File size: 5,875 Bytes
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 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 | import OAI.MathematicalPhysics.DefocusingNLS.Profile.RadianFourierDerivatives
import OAI.MathematicalPhysics.DefocusingNLS.Linear.SchwartzSampledNonlinearity
import OAI.MathematicalPhysics.DefocusingNLS.Nonlinear.CutoffNormalizedEquation
import OAI.MathematicalPhysics.DefocusingNLS.Nonlinear.CutoffProfileModeDerivative
namespace OAI
/-! # The exact Fourier transform of the normalized cutoff defect -/
open scoped SchwartzMap Laplacian ContDiff
namespace DefocusingNLS
local notation "E" => EuclideanSpace ℝ (Fin 12)
noncomputable def physicalSchwartzCoefficient (L : ℝ) (n : frequencyLattice) :
𝓢(E, ℂ) →ₗ[ℂ] ℂ where
toFun ψ := schwartzLatticeCoefficient L (radianFourierKernel ψ) n
map_add' ψ φ := by
change (((2 * Real.pi * L) ^ 12)⁻¹ : ℝ) *
(radianFourierCLM (ψ + φ)) (L⁻¹ • (n : E)) =
(((2 * Real.pi * L) ^ 12)⁻¹ : ℝ) * (radianFourierCLM ψ) (L⁻¹ • (n : E)) +
(((2 * Real.pi * L) ^ 12)⁻¹ : ℝ) * (radianFourierCLM φ) (L⁻¹ • (n : E))
rw [map_add]
simp only [add_apply, mul_add]
map_smul' c ψ := by
change (((2 * Real.pi * L) ^ 12)⁻¹ : ℝ) *
(radianFourierCLM (c • ψ)) (L⁻¹ • (n : E)) =
c * ((((2 * Real.pi * L) ^ 12)⁻¹ : ℝ) * (radianFourierCLM ψ) (L⁻¹ • (n : E)))
rw [map_smul radianFourierCLM]
simp only [smul_apply, smul_eq_mul]
ring
@[simp] theorem physicalSchwartzCoefficient_apply (L : ℝ) (n : frequencyLattice)
(ψ : 𝓢(E, ℂ)) :
physicalSchwartzCoefficient L n ψ =
schwartzLatticeCoefficient L (radianFourierKernel ψ) n := rfl
theorem physicalSchwartzCoefficient_laplacian (ψ : 𝓢(E, ℂ)) (n : frequencyLattice) :
physicalSchwartzCoefficient 1 n (Δ ψ) =
-((‖n‖ ^ 2 : ℝ) : ℂ) * physicalSchwartzCoefficient 1 n ψ := by
change (((2 * Real.pi * 1) ^ 12)⁻¹ : ℝ) *
radianFourierKernel (Δ ψ) (1⁻¹ • (n : E)) = _
simp only [inv_one, one_smul, radianFourierKernel_laplacian,
physicalSchwartzCoefficient_apply, schwartzLatticeCoefficient, Submodule.norm_coe]
ring
theorem physicalSchwartzCoefficient_dilation (R : ℝ) (hR : 0 < R)
(ψ : 𝓢(E, ℂ)) (n : frequencyLattice) :
physicalSchwartzCoefficient R n (schwartzPhysicalDilation 0 R hR ψ) =
physicalSchwartzCoefficient 1 n ψ := by
simp only [physicalSchwartzCoefficient_apply]
rw [radianFourierKernel_schwartzPhysicalDilation,
schwartzLatticeCoefficient_homogeneousDilation 0 R R hR hR]
simp only [mul_zero, Real.rpow_zero, Complex.ofReal_one, one_mul, div_self hR.ne']
noncomputable def cutoffVelocitySchwartz (χ : 𝓢(E, ℂ))
(hχ : HasCompactSupport (χ : E → ℂ)) (Q : E → ℂ) (hQ : ContDiff ℝ ∞ Q)
(R : ℝ) : 𝓢(E, ℂ) :=
(hχ.mul_right (f' := fun x => fderiv ℝ Q (R • x) ((R / 2) • x))).toSchwartzMap
