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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 89 90 91 | import OAI.MathematicalPhysics.DefocusingNLS.Profile.NormalizedSlowDerivative
import OAI.MathematicalPhysics.DefocusingNLS.Profile.SlowBoundaryEquation
namespace OAI
/-! The differential equation for the actual normalized outgoing H solution. -/
namespace DefocusingNLS
noncomputable def normalizedSlowFirst (q : ℂ) (m : ℕ) (x : ℂ) : ℂ :=
q*x^(q-1)*regularizedSlowSolution q m x+
x^q*deriv (regularizedSlowSolution q m) x
theorem hasDerivAt_normalizedSlowFirst_value (q : ℂ) (m : ℕ) (x : ℂ)
(hq : -1 < q.re) (hx : x ∈ Complex.slitPlane) :
HasDerivAt (normalizedSlowSolution q m) (normalizedSlowFirst q m x) x := by
have hf := (analyticOnNhd_regularizedSlowSolution_slit q m hq x hx).differentiableAt.hasDerivAt
convert ((hasDerivAt_id x).cpow_const (c := q) hx).mul hf using 1
· rfl
· simp only [normalizedSlowFirst,id_eq,mul_one]
theorem hasDerivAt_normalizedSlowFirst (q : ℂ) (m : ℕ) (x : ℂ)
(hq : -1 < q.re) (hx : x ∈ Complex.slitPlane) :
HasDerivAt (normalizedSlowFirst q m)
(q*(q-1)*x^(q-2)*regularizedSlowSolution q m x+
2*q*x^(q-1)*deriv (regularizedSlowSolution q m) x+
x^q*deriv (deriv (regularizedSlowSolution q m)) x) x := by
have ha := analyticOnNhd_regularizedSlowSolution_slit q m hq
have hf := (ha x hx).differentiableAt.hasDerivAt
have hdf := (ha.deriv x hx).differentiableAt.hasDerivAt
have hp := (hasDerivAt_id x).cpow_const (c := q) hx
have hp1 := (hasDerivAt_id x).cpow_const (c := q-1) hx
convert ((hp1.const_mul q).mul hf).add (hp.mul hdf) using 1
· rfl
· simp only [id_eq,mul_one]
rw [show q-1-1=q-2 by ring]
ring
theorem normalizedSlowFirst_eq_shift (q : ℂ) (m : ℕ) (x : ℂ)
(hq : -1 < q.re) (hx : 0 ≤ x.re) (hx0 : x ≠ 0) :
normalizedSlowFirst q m x =
-q*((m : ℂ)-1-q)/x^2*normalizedSlowSolution (q+1) m x := by
have hs : x ∈ Complex.slitPlane := by
apply Complex.mem_slitPlane_iff.mpr
rcases hx.eq_or_lt with h | h
· right
intro hi
exact hx0 (Complex.ext h.symm hi)
· exact Or.inl h
exact (hasDerivAt_normalizedSlowFirst_value q m x hq hs).unique
(hasDerivAt_normalizedSlowSolution q m x hq hx hx0)
theorem normalizedSlowSolution_equation (q : ℂ) (m : ℕ) (x : ℂ)
(hq : -1 < q.re) (hx : 0 ≤ x.re) (hx0 : x ≠ 0) :
x^2*deriv (normalizedSlowFirst q m) x+
(((m : ℂ)-2*q)*x-x^2)*normalizedSlowFirst q m x+
q*(q+1-m)*normalizedSlowSolution q m x=0 := by
have hs : x ∈ Complex.slitPlane := by
apply Complex.mem_slitPlane_iff.mpr
rcases hx.eq_or_lt with h | h
· right
intro hi
exact hx0 (Complex.ext h.symm hi)
· exact Or.inl h
rw [(hasDerivAt_normalizedSlowFirst q m x hq hs).deriv]
have he1 : x^(q-1)*x=x^q := by
calc
_ = x^(q-1)*x^(1 : ℂ) := by rw [Complex.cpow_one]
_ = x^((q-1)+1) := (Complex.cpow_add _ _ hx0).symm
_ = _ := by congr 1; ring
have he2 : x^(q-2)*x^2=x^q := by
calc
_ = x^(q-2)*x^(2 : ℂ) := by rw [Complex.cpow_two]
_ = x^((q-2)+2) := (Complex.cpow_add _ _ hx0).symm
_ = _ := by congr 1; ring
have hK := regularizedSlowSolution_kummer_equation_closed q m x hq hx hx0
dsimp only [normalizedSlowFirst,normalizedSlowSolution]
calc
_ = x^q*x*(x*deriv (deriv (regularizedSlowSolution q m)) x+
((m : ℂ)-x)*deriv (regularizedSlowSolution q m) x-q*regularizedSlowSolution q m x) := by
linear_combination
q*(q-1)*regularizedSlowSolution q m x*he2 +
(q*((m : ℂ)-2*q)*regularizedSlowSolution q m x+
2*q*x*deriv (regularizedSlowSolution q m) x)*he1 -
q*regularizedSlowSolution q m x*he1*x
_ = 0 := by rw [hK,mul_zero]
end DefocusingNLS
end OAI
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