Pokhozhaev's identity


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 :ru:%D0%9F%D0%BE%D1%85%D0%BE%D0%B6%D0%B0%D0%B5%D0%B2,_%D0%A1%D1%82%D0%B0%D0%BD%D0%B8%D1%81%D0%BB%D0%B0%D0%B2_%D0%98%D0%B2%D0%B0%D0%BD%D0%BE%D0%B2%D0%B8%D1%87|S.I. Pokhozhaev 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.
Let be continuous and real-valued, with.
Denote.
Let
be a solution to the equation
in the sense of distributions.
Then satisfies the relation

The Pokhozhaev identity for the stationary nonlinear Dirac equation

Let
and let and be the self-adjoint Dirac matrices of size :
Let be the massless Dirac operator.
Let be continuous and real-valued, with.
Denote.
Let 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