openAI-math-dataset / lean /OAI /MathematicalPhysics /DefocusingNLS /Certificates /ContinuousDiagonalStableOrbit.lean
| import OAI.MathematicalPhysics.DefocusingNLS.Nonlinear.ContinuousCutoffStableOrbit | |
| import OAI.MathematicalPhysics.DefocusingNLS.Certificates.ContinuousDiagonalLinearStep | |
| import OAI.MathematicalPhysics.DefocusingNLS.Nonlinear.FiniteCoordinateComplete | |
| namespace OAI | |
| /-! # Continuous finite symmetry coordinates select global stable cutoff orbits -/ | |
| open scoped SchwartzMap ContDiff NNReal | |
| namespace DefocusingNLS | |
| local notation "E" => EuclideanSpace ℝ (Fin 12) | |
| local notation "Radius" => {L : ℝ // 1 ≤ L} | |
| attribute [local irreducible] diagonalRealCoordinates | |
| def HasContinuousFiniteDiagonalCutoffStableOrbits {V : Type*} | |
| [NormedAddCommGroup V] [NormedSpace ℂ V] [FiniteDimensional ℂ V] | |
| (a b k : ℝ) (ha : 0 < a) (ha1 : a < 1) (hk : 8 < k) (m : ℕ) | |
| (χ : 𝓢(E, ℂ)) (hχ : HasCompactSupport (χ : E → ℂ)) | |
| (Qp : E → ℂ) (hQp : ContDiff ℝ ∞ Qp) (π : HomogeneousY a k →L[ℝ] V) : Prop := | |
| ∃ G : V →L[ℂ] V, | |
| ∃ hspan : (⨆ lam : ℂ, Module.End.eigenspace G.toLinearMap lam) = ⊤, | |
| ∃ hspec : ∀ (lam : ℂ) (v : V), v ≠ 0 → G v = lam • v → | |
| lam = 0 ∨ lam = 1 ∨ lam = 1 / 2, | |
| HasContinuousCutoffStableOrbits (F := (diagonalRealCoordinates π G hspan hspec).range) | |
| a b k ha ha1 hk m χ hχ Qp hQp | |
| theorem continuousCutoffStableOrbits_of_linear_steps {V : Type*} | |
| [NormedAddCommGroup V] [NormedSpace ℂ V] [FiniteDimensional ℂ V] | |
| (a b k : ℝ) (ha : 0 < a) (ha1 : a < 1) (hk : 8 < k) (m : ℕ) | |
| (χ : 𝓢(E, ℂ)) (hχ : HasCompactSupport (χ : E → ℂ)) | |
| (Qp : E → ℂ) (hQp : ContDiff ℝ ∞ Qp) (Q : ℝ) (hQ : 0 ≤ Q) | |
| (hqb : ∀ L : Radius, ‖cutoffProfileCoefficient a k ha1 hk χ hχ Qp hQp L‖ ≤ Q) | |
| (π : HomogeneousY a k →L[ℝ] V) | |
| (hlinear : HasContinuousFiniteDiagonalTorusLinearSteps π | |
| (cutoffProfileEndpoint a b k ha ha1 hk m χ hχ Qp hQp Q hQ hqb)) | |
| (hnonlinear : ∀ T : ℝ≥0, HasContinuousCutoffNonlinearSteps a b k ha ha1 hk m χ hχ Qp hQp Q hQ hqb T) : | |
| HasContinuousFiniteDiagonalCutoffStableOrbits a b k ha ha1 hk m χ hχ Qp hQp π := by | |
| obtain ⟨G, hspan, hspec, T₀, _, hT₀⟩ := hlinear | |
| let : FiniteDimensional ℝ (SymmetryCoordinates G) := | |
| FiniteDimensional.trans ℝ ℂ (SymmetryCoordinates G) | |
| let F := (diagonalRealCoordinates π G hspan hspec).range | |
| let : CompleteSpace F := finiteCoordinate_complete (diagonalRealCoordinates π G hspan hspec) | |
| obtain ⟨M₀, hM₀, hM⟩ := exists_cutoffStep_geometric_ratio a ha | |
| let T : ℝ≥0 := ⟨max T₀ M₀, hM₀.le.trans (le_max_right _ _)⟩ | |
| have hT : 0 < (T : ℝ) := hM₀.trans_le (le_max_right _ _) | |
| have hr : 2 * Real.exp (-(2 + a) * T / 2) ≤ 1 := | |
| (hM T (le_max_right _ _)).trans (by norm_num) | |
| refine ⟨G, hspan, hspec, ?_⟩ | |
| exact exists_continuous_cutoffStable_orbits (F := F) a b k ha ha1 hk m χ hχ Qp hQp Q hQ hqb T hT hr | |
| (hT₀ T (le_max_left _ _)) (hnonlinear T) | |
| end DefocusingNLS | |
| end OAI | |