Past exam of the mathematics course of the University of Cambridge / 2007 / iii / Paper 84 / 2

To first order, $|1+u+iv{|}^2=1+2u$ and $(1-|\psi {|}^2)\psi =-2u$. Equating real and imaginary parts of the Gross–Pitaevskii equation therefore givesDifferentiate the second equation in time and use the first to eliminate $v$:For a Fourier mode with $k=|\mathbf{k}|$, the dispersion relation isThis is the normalized Bogoliubov spectrum. At small wavenumber the positive branch has sound speed ${\lim }_{k\to 0}{\omega }_{+}(k)/k=1/\sqrt{2}$; at large wavenumber it approaches the free-particle quadratic dispersion. The linear frequencies are real, confirming stability of the repulsive uniform bulk.

Introduce the translating coordinate $Z=z-Ut$ and put $\psi (\mathbf{x},t)=\Psi (x,y,Z)$. Then ${\psi }_t=-U{\Psi