In mathematical physics, the Degasperis–Procesi equation
u
t
−
u
x
x
t
+
2
κ
u
x
+
4
u
u
x
=
3
u
x
u
x
x
+
u
u
x
x
x
is one of only two exactly solvable equations in the following family of third-order, non-linear, dispersive PDEs:
u
t
−
u
x
x
t
+
2
κ
u
x
+
(
b
+
1
)
u
u
x
=
b
u
x
u
x
x
+
u
u
x
x
x
,
where
κ
and b are real parameters (b=3 for the Degasperis–Procesi equation). It was discovered by Degasperis and Procesi in a search for integrable equations similar in form to the Camassa–Holm equation, which is the other integrable equation in this family (corresponding to b=2); that those two equations are the only integrable cases has been verified using a variety of different integrability tests. Although discovered solely because of its mathematical properties, the Degasperis–Procesi equation (with
κ
>
0
) has later been found to play a similar role in water wave theory as the Camassa–Holm equation.
Among the solutions of the Degasperis–Procesi equation (in the special case
κ
=
0
) are the so-called multipeakon solutions, which are functions of the form
u
(
x
,
t
)
=
∑
i
=
1
n
m
i
(
t
)
e
−
|
x
−
x
i
(
t
)
|
where the functions
m
i
and
x
i
satisfy
x
˙
i
=
∑
j
=
1
n
m
j
e
−
|
x
i
−
x
j
|
,
m
˙
i
=
2
m
i
∑
j
=
1
n
m
j
sgn
(
x
i
−
x
j
)
e
−
|
x
i
−
x
j
|
.
These ODEs can be solved explicitly in terms of elementary functions, using inverse spectral methods.
When
κ
>
0
the soliton solutions of the Degasperis–Procesi equation are smooth; they converge to peakons in the limit as
κ
tends to zero.
The Degasperis–Procesi equation (with
κ
=
0
) is formally equivalent to the (nonlocal) hyperbolic conservation law
∂
t
u
+
∂
x
[
u
2
2
+
G
2
∗
3
u
2
2
]
=
0
,
where
G
(
x
)
=
exp
(
−
|
x
|
)
, and where the star denotes convolution with respect to x. In this formulation, it admits weak solutions with a very low degree of regularity, even discontinuous ones (shock waves). In contrast, the corresponding formulation of the Camassa–Holm equation contains a convolution involving both
u
2
and
u
x
2
, which only makes sense if u lies in the Sobolev space
H
1
=
W
1
,
2
with respect to x. By the Sobolev embedding theorem, this means in particular that the weak solutions of the Camassa–Holm equation must be continuous with respect to x.
