoingPolynomial (νp-2*lam) (νm-2*lam) η m P c j
circularCoefficientForcing A B (boundedRadialPolynomial U.1) (boundedRadialPolynomial U.2)+
spectralNormalizedResidual (νp-2*lam) (νm-2*lam) η m P c j
theorem spectralAnalyticTailData_analyticAt (νp νm η : ℂ) (m : ℕ)
(P : ℂ[X]) (c : ℂ × ℂ) (j : ℕ) (A B : ℝ →ᵇ ℂ) (z : ℂ) :
AnalyticAt ℂ (spectralAnalyticTailData νp νm η m P c j A B) z := by
have hp : AnalyticAt ℂ (fun lam : ℂ => νp-2*lam) z :=
analyticAt_const.sub (analyticAt_const.mul analyticAt_id)
have hm : AnalyticAt ℂ (fun lam : ℂ => νm-2*lam) z :=
analyticAt_const.sub (analyticAt_const.mul analyticAt_id)
have hc (k : ℕ) := spectralOutgoingPolynomial_coeff_analytic
(fun lam => νp-2*lam) (fun lam => νm-2*lam) η m P c z hp hm j k
have hd (lam : ℂ) := spectralOutgoingPolynomial_degree (νp-2*lam) (νm-2*lam) η m P c j
exact (circularCoefficientForcing_analyticAt A B _ _ z
(boundedRadialPolynomial_analyticAt _ j z (fun lam => (hd lam).1) (fun k => (hc k).1))
(boundedRadialPolynomial_analyticAt _ j z (fun lam => (hd lam).2) (fun k => (hc k).2))).add
(spectralNormalizedResidual_analyticAt _ _ η m P c j z hp hm)
theorem spectralAnalyticTailData_normalized (νp νm η : ℂ) (m : ℕ)
(P : ℂ[X]) (c : ℂ × ℂ) (j : ℕ) (A B q : ℝ →ᵇ ℂ)
(hAB : ∀ t, 0 ≤ t →
A t=(Real.exp ((2*(j : ℝ))*t) : ℂ)*(spectralDiagonalCoefficient m (q t)-
spectralDiagonalCoefficient m (boundedRadialPolynomial P t)) ∧
B t=(Real.exp ((2*(j : ℝ))*t) : ℂ)*(spectralCrossCoefficient m (q t)-
spectralCrossCoefficient m (boundedRadialPolynomial P t)))
(lam : ℂ) (t : ℝ) (ht : 0 ≤ t) :
let U := spectralOutgoingPolynomial (νp-2*lam) (νm-2*lam) η m P c j
Real.exp (-(2*(j : ℝ))*t) •
circularTailEvaluation (spectralAnalyticTailData νp νm η m P c j A B lam) t=
(circularBoundedField (νp-2*lam) (νm-2*lam) η m (q t) (circularPolynomialJet U t)-
circularBoundedField (νp-2*lam) (νm-2*lam) η m
(radialExteriorPolynomialFunction P t) (circularPolynomialJet U t))-
circularPolynomialResidualJet (νp-2*lam) (νm-2*lam) η m P U t := by
intro U
have he := circularCoefficientForcing_normalized A B
(boundedRadialPolynomial U.1) (boundedRadialPolynomial U.2)
(νp-2*lam) (νm-2*lam) η m (2*(j : ℝ)) t (q t) (boundedRadialPolynomial P t)
(circularPolynomialJet U t)
(boundedRadialPolynomial_nonneg U.1 t ht) (boundedRadialPolynomial_nonneg U.2 t ht)
(hAB t ht).1 (hAB t ht).2
have hexp : Real.exp (-(2*(j : ℝ))*t)*Real.exp ((2*(j : ℝ))*t)=1 := by
rw [← Real.exp_add]
simp
change Real.exp (-(2*(j : ℝ))*t) •
(circularTailEvaluation (circularCoefficientForcing A B
(boundedRadialPolynomial U.1) (boundedRadialPolynomial U.2)) t+
circularTailEvaluation (spectralNormalizedResidual (νp-2*lam) (νm-2*lam) η m P c j) t)=_
rw [smul_add,he,smul_smul,hexp,one_smul,spectralNormalizedResidual_eq _ _ _ _ _ _ _ t ht,
boundedRadialPolynomial_nonneg P t ht]
rfl
end DefocusingNLS
end OAI
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