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 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

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Here is a general form due to H. Berestycki and P.-L. Lions.[2]

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

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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.[4] 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

 

See also

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References

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  1. ^ Pokhozhaev, S.I. (1965). "On the eigenfunctions of the equation  ". Dokl. Akad. Nauk SSSR. 165: 36–39.
  2. ^ Berestycki, H. and Lions, P.-L. (1983). "Nonlinear scalar field equations, I. Existence of a ground state". Arch. Rational Mech. Anal. 82 (4): 313–345. Bibcode:1983ArRMA..82..313B. doi:10.1007/BF00250555. S2CID 123081616.{{cite journal}}: CS1 maint: multiple names: authors list (link)
  3. ^ Esteban, M. and Séré, E. (1995). "Stationary states of the nonlinear Dirac equation: A variational approach". Commun. Math. Phys. 171 (2): 323–350. doi:10.1007/BF02099273. S2CID 120901245.{{cite journal}}: CS1 maint: multiple names: authors list (link)
  4. ^ Boussaid, N. and Comech, A. (2019). Nonlinear Dirac equation. Spectral stability of solitary waves. Mathematical Surveys and Monographs. Vol. 244. American Mathematical Society. doi:10.1090/surv/244. ISBN 978-1-4704-4395-5. S2CID 216380644.{{cite book}}: CS1 maint: multiple names: authors list (link)