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 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.[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

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

The Pokhozhaev identity for the stationary nonlinear Dirac equation

Let ${\displaystyle n\in \mathbb {N} ,\,N\in \mathbb {N} }$ 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

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

See also

• Virial theorem
• Derrick's theorem

References

1. Pokhozhaev, S.I. (1965). "On the eigenfunctions of the equation ${\displaystyle \Delta u+\lambda f(u)=0}$". 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. doi:10.1007/BF00250555.CS1 maint: multiple names: authors list (link)