Pokhozhaev's identity

Pokhozhaev's identity is an integral relation satisfied by stationary localized solutions to a nonlinear wave equation or nonlinear Klein–Gordon equation. It was obtained by S.I. Pokhozhaev and is similar to the Virial theorem. This relation is also known as D.H. Derrick's theorem.

The Pokhozhaev identity for the stationary nonlinear Schrödinger equation

Here is a slightly more general form obtained by H. Berestycki and P.-L. Lions.

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

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

be a solution to the equation

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

in the sense of distributions. Then $u$ satisfies the relation

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

The Pokhozhaev identity for the stationary nonlinear Dirac equation

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

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

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

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

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

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

Then $\phi$ satisfies the relation

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

See also

• Virial identity
• Derrick's theorem