openAI-math-dataset / lean /OAI /MathematicalPhysics /DefocusingNLS /Spectrum /SpectralCanonicalOutgoing.lean
| import OAI.MathematicalPhysics.DefocusingNLS.Spectrum.SpectralAllOrders | |
| import OAI.MathematicalPhysics.DefocusingNLS.Spectrum.SpectralHolomorphicAllOrders | |
| import OAI.MathematicalPhysics.DefocusingNLS.Certificates.ContinuousHalfLineBound | |
| import OAI.MathematicalPhysics.DefocusingNLS.Profile.RadialExteriorAllOrders | |
| namespace OAI | |
| /-! Applying the all-order slow-column construction to the canonical nonlinear profile. -/ | |
| open Polynomial Filter Set | |
| open scoped BoundedContinuousFunction | |
| namespace DefocusingNLS | |
| local notation "E₄" => (ℂ × ℂ) × (ℂ × ℂ) | |
| theorem canonical_circular_expansion_data (ν b : ℂ) (n : ℕ) (L : ℝ) | |
| (hX : HasRadialExterior ν n b L) (hb : b ≠ 0) : | |
| ∃ q : ℝ →ᵇ ℂ, | |
| (∀ t, 0 ≤ t → q t=(radialExteriorCanonical ν n b L t).1) ∧ | |
| ∀ J : ℕ, ∃ j : ℕ, J ≤ j ∧ ∃ e : ℝ →ᵇ ℂ, ∀ t, 0 ≤ t → | |
| q t-radialExteriorPolynomialFunction (radialExteriorExpansion ν n b j) t= | |
| (Real.exp (-(2*(j : ℝ))*t) : ℂ)*e t := by | |
| obtain ⟨hqcont,hqexp,_⟩ := radialExteriorCanonical_spec hX | |
| obtain ⟨q,hq⟩ := continuous_tendsto_halfLine_bounded | |
| (fun t => (radialExteriorCanonical ν n b L t).1) hqcont.fst b hqexp.tendsto.fst_nhds | |
| refine ⟨q,hq,?_⟩ | |
| intro J | |
| obtain ⟨j,hJj,v,T,hv⟩ := radialExteriorCanonical_allOrders ν b n L hX hb J | |
| let P := radialExteriorExpansion ν n b j | |
| let κ : ℝ := 2*(j : ℝ) | |
| let f : ℝ → ℂ := fun t => (Real.exp (κ*t) : ℂ)* | |
| (q t-radialExteriorPolynomialFunction P t) | |
| have hf : Continuous f := by | |
| dsimp [f,radialExteriorPolynomialFunction] | |
| fun_prop | |
| have hef : ∀ᶠ t in atTop, ‖f t‖ ≤ ‖v‖ := by | |
| filter_upwards [eventually_ge_atTop (max T 0)] with t ht | |
| have htT : T ≤ t := (le_max_left _ _).trans ht | |
| have ht0 : 0 ≤ t := (le_max_right _ _).trans ht | |
| have he := congrArg Prod.fst (hv t htT) | |
| simp only [radialPolynomialJet,Prod.fst_add,radialExteriorUnweight, | |
| Prod.smul_fst,Complex.real_smul] at he | |
| have hexp : (Real.exp (κ*t) : ℂ)*(Real.exp (-κ*t) : ℂ)=1 := by | |
| rw [← Complex.ofReal_mul,← Real.exp_add] | |
| simp | |
| have heq : f t=(v t).1 := by | |
| dsimp [f] | |
| rw [hq t ht0,he] | |
| change (Real.exp (κ*t) : ℂ)* | |
| ((radialExteriorPolynomialFunction P t+(Real.exp (-κ*t) : ℂ)*(v t).1)- | |
| radialExteriorPolynomialFunction P t)=(v t).1 | |
| rw [add_sub_cancel_left,← mul_assoc,hexp,one_mul] | |
| rw [heq] | |
| exact (norm_fst_le (v t)).trans (v.norm_coe_le_norm t) | |
| obtain ⟨e,he⟩ := exists_bounded_extension_nonneg f hf ⟨‖v‖,hef⟩ | |
| refine ⟨j,hJj,e,?_⟩ | |
| intro t ht | |
| rw [he t ht] | |
