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 and is similar to the virial theorem. This relation is also known as G.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.
Let formula_1 be continuous and real-valued, with formula_2.
Denote formula_3.
Let
formula_4
be a solution to the equation
formula_5,
in the sense of distributions. 
Then formula_6 satisfies the relation
formula_7
The Pokhozhaev identity for the stationary nonlinear Dirac equation.
There is a form of the virial identity for the stationary nonlinear Dirac equation in three spatial dimensions (and also the Maxwell-Dirac equations) and in arbitrary spatial dimension.
Let formula_8
and let formula_9 and formula_10 be the self-adjoint Dirac matrices of size formula_11:
formula_12
Let formula_13 be the massless Dirac operator.
Let formula_1 be continuous and real-valued, with formula_2.
Denote formula_3.
Let formula_17 be a spinor-valued solution that satisfies the stationary form of the nonlinear Dirac equation,
formula_18
in the sense of distributions,
with some formula_19.
Assume that
formula_20
Then formula_21 satisfies the relation
formula_22