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 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.

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

Let ${\displaystyle g(s)}$ be continuous and real-valued, with ${\displaystyle g(0)=0}$. Denote ${\displaystyle G(s)=\int _{0}^{s}g(t)\,dt}$. Let

${\displaystyle u\in L_{\mathrm {loc} }^{\infty }(\mathbb {R} ^{n}),\qquad \nabla u\in L^{2}(\mathbb {R} ^{n}),\qquad G(u)\in L^{1}(\mathbb {R} ^{n}),\qquad n\in \mathbb {N} ,}$

be a solution to the equation

${\displaystyle -\nabla ^{2}u=g(u)}$,

in the sense of distributions. Then ${\displaystyle u}$ satisfies the relation

${\displaystyle {\frac {n-2}{2}}\int _{\mathbb {R} ^{n}}|\nabla u(x)|^{2}\,dx=n\int _{\mathbb {R} ^{n}}G(u(x))\,dx.}$

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 ${\displaystyle n\in \mathbb {N} ,\,N\in \mathbb {N} }$ and let ${\displaystyle \alpha ^{i},\,1\leq i\leq n}$ and ${\displaystyle \beta }$ be the self-adjoint Dirac matrices of size ${\displaystyle N\times N}$:

${\displaystyle \alpha ^{i}\alpha ^{j}+\alpha ^{j}\alpha ^{i}=2\delta _{ij}I_{N},\quad \beta ^{2}=I_{N},\quad \alpha ^{i}\beta +\beta \alpha ^{i}=0,\quad 1\leq i,j\leq n.}$

Let ${\displaystyle D_{0}=-\mathrm {i} \alpha \cdot \nabla =-\mathrm {i} \sum _{i=1}^{n}\alpha ^{i}{\frac {\partial }{\partial x^{i}}}}$ be the massless Dirac operator. Let ${\displaystyle g(s)}$ be continuous and real-valued, with ${\displaystyle g(0)=0}$. Denote ${\displaystyle G(s)=\int _{0}^{s}g(t)\,dt}$. Let ${\displaystyle \phi \in L_{\mathrm {loc} }^{\infty }(\mathbb {R} ^{n},\mathbb {C} ^{N})}$ be a spinor-valued solution that satisfies the stationary form of the nonlinear Dirac equation,

${\displaystyle \omega \phi =D_{0}\phi +g(\phi ^{\ast }\beta \phi )\beta \phi ,}$

in the sense of distributions, with some ${\displaystyle \omega \in \mathbb {R} }$. Assume that

${\displaystyle \phi \in H^{1}(\mathbb {R} ^{n},\mathbb {C} ^{N}),\qquad G(\phi ^{\ast }\beta \phi )\in L^{1}(\mathbb {R} ^{n}).}$

Then ${\displaystyle \phi }$ satisfies the relation

${\displaystyle \omega \int _{\mathbb {R} ^{n}}\phi (x)^{\ast }\phi (x)\,dx={\frac {n-1}{n}}\int _{\mathbb {R} ^{n}}\phi (x)^{\ast }D_{0}\phi (x)\,dx+\int _{\mathbb {R} ^{n}}G(\phi (x)^{\ast }\beta \phi (x))\,dx.}$

• Virial theorem
• Derrick's theorem