In mathematical physics or theory of partial differential equations, the solitary wave solution of the form
u
(
x
,
t
)
=
e
−
i
ω
t
ϕ
(
x
)
is said to be orbitally stable if any solution with the initial data sufficiently close to
ϕ
(
x
)
forever remains in a given small neighborhood of the trajectory of
e
−
i
ω
t
ϕ
(
x
)
.
Formal definition is as follows. Let us consider the dynamical system
i
d
u
d
t
=
A
(
u
)
,
u
(
t
)
∈
X
,
t
∈
R
,
with
X
a Banach space over
C
, and
A
:
X
→
X
. We assume that the system is
U
(
1
)
-invariant, so that
A
(
e
i
s
u
)
=
e
i
s
A
(
u
)
for any
u
∈
X
and any
s
∈
R
.
Assume that
ω
ϕ
=
A
(
ϕ
)
, so that
u
(
t
)
=
e
−
i
ω
t
ϕ
is a solution to the dynamical system. We call such solution a solitary wave.
We say that the solitary wave
e
−
i
ω
t
ϕ
is orbitally stable if for any
ϵ
>
0
there is
δ
>
0
such that for any
v
0
∈
X
with
∥
ϕ
−
v
0
∥
X
<
δ
there is a solution
v
(
t
)
defined for all
t
≥
0
such that
v
(
0
)
=
v
0
, and such that this solution satisfies
sup
t
≥
0
inf
s
∈
R
∥
v
(
t
)
−
e
i
s
ϕ
∥
X
<
ϵ
.
According to , the solitary wave solution
e
−
i
ω
t
ϕ
ω
(
x
)
to the nonlinear Schrödinger equation
i
∂
∂
t
u
=
−
∂
2
∂
x
2
u
+
g
(
|
u
|
2
)
u
,
u
(
x
,
t
)
∈
C
,
x
∈
R
,
t
∈
R
,
where
g
is a smooth real-valued function, is orbitally stable if the Vakhitov–Kolokolov stability criterion is satisfied:
d
d
ω
Q
(
ϕ
ω
)
<
0
,
where
Q
(
u
)
=
1
2
∫
R
|
u
(
x
,
t
)
|
2
d
x
is the charge of the solution
u
(
x
,
t
)
, which is conserved in time (at least if the solution
u
(
x
,
t
)
is sufficiently smooth).
It was also shown, that if
d
d
ω
Q
(
ω
)
<
0
at a particular value of
ω
, then the solitary wave
e
−
i
ω
t
ϕ
ω
(
x
)
is Lyapunov stable, with the Lyapunov function given by
L
(
u
)
=
E
(
u
)
−
ω
Q
(
u
)
+
Γ
(
Q
(
u
)
−
Q
(
ϕ
ω
)
)
2
, where
E
(
u
)
=
1
2
∫
R
(
|
∂
u
∂
x
|
2
+
G
(
|
u
|
2
)
)
d
x
is the energy of a solution
u
(
x
,
t
)
, with
G
(
y
)
=
∫
0
y
g
(
z
)
d
z
the antiderivative of
g
, as long as the constant
Γ
>
0
is chosen sufficiently large.
