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fac6777 | 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 | import OAI.MathematicalPhysics.DefocusingNLS.Nonlinear.CutoffTransport
namespace OAI
/-! # The cutoff defect in the fixed normalized spatial coordinate -/
open scoped Laplacian ContDiff
namespace DefocusingNLS
local notation "E" => EuclideanSpace ℝ (Fin 12)
theorem laplacian_complex_dilation (f : E → ℂ) (hf : ContDiff ℝ ∞ f)
(r : ℝ) (x : E) :
Δ (fun y => f (r • y)) x = (r ^ (2 : ℕ) : ℝ) • Δ f (r • x) := by
simp only [InnerProductSpace.laplacian_eq_iteratedFDeriv_orthonormalBasis _
(EuclideanSpace.basisFun (Fin 12) ℝ)]
have hd : iteratedFDeriv ℝ 2 (fun y => f (r • y)) x =
r ^ (2 : ℕ) • iteratedFDeriv ℝ 2 f (r • x) := by
simpa only using! congrFun (iteratedFDeriv_comp_const_smul r
(hf.of_le (by simp : (2 : ℕ∞ω) ≤ ∞))) x
rw [hd, Finset.smul_sum]
apply Finset.sum_congr rfl
intro j _
simp only [smul_apply]
/-- Exact fixed-coordinate equation, with both the profile defect and cutoff residual visible. -/
theorem cutoff_normalized_similarity_defect (a b R : ℝ) (hR : 0 < R) (m : ℕ)
(χ : E → ℝ) (hχ : ContDiff ℝ ∞ χ) (Q : E → ℂ) (hQ : ContDiff ℝ ∞ Q) (x : E) :
Complex.I * ((χ x : ℂ) * fderiv ℝ Q (R • x) ((R / 2) • x)) +
((R ^ (2 : ℕ))⁻¹ : ℝ) • Δ (fun z => (χ z : ℂ) * Q (R • z)) x +
(Complex.I * (a : ℂ) + (b : ℂ)) * ((χ x : ℂ) * Q (R • x)) -
oddPowerNonlinearity m ((χ x : ℂ) * Q (R • x)) =
(χ x : ℂ) * stationarySimilarityDefect a b m Q (R • x) +
cutoffResidual m R χ Q (R • x) := by
let ψ : E → ℂ := fun y => (χ (R⁻¹ • y) : ℂ) * Q y
have hψ : ContDiff ℝ ∞ ψ :=
(Complex.ofRealCLM.contDiff.comp (hχ.comp (contDiff_id.const_smul R⁻¹))).mul hQ
have hscale : (fun z => (χ z : ℂ) * Q (R • z)) = fun z => ψ (R • z) := by
funext z
simp only [ψ, smul_smul, inv_mul_cancel₀ hR.ne', one_smul]
have hlap : ((R ^ (2 : ℕ))⁻¹ : ℝ) •
Δ (fun z => (χ z : ℂ) * Q (R • z)) x = Δ ψ (R • x) := by
rw [hscale, laplacian_complex_dilation ψ hψ, smul_smul,
inv_mul_cancel₀ (pow_ne_zero _ hR.ne'), one_smul]
have harg : R⁻¹ • (R • x) = x := by
rw [smul_smul, inv_mul_cancel₀ hR.ne', one_smul]
have hsp := cutoffResidual_spatial_identity m R χ Q hχ hQ (R • x)
rw [harg] at hsp
change Δ ψ (R • x) - oddPowerNonlinearity m ((χ x : ℂ) * Q (R • x)) = _ at hsp
have hvec : (1 / 2 : ℝ) • (R • x) = (R / 2) • x := by
rw [smul_smul]
congr 1
ring
rw [hlap]
unfold stationarySimilarityDefect
rw [hvec]
linear_combination hsp
end DefocusingNLS
end OAI
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