Propriétés algébriques des opérateurs pseudodifférentiels

dui
dt =Pi(u, u0, u00, ...), i ∈I={1, ..., l}
ui(t, x)M1
t x ∈M Pi
u= (ui)i∈Iu(n)= (∂nui
∂xn)i∈I
du
dt =u000 + 6uu0
ui(n)
AI=C[ui(n)|i∈I, n ≥0]
∂ ∂(ui(n)) = ui(n+1)
R AI
∂
∂ui(n)
AIf∈R∂f
∂ui(n)= 0
(i, n)R ∂ AI
∂=X
i,n
ui(n+1) ∂
∂ui(n)
AIf∈AI
Q
C[√u−1, u, u0, ...]

(0.1) XP
R
XP=X
i,n
(∂nPi)∂
∂ui(n)
XP(f)f t
(0.1) [∂, XP]=0
R
XPP∈Rl
Der(R)
[XP, XQ] = XXP(Q)−XQ(P)
R/∂R f ∈R
R/∂R Rf
Rf
(0.1)
Zdf
dt =ZXP(f)=0
f
Zδf
δu.P = 0
δf
δu ∈Rlf
(δf
δu)i=δf δui
=X
n
(−∂)n∂f
∂ui(n)
f1=u , f2=u2, f3=−(u0)2+ 2u3
df1
dt = (u00 + 3u2)0

df2
dt = (2uu00 −(u0)2+ 4u3)0
df3
dt = (9u4+ 6u2u00 + (u00)2−12u(u0)2−2u0u000)0
F∈Rl
DF(DF(P))k=Dfk(P) =
XP(fk)
Ω(R) = e Ω(R)/∂ e Ω(R)
Ω0R/∂R Ω1RlΩ2
Rf∈R/∂R F ∈Rl
d(Zf) = δf
δu , d(F) = DF−DF∗
eu=u+τG
Gdeu
dt =F(eu, eu0, ...)τ
deu
dt −F(eu, eu0, ...) = τ(XF(G)−XG(F)) + o(τ)
G∈Rl
[XF, XG] = 0 ut=F
uτ=G
(Gn)n≥0
[XGn, XGm]=0 ∀(n, m)∈N2

1 3 5
G1=u0
G3=u000 + 6uu0=F
G5=u(5) + 10uu000 + 20u0u00 + 30u2u0
ut=P
A D D
D(ab) = D(a)b+aD(b)∀(a, b)∈A2
a0=D(a)a∈A A
∂
A[∂] = {X
n≤N
an∂n|N∈N, an∈A}
A
∂.a =a.∂ +a0∀a∈A