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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 | import OAI.MathematicalPhysics.DefocusingNLS.Profile.SlowHigherDerivatives
namespace OAI
/-! # The differential recurrence of the normalized outgoing solution -/
open Filter Topology
namespace DefocusingNLS
noncomputable def normalizedSlowSolution (q : ℂ) (m : ℕ) (x : ℂ) : ℂ :=
x ^ q * regularizedSlowSolution q m x
theorem hasDerivAt_normalizedSlowSolution (q : ℂ) (m : ℕ) (x : ℂ)
(hq : -1 < q.re) (hx : 0 ≤ x.re) (hx0 : x ≠ 0) :
HasDerivAt (normalizedSlowSolution q m)
(-q * ((m : ℂ) - 1 - q) / x ^ 2 * normalizedSlowSolution (q + 1) m x) x := by
have hslit : x ∈ Complex.slitPlane := by
rcases hx.eq_or_lt with hr | hr
· apply Complex.mem_slitPlane_iff.mpr
right
intro hi
exact hx0 (Complex.ext hr.symm hi)
· exact Complex.mem_slitPlane_iff.mpr (Or.inl hr)
have hp := (hasDerivAt_id x).cpow_const (c := q) hslit
have hh := hasDerivAt_regularizedSlowSolution_shift_closed q m x hq hx hx0
have hc := regularizedSlowSolution_contiguous q m x hq hx hx0
have he : x ^ (q + 1) = x ^ (q - 1) * x ^ (2 : ℕ) := by
calc
x ^ (q + 1) = x ^ ((q - 1) + 2) := by congr 1; ring
_ = _ := by rw [Complex.cpow_add _ _ hx0, Complex.cpow_two]
have he1 : x ^ q = x ^ (q - 1) * x := by
calc
x ^ q = x ^ ((q - 1) + 1) := by congr 1; ring
_ = _ := by rw [Complex.cpow_add _ _ hx0, Complex.cpow_one]
convert! hp.mul hh using 1
simp only [normalizedSlowSolution, id_eq, he, he1, mul_one]
rw [hc]
field_simp
ring
/-- The normalized solution is uniformly bounded for large arguments in the
closed right half-plane, including the two imaginary rays. -/
theorem normalizedSlowSolution_bounded (q : ℂ) (m : ℕ) (hq : -1 < q.re) :
∃ C : ℝ, 0 ≤ C ∧ ∀ x : ℂ, 0 ≤ x.re → 1 ≤ ‖x‖ →
‖normalizedSlowSolution q m x‖ ≤ C := by
obtain ⟨A, hA, hb⟩ := regularizedSlowSolution_first_remainder q m hq
refine ⟨A + 1, by linarith, ?_⟩
intro x hx hn
calc
‖normalizedSlowSolution q m x‖ ≤ ‖normalizedSlowSolution q m x - 1‖ + 1 := by
simpa only [norm_one] using norm_le_norm_sub_add (normalizedSlowSolution q m x) 1
_ ≤ A / ‖x‖ + 1 := add_le_add (hb x hx hn) le_rfl
_ ≤ A + 1 := add_le_add (div_le_self hA hn) le_rfl
end DefocusingNLS
end OAI
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