| change q t-radialExteriorPolynomialFunction P t= | |
| (Real.exp (-κ*t) : ℂ)*((Real.exp (κ*t) : ℂ)* | |
| (q t-radialExteriorPolynomialFunction P t)) | |
| have hexp : (Real.exp (-κ*t) : ℂ)*(Real.exp (κ*t) : ℂ)=1 := by | |
| rw [← Complex.ofReal_mul,← Real.exp_add] | |
| simp | |
| rw [← mul_assoc,hexp,one_mul] | |
| theorem canonical_circular_allOrders (ν νp νm η b : ℂ) (n : ℕ) (hn : 1 ≤ n) | |
| (L : ℝ) (hX : HasRadialExterior ν n b L) (hb : b ≠ 0) (c : ℂ × ℂ) : | |
| ∃ Y : ℝ → E₄, | |
| (∀ t, 0 ≤ t → HasDerivAt Y | |
| (circularLeadingField t (Y t)+ | |
| circularBoundedField νp νm η n (radialExteriorCanonical ν n b L t).1 (Y t)) t) ∧ | |
| Tendsto Y atTop (nhds ((c.1,0),(c.2,0))) ∧ | |
| ∀ J : ℕ, ∃ j : ℕ, J ≤ j ∧ ∃ v : CircularTailSpace, ∀ t, 0 ≤ t → | |
| Y t=circularPolynomialJet (spectralOutgoingPolynomial νp νm η n | |
| (radialExteriorExpansion ν n b j) c j) t+circularUnweight (2*(j : ℝ)) v t := by | |
| obtain ⟨q,hq,hP⟩ := canonical_circular_expansion_data ν b n L hX hb | |
| obtain ⟨Y,hY,hlim,he⟩ := exists_circular_allOrders ν νp νm η b n hn c q hP | |
| refine ⟨Y,?_,hlim,he⟩ | |
| intro t ht | |
| simpa only [hq t ht] using hY t ht | |
| theorem canonical_circular_outgoing (ν νp νm η b : ℂ) (n : ℕ) (hn : 1 ≤ n) | |
| (L : ℝ) (hX : HasRadialExterior ν n b L) (hb : b ≠ 0) (c : ℂ × ℂ) : | |
| ∃ Y : ℝ → E₄, | |
| (∀ t, 0 ≤ t → HasDerivAt Y | |
| (circularLeadingField t (Y t)+ | |
| circularBoundedField νp νm η n (radialExteriorCanonical ν n b L t).1 (Y t)) t) ∧ | |
| Tendsto Y atTop (nhds ((c.1,0),(c.2,0))) := by | |
| obtain ⟨Y,hY,hlim,_⟩ := canonical_circular_allOrders ν νp νm η b n hn L hX hb c | |
| exact ⟨Y,hY,hlim⟩ | |
| theorem canonical_holomorphic_circular_allOrders (ν νp νm η b : ℂ) | |
| (n : ℕ) (hn : 1 ≤ n) (L : ℝ) (hX : HasRadialExterior ν n b L) (hb : b ≠ 0) | |
| (c : ℂ × ℂ) : | |
| ∃ Y : ℂ → ℝ → E₄, | |
| (∀ lam t, 0 ≤ t → HasDerivAt (Y lam) | |
| (circularLeadingField t (Y lam t)+circularBoundedField (νp-2*lam) (νm-2*lam) η n | |
| (radialExteriorCanonical ν n b L t).1 (Y lam t)) t) ∧ | |
| (∀ lam, Tendsto (Y lam) atTop (nhds ((c.1,0),(c.2,0)))) ∧ | |
| (∀ z t, 0 ≤ t → AnalyticAt ℂ (fun lam => Y lam t) z) ∧ | |
| (∀ lam J, ∃ j : ℕ, J ≤ j ∧ ∃ v : CircularTailSpace, ∀ t, 0 ≤ t → | |
| Y lam t=circularPolynomialJet | |
| (spectralOutgoingPolynomial (νp-2*lam) (νm-2*lam) η n | |
| (radialExteriorExpansion ν n b j) c j) t+circularUnweight (2*(j : ℝ)) v t) := by | |
| obtain ⟨q,hq,hP⟩ := canonical_circular_expansion_data ν b n L hX hb | |
| obtain ⟨Y,hY,hlim,ha,he⟩ := exists_holomorphic_circular_allOrders ν νp νm η b n hn c q hP | |
| refine ⟨Y,?_,hlim,ha,he⟩ | |
| intro lam t ht | |
| simpa only [hq t ht] using hY lam t ht | |
| end DefocusingNLS | |
| end OAI | |