openAI-math-dataset / lean /OAI /MathematicalPhysics /DefocusingNLS /Spectrum /SpectralAllOrders.lean
| import OAI.MathematicalPhysics.DefocusingNLS.Spectrum.SpectralExpansionUniqueness | |
| import OAI.MathematicalPhysics.DefocusingNLS.Spectrum.SpectralOutgoingExistence | |
| namespace OAI | |
| /-! One exact outgoing solution realizes every sufficiently accurate profile expansion. -/ | |
| open Polynomial Filter | |
| open scoped BoundedContinuousFunction | |
| namespace DefocusingNLS | |
| local notation "E₄" => (ℂ × ℂ) × (ℂ × ℂ) | |
| theorem exists_circular_allOrders (ν νp νm η b : ℂ) (n : ℕ) (hn : 1 ≤ n) | |
| (c : ℂ × ℂ) (q : ℝ →ᵇ ℂ) | |
| (hP : ∀ 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) : | |
| ∃ Y : ℝ → E₄, | |
| (∀ t, 0 ≤ t → HasDerivAt Y | |
| (circularLeadingField t (Y t)+circularBoundedField νp νm η n (q t) (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 | |
| let C := circularFieldBound νp νm η n ‖q‖ | |
| obtain ⟨N,hN⟩ := exists_nat_gt (C+1) | |
| have hgap (j : ℕ) (hj : N ≤ j) : C < 2*(j : ℝ) := by | |
| have hj' : (N : ℝ) ≤ j := by exact_mod_cast hj | |
| have hpos : (0 : ℝ) ≤ j := Nat.cast_nonneg j | |
| linarith | |
| obtain ⟨j₀,hNj₀,e₀,he₀⟩ := hP N | |
| let P₀ := radialExteriorExpansion ν n b j₀ | |
| let U₀ := spectralOutgoingPolynomial νp νm η n P₀ c j₀ | |
| obtain ⟨v₀,hv₀⟩ := exists_circular_outgoing_expansion νp νm η n hn P₀ c j₀ ‖q‖ q | |
| q.continuous q.norm_coe_le_norm (hgap j₀ hNj₀) e₀ he₀ | |
| let Y₀ := fun t => circularPolynomialJet U₀ t+circularUnweight (2*(j₀ : ℝ)) v₀ t | |
| have hlim : Tendsto Y₀ atTop (nhds ((c.1,0),(c.2,0))) := by | |
| have hjpos : 0 < 2*(j₀ : ℝ) := lt_of_le_of_lt | |
| (circularFieldBound_nonneg νp νm η n ‖q‖) (hgap j₀ hNj₀) | |
| have hl := (circularPolynomialJet_tendsto U₀).add | |
| (circularUnweight_tendsto (2*(j₀ : ℝ)) hjpos v₀) | |
| have hc := spectralOutgoingPolynomial_constant νp νm η n P₀ c j₀ | |
| simpa only [Y₀,U₀,hc.1,hc.2,add_zero] using hl | |
| refine ⟨Y₀,hv₀,hlim,?_⟩ | |
| intro J | |
| obtain ⟨j,hj,e,he⟩ := hP (max J j₀) | |
| have hj₀j : j₀ ≤ j := (le_max_right _ _).trans hj | |
| have hNj : N ≤ j := hNj₀.trans hj₀j | |
| let P := radialExteriorExpansion ν n b j | |
| let U := spectralOutgoingPolynomial νp νm η n P c j | |
| obtain ⟨v,hv⟩ := exists_circular_outgoing_expansion νp νm η n hn P c j ‖q‖ q | |
| q.continuous q.norm_coe_le_norm (hgap j hNj) e he | |
| let Y := fun t => circularPolynomialJet U t+circularUnweight (2*(j : ℝ)) v t | |
| have hsame := circular_outgoing_orders_unique ν νp νm η b n hn c j₀ j hj₀j v₀ v | |
| ‖q‖ 0 q Y₀ Y (fun t _ => q.norm_coe_le_norm t) hv₀ hv | |
| (fun _ _ => rfl) (fun _ _ => rfl) (hgap j₀ hNj₀) | |
| refine ⟨j,(le_max_left _ _).trans hj,v,?_⟩ | |
| intro t ht | |
| exact hsame t (by simpa only [max_self] using ht) | |
| end DefocusingNLS | |
| end OAI | |