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For a finite-energy Gross–Pitaevskii solitary wave, write $E=E_z+E_{\perp }+E_{\mathrm{pot}}$ with $E_z=\frac{1}{2}\int |{\psi }_z{|}^2$. The longitudinal dilation ${\psi }_{\lambda }(x_{\perp },z)=\psi (x_{\perp },\lambda z)$ leaves its renormalized momentum of a condensate invariant and changes energy to $\lambda E_z+{\lambda }^{-1}(E_{\perp }+E_{\mathrm{pot}})$. Stationarity of $E-Up$ at $\lambda =1$ implies $E_z=E_{\perp }+E_{\mathrm{pot}}$ and hence the displayed identity. This is a directional Pohozaev identity, valid when the integrals and variation boundary terms converge.

Split the solitary-wave energy into longitudinal, transverse and interaction contributions,Consider the longitudinal dilation ${\psi }_{\lambda }(x,y,z)=\psi (x,y,\lambda z)$ for $\lambda >0$. Changing variables to $Z=\lambda z$ givesThe renormalized momentum of a condensate is unchanged: its single $z$ derivative contributes $\lambda$, cancelling the measure factor ${\lambda }^{-1}$. Since the wave is a critical point of $E-Up$, differentiating this admissible scaling variation at $\lambda =1$ givesTherefore the longitudinal scaling identity for a condensate solitary wave yieldsThis directional Pohozaev identity assumes the finite energy and sufficient decay to justify the dilation variation; it can equivalently be obtained with cutoffs followed by a limit. It is an identity for solitary-wave critical points, not for arbitrary localized fields.

A Pohozaev identity is an integral relation obtained by testing a nonlinear partial differential equation against a scaling direction such as $x\cdot \nabla Q$. For a sufficiently regular, rapidly decaying real solution ofwrite $A=\int |\nabla Q{|}^2$, $B=\int |Q{|}^2$ and $P=\int |Q{|}^{p+1}$. The identity isOne derivation uses the stationary energy functional $S(Q)=A/2+B/2-P/(p+1)$. Evaluate it on $Q(x/\lambda )$: the three terms scale as ${\lambda }^{d-2}$, ${\lambda }^d$ and ${\lambda }^d$. Differentiating at $\lambda =1$ gives the displayed formula, with decay justifying the scaling variation. Direct integration by parts with $x\cdot \nabla Q$ gives the same identity.

Testing against $Q$ also gives $A+B=P$. Combining the two relations at $p=1+4/d$ yields the Pohozaev identity for the mass-critical NLS ground state. The usefulness is that a differential equation imposes exact constraints on global kinetic, mass and nonlinear integrals.

Testing the ground-state equation against $Q$ and applying the Pohozaev identity givesConsequently $E(Q)=0$ and $\inf J=\frac{d}{d+2}\Vert Q{\Vert }_2^{4/d}$, where $J$ is the Weinstein functional.