Pokhozhaev's identity is an integral relation satisfied by stationary localized solutions to a nonlinear Schrödinger equation or nonlinear Klein–Gordon equation.
Pokhozhaev[1] 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.
Here is a general form due to H. Berestycki and P.-L.
Lions.
be continuous and real-valued, with
Let be a solution to the equation in the sense of distributions.
satisfies the relation There is a form of the virial identity for the stationary nonlinear Dirac equation in three spatial dimensions (and also the Maxwell-Dirac equations)[3] and in arbitrary spatial dimension.
β
be the self-adjoint Dirac matrices of size
be the massless Dirac operator.
be continuous and real-valued, with
Denote
{\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, in the sense of distributions, with some
Assume that Then
satisfies the relation