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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 | import OAI.MathematicalPhysics.DefocusingNLS.Profile.CartesianDirectionalCalculus
namespace OAI
/-! The dilation transport and its exact commutator with the Laplacian. -/
open scoped ContDiff Laplacian
namespace DefocusingNLS
local notation "E" => EuclideanSpace ℝ (Fin 12)
noncomputable def cartesianTransport (f : E → ℂ) (x : E) : ℂ :=
fderiv ℝ f x ((1/2 : ℝ) • x)
theorem cartesianTransport_contDiff (f : E → ℂ) (hf : ContDiff ℝ ∞ f) :
ContDiff ℝ ∞ (cartesianTransport f) :=
(contDiff_infty_iff_fderiv.mp hf).2.clm_apply (contDiff_id.const_smul (1/2 : ℝ))
theorem cartesianDerivative_cartesianTransport (f : E → ℂ) (hf : ContDiff ℝ ∞ f)
(v : E) :
cartesianDerivative v (cartesianTransport f)=fun x =>
cartesianTransport (cartesianDerivative v f) x+
(1/2 : ℂ)*cartesianDerivative v f x := by
funext x
exact cartesianDerivative_transport f hf x v
theorem cartesianTransport_sum (F : Fin 12 → E → ℂ)
(hF : ∀ j, ContDiff ℝ ∞ (F j)) :
cartesianTransport (fun x => ∑ j : Fin 12, F j x)=
fun x => ∑ j : Fin 12, cartesianTransport (F j) x := by
funext x
exact congrFun (cartesianDerivative_sum F hF ((1/2 : ℝ) • x)) x
theorem cartesianDerivative_twice_transport (f : E → ℂ) (hf : ContDiff ℝ ∞ f)
(v x : E) :
cartesianDerivative v (cartesianDerivative v (cartesianTransport f)) x=
cartesianTransport (cartesianDerivative v (cartesianDerivative v f)) x+
cartesianDerivative v (cartesianDerivative v f) x := by
have hd := cartesianDerivative_contDiff f hf v
have ht := cartesianTransport_contDiff (cartesianDerivative v f) hd
rw [cartesianDerivative_cartesianTransport f hf v]
have hs : cartesianDerivative v (fun y =>
cartesianTransport (cartesianDerivative v f) y+
(1/2 : ℂ)*cartesianDerivative v f y) x=
cartesianDerivative v (cartesianTransport (cartesianDerivative v f)) x+
(1/2 : ℂ)*cartesianDerivative v (cartesianDerivative v f) x := by
change fderiv ℝ (fun y => cartesianTransport (cartesianDerivative v f) y+
(1/2 : ℂ)*cartesianDerivative v f y) x v=
fderiv ℝ (cartesianTransport (cartesianDerivative v f)) x v+
(1/2 : ℂ)*fderiv ℝ (cartesianDerivative v f) x v
rw [fderiv_fun_add (ht.differentiable (by simp) x)
((hd.differentiable (by simp) x).const_mul (1/2 : ℂ)),
fderiv_const_mul (hd.differentiable (by simp) x)]
simp only [add_apply,smul_apply,smul_eq_mul]
rw [hs,cartesianDerivative_cartesianTransport _ hd v]
dsimp only
ring
theorem laplacian_cartesianTransport (f : E → ℂ) (hf : ContDiff ℝ ∞ f) :
Δ (cartesianTransport f)=fun x => cartesianTransport (Δ f) x+Δ f x := by
let e := EuclideanSpace.basisFun (Fin 12) ℝ
rw [laplacian_eq_cartesianDerivatives _ (cartesianTransport_contDiff f hf)]
have hi (x : E) :
(∑ j : Fin 12, cartesianDerivative (e j)
(cartesianDerivative (e j) (cartesianTransport f)) x)=
(∑ j : Fin 12, cartesianTransport
(cartesianDerivative (e j) (cartesianDerivative (e j) f)) x)+
∑ j : Fin 12, cartesianDerivative (e j) (cartesianDerivative (e j) f) x := by
simp_rw [cartesianDerivative_twice_transport f hf]
exact Finset.sum_add_distrib
funext x
rw [hi]
rw [laplacian_eq_cartesianDerivatives f hf,
cartesianTransport_sum _ (fun j => cartesianDerivative_contDiff _
(cartesianDerivative_contDiff f hf (e j)) (e j))]
end DefocusingNLS
end OAI
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