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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 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 | import OAI.MathematicalPhysics.DefocusingNLS.Profile.NormalizedSlowCone
namespace OAI
/-! # Identification of the nonsingular tail with division by `q` -/
open MeasureTheory Set Polynomial
namespace DefocusingNLS
theorem shiftedSlowDerivative_parameter_step (q : ℂ) (m : ℕ) (s : ℂ) (t : ℝ)
(hq : -1 < q.re) (hsre : s.re = 0) (ht : 0 < t) :
shiftedSlowDerivative 1 q m s t = -q * shiftedSlowDerivative 0 (q + 1) (m + 1) s t := by
have hx : 0 < ((t : ℂ) - s).re := by simpa [hsre] using ht
simpa only [shiftedSlowDerivative, iteratedDeriv_succ, iteratedDeriv_zero] using
(hasDerivAt_regularizedSlowSolution_shift q m ((t : ℂ) - s) hq hx).deriv
theorem shiftedSlowSecondDerivative_parameter_step (q : ℂ) (m : ℕ) (s : ℂ) (t : ℝ)
(hq : -1 < q.re) (hsre : s.re = 0) (ht : 0 < t) :
shiftedSlowDerivative 2 q m s t = -q * shiftedSlowDerivative 1 (q + 1) (m + 1) s t := by
have hx : 0 < ((t : ℂ) - s).re := by simpa [hsre] using ht
have hq' : -1 < (q + 1).re := by change -1 < q.re + 1; linarith
have h1 := (hasDerivAt_regularizedSlowSolution_shift (q + 1) (m + 1)
((t : ℂ) - s) hq' hx).deriv
have h2 := (hasDerivAt_deriv_regularizedSlowSolution q m ((t : ℂ) - s) hq hx).deriv
simp only [shiftedSlowDerivative, iteratedDeriv_succ, iteratedDeriv_zero]
rw [h1, h2]
have he : q + 1 + 1 = q + 2 := by ring
rw [he, Nat.add_assoc]
ring
theorem slowDerivative_coefficient_parameter_step (q : ℂ) (m : ℕ) (s : ℂ) (n : ℕ)
(hq : -1 < q.re) (hsre : s.re = 0) :
laguerreCoefficientIntegral (shiftedSlowDerivative 1 q m s) n =
-q * slowLaguerreCoefficient (q + 1) (m + 1) s n := by
unfold slowLaguerreCoefficient laguerreCoefficientIntegral
rw [← integral_const_mul]
apply setIntegral_congr_fun measurableSet_Ioi
intro t ht
simp only [laguerreIntegrand,
shiftedSlowDerivative_parameter_step q m s t hq hsre ht]
ring
theorem slowSecondDerivative_coefficient_parameter_step (q : ℂ) (m : ℕ) (s : ℂ) (n : ℕ)
(hq : -1 < q.re) (hsre : s.re = 0) :
laguerreCoefficientIntegral (shiftedSlowDerivative 2 q m s) n =
-q * laguerreCoefficientIntegral (shiftedSlowDerivative 1 (q + 1) (m + 1) s) n := by
unfold laguerreCoefficientIntegral
rw [← integral_const_mul]
apply setIntegral_congr_fun measurableSet_Ioi
intro t ht
simp only [laguerreIntegrand,
shiftedSlowSecondDerivative_parameter_step q m s t hq hsre ht]
ring
theorem q_mul_normalizedSlowB (q : ℂ) (m : ℕ) (s : ℂ) (n : ℕ)
(hq : -1 < q.re) (hsre : s.re = 0) (hsim : s.im ≠ 0) :
q * normalizedSlowB q m s (n + 1) = slowLaguerreB q m s (n + 1) := by
rw [slowLaguerreB_eq_derivative_coefficient q m s n hq hsre hsim,
slowDerivative_coefficient_parameter_step q m s n hq hsre]
simp only [normalizedSlowB, Nat.add_sub_cancel]
ring
theorem q_mul_normalizedSlowC (q : ℂ) (m : ℕ) (s : ℂ) (n : ℕ)
(hq : -1 < q.re) (hsre : s.re = 0) (hsim : s.im ≠ 0) :
q * normalizedSlowC q m s n = slowLaguerreC q m s n := by
have hq' : -1 < (q + 1).re := by change -1 < q.re + 1; linarith
rw [slowLaguerreC_eq_derivative_coefficients q m s n hq hsre hsim,
slowSecondDerivative_coefficient_parameter_step q m s n hq hsre,
slowDerivative_coefficient_parameter_step q m s n hq hsre]
have hb := slowLaguerreB_succ (q + 1) (m + 1) s n
rw [slowLaguerreB_eq_derivative_coefficient (q + 1) (m + 1) s n hq' hsre hsim] at hb
unfold normalizedSlowC
linear_combination -q * hb
theorem q_mul_normalizedSlowCoefficient (q : ℂ) (m : ℕ) (s : ℂ) (n : ℕ)
(hq : -1 < q.re) (hsre : s.re = 0) (hsim : s.im ≠ 0) :
q * normalizedSlowCoefficient q m s (n + 1) = slowLaguerreCoefficient q m s (n + 1) := by
have h := slowLaguerreB_succ q m s (n + 1)
rw [← q_mul_normalizedSlowB q m s (n + 1) hq hsre hsim,
← q_mul_normalizedSlowB q m s n hq hsre hsim] at h
simp only [normalizedSlowB, normalizedSlowCoefficient, Nat.add_sub_cancel] at h ⊢
linear_combination -h
end DefocusingNLS
end OAI
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