(χ.smooth'.mul (((contDiff_infty_iff_fderiv.mp hQ).2.comp
(contDiff_id.const_smul R)).clm_apply (contDiff_id.const_smul (R / 2))))
@[simp] theorem cutoffVelocitySchwartz_apply (χ : 𝓢(E, ℂ))
(hχ : HasCompactSupport (χ : E → ℂ)) (Q : E → ℂ) (hQ : ContDiff ℝ ∞ Q)
(R : ℝ) (x : E) :
cutoffVelocitySchwartz χ hχ Q hQ R x = χ x * fderiv ℝ Q (R • x) ((R / 2) • x) := rfl
noncomputable def normalizedCutoffDefectSchwartz (a b R : ℝ) (m : ℕ)
(χ : 𝓢(E, ℂ)) (hχ : HasCompactSupport (χ : E → ℂ))
(Q : E → ℂ) (hQ : ContDiff ℝ ∞ Q) : 𝓢(E, ℂ) :=
let ψ := compactDilationFamily χ hχ Q hQ R
Complex.I • cutoffVelocitySchwartz χ hχ Q hQ R +
(((R ^ (2 : ℕ))⁻¹ : ℝ) : ℂ) • Δ ψ +
(Complex.I * (a : ℂ) + (b : ℂ)) • ψ - schwartzOddPower m ψ
theorem hasCompactSupport_complexCutoff (χ : 𝓢(E, ℝ))
(hχ : HasCompactSupport (χ : E → ℝ)) :
HasCompactSupport (χ.postcompCLM Complex.ofRealCLM : E → ℂ) :=
hχ.comp_left (g := fun r : ℝ => (r : ℂ)) rfl
theorem normalizedCutoffDefectSchwartz_apply (a b R : ℝ) (hR : 0 < R) (m : ℕ)
(χ : 𝓢(E, ℝ)) (hχ : HasCompactSupport (χ : E → ℝ))
(Q : E → ℂ) (hQ : ContDiff ℝ ∞ Q) (x : E) :
normalizedCutoffDefectSchwartz a b R m (χ.postcompCLM Complex.ofRealCLM)
(hasCompactSupport_complexCutoff χ hχ) Q hQ x =
(χ x : ℂ) * stationarySimilarityDefect a b m Q (R • x) +
cutoffResidual m R χ Q (R • x) := by
have hφ : (compactDilationFamily (χ.postcompCLM Complex.ofRealCLM)
(hasCompactSupport_complexCutoff χ hχ) Q hQ R : E → ℂ) =
fun z => (χ z : ℂ) * Q (R • z) := rfl
have h := cutoff_normalized_similarity_defect a b R hR m χ χ.smooth' Q hQ x
simpa only [normalizedCutoffDefectSchwartz, add_apply, sub_apply, smul_apply,
smul_eq_mul, cutoffVelocitySchwartz_apply, compactDilationFamily_apply,
SchwartzMap.postcompCLM_apply, Complex.ofRealCLM_apply,
SchwartzMap.laplacian_apply, schwartzOddPower_apply, Complex.real_smul, hφ] using h
theorem physicalSchwartzCoefficient_cutoffDefect (a b R : ℝ) (m : ℕ)
(χ : 𝓢(E, ℂ)) (hχ : HasCompactSupport (χ : E → ℂ))
(Q : E → ℂ) (hQ : ContDiff ℝ ∞ Q) (n : frequencyLattice) :
physicalSchwartzCoefficient 1 n (normalizedCutoffDefectSchwartz a b R m χ hχ Q hQ) =
Complex.I * physicalSchwartzCoefficient 1 n (cutoffVelocitySchwartz χ hχ Q hQ R) +
(-(((R ^ (2 : ℕ))⁻¹ : ℝ) : ℂ) * ((‖n‖ ^ 2 : ℝ) : ℂ) +
Complex.I * (a : ℂ) + (b : ℂ)) *
physicalSchwartzCoefficient 1 n (compactDilationFamily χ hχ Q hQ R) -
physicalSchwartzCoefficient 1 n (schwartzOddPower m (compactDilationFamily χ hχ Q hQ R)) := by
rw [normalizedCutoffDefectSchwartz, map_sub, map_add, map_add,
map_smul, map_smul, map_smul, physicalSchwartzCoefficient_laplacian]
simp only [smul_eq_mul]
ring
end DefocusingNLS
end OAI
|