Pokhozhaev's identity is an integral relation satisfied by stationary localized solutions to a nonlinear Schrödinger equation or nonlinear Klein–Gordon equation . It was obtained by S.I. Pokhozhaev [1] and is similar to the virial theorem . This relation is also known as D.H. Derrick's theorem . Similar identities can be derived for other equations of mathematical physics.
The Pokhozhaev identity for the stationary nonlinear Schrödinger equation [ ]
Here is a general form due to H. Berestycki and P.-L. Lions .[2]
Let
g
(
s
)
{\displaystyle g(s)}
be continuous and real-valued, with
g
(
0
)
=
0
{\displaystyle g(0)=0}
.
Denote
G
(
s
)
=
∫
0
s
g
(
t
)
d
t
{\displaystyle G(s)=\int _{0}^{s}g(t)\,dt}
.
Let
u
∈
L
l
o
c
∞
(
R
n
)
,
∇
u
∈
L
2
(
R
n
)
,
G
(
u
)
∈
L
1
(
R
n
)
,
n
∈
N
,
{\displaystyle u\in L_{\mathrm {loc} }^{\infty }(\mathbb {R} ^{n}),\qquad \nabla u\in L^{2}(\mathbb {R} ^{n}),\qquad G(u)\in L^{1}(\mathbb {R} ^{n}),\qquad n\in \mathbb {N} ,}
be a solution to the equation
−
∇
2
u
=
g
(
u
)
{\displaystyle -\nabla ^{2}u=g(u)}
,
in the sense of distributions .
Then
u
{\displaystyle u}
satisfies the relation
(
n
−
2
)
∫
R
n
|
∇
u
(
x
)
|
2
d
x
=
n
∫
R
n
G
(
u
(
x
)
)
d
x
.
{\displaystyle (n-2)\int _{\mathbb {R} ^{n}}|\nabla u(x)|^{2}\,dx=n\int _{\mathbb {R} ^{n}}G(u(x))\,dx.}
The Pokhozhaev identity for the stationary nonlinear Dirac equation [ ]
Let
n
∈
N
,
N
∈
N
{\displaystyle n\in \mathbb {N} ,\,N\in \mathbb {N} }
and let
α
i
,
1
≤
i
≤
n
{\displaystyle \alpha ^{i},\,1\leq i\leq n}
and
β
{\displaystyle \beta }
be the self-adjoint Dirac matrices of size
N
×
N
{\displaystyle N\times N}
:
α
i
α
j
+
α
j
α
i
=
2
δ
i
j
I
N
,
β
2
=
I
N
,
α
i
β
+
β
α
i
=
0
,
1
≤
i
,
j
≤
n
.
{\displaystyle \alpha ^{i}\alpha ^{j}+\alpha ^{j}\alpha ^{i}=2\delta _{ij}I_{N},\quad \beta ^{2}=I_{N},\quad \alpha ^{i}\beta +\beta \alpha ^{i}=0,\quad 1\leq i,j\leq n.}
Let
D
0
=
−
i
α
⋅
∇
=
−
i
∑
i
=
1
n
α
i
∂
∂
x
i
{\displaystyle D_{0}=-\mathrm {i} \alpha \cdot \nabla =-\mathrm {i} \sum _{i=1}^{n}\alpha ^{i}{\frac {\partial }{\partial x^{i}}}}
be the massless Dirac operator .
Let
g
(
s
)
{\displaystyle g(s)}
be continuous and real-valued, with
g
(
0
)
=
0
{\displaystyle g(0)=0}
.
Denote
G
(
s
)
=
∫
0
s
g
(
t
)
d
t
{\displaystyle G(s)=\int _{0}^{s}g(t)\,dt}
.
Let
ϕ
∈
L
l
o
c
∞
(
R
n
,
C
N
)
{\displaystyle \phi \in L_{\mathrm {loc} }^{\infty }(\mathbb {R} ^{n},\mathbb {C} ^{N})}
be a spinor -valued solution that satisfies the stationary form of the nonlinear Dirac equation ,
ω
ϕ
=
D
0
ϕ
+
g
(
ϕ
∗
β
ϕ
)
β
ϕ
,
{\displaystyle \omega \phi =D_{0}\phi +g(\phi ^{\ast }\beta \phi )\beta \phi ,}
in the sense of distributions ,
with some
ω
∈
R
{\displaystyle \omega \in \mathbb {R} }
.
Assume that
ϕ
∈
H
1
(
R
n
,
C
N
)
,
G
(
ϕ
∗
β
ϕ
)
∈
L
1
(
R
n
)
.
{\displaystyle \phi \in H^{1}(\mathbb {R} ^{n},\mathbb {C} ^{N}),\qquad G(\phi ^{\ast }\beta \phi )\in L^{1}(\mathbb {R} ^{n}).}
Then
ϕ
{\displaystyle \phi }
satisfies the relation
ω
∫
R
n
ϕ
(
x
)
∗
ϕ
(
x
)
d
x
=
n
−
1
n
∫
R
n
ϕ
(
x
)
∗
D
0
ϕ
(
x
)
d
x
+
∫
R
n
G
(
ϕ
(
x
)
∗
β
ϕ
(
x
)
)
d
x
.
{\displaystyle \omega \int _{\mathbb {R} ^{n}}\phi (x)^{\ast }\phi (x)\,dx={\frac {n-1}{n}}\int _{\mathbb {R} ^{n}}\phi (x)^{\ast }D_{0}\phi (x)\,dx+\int _{\mathbb {R} ^{n}}G(\phi (x)^{\ast }\beta \phi (x))\,dx.}
See also [ ]
References [ ]