Stability of Peakons for the Degasperis-Procesi Equation

It is found that the first term
I =
∫ ξi
η i−1
{− (∂ xx v) 3 + 12 (∂ xx v) 2 v + 27∂ xx v (∂ x v) 2 − 108v∂ xx v∂ x v + 60v 2 ∂ xx v
∫ ξi
− 54 (∂ x v) 3 + 216 (∂ x v) 2 v − 216v 2 ∂ x v + 64v 3 }dx
= {− (∂ xx v) 3 + 12 (∂ xx v) 2 v + 54 (∂ x v) 3 + 60v 2 ∂ xx v − 54 (∂ x v) 3
η i−1
− 108v 2 ∂ xx v + 64v 3 }dx − 72
(v (ξ i ) 3 − v (η i−1 ) 3)
∫ ξi [
= − (∂ xx v) 3 + 12 (∂ xx v) 2 v − 48v 2 ∂ xx v + 64v 3] (
dx − 72 v (ξ i ) 3 − v (η i−1 ) 3) ,
η i−1
where use has been made of the following integral identities due to integration
by parts and ∂ x v (ξ i ) = ∂ x v (η i−1 ) = 0,
∫ ξi
∂ xx v (∂ x v) 2 dx = 1 ∫ ξi (
∂ x (∂ x v) 3) dx = 0
η i−1
3 η i−1
∫ ξi
η i−1
v∂ xx v∂ x v dx =
∫ ξi
η i−1
(∂ x v) 2 v dx =
∫ ξi
η i−1
v 2 ∂ x vdx =
∫ ξi
η i−1
v∂ x
( 1
2 (∂ xv) 2 )
dx = − 1 2
∫ ξi
∫ ξi
η i−1
∂ x v∂ x
( 1
2 v2 )
dx = − 1 2
η i−1
1
3 ∂ x
(
v
3 ) dx = 1 3
∫ ξi
∫ ξi
η i−1
(∂ x v) 3 dx,
v 2 ∂ xx vdx,
η i−1
(
v (ξ i ) 3 − v (η i−1 ) 3) .
Similarly,
∫ ηi [
II = − (∂ xx v) 3 + 12 (∂ xx v) 2 v − 48v 2 ∂ xx v + 64v 3] dx+72
(v (η i ) 3 − v (ξ i ) 3)
ξ i
and thus
∫ ηi
η i−1
h (x) g 2 (x)dx =
∫ ηi [
− (∂ xx v) 3 + 12 (∂ xx v) 2 v − 48v 2 ∂ xx v + 64v 3] dx
η i−1
(
− 144v (ξ i ) 3 + 72 v (η i−1 ) 3 + v (η i ) 3) .
By adding up the above integral from 1 to n, we get
∫
∫ [
h(x)g 2 (x)dx = − (∂ xx v) 3 + 12 (∂ xx v) 2 v − 48v 2 ∂ xx v + 64v 3] dx
R
R
n∑
n∑
− 144 v (ξ i ) 3 + 72
(v (η i−1 ) 3 + v (η i ) 3)
i=1
= E 3 (u) − 144
( n
∑
i=1
i=1
n−1
∑
Mi 3 −
i=1
m 3 i
)
.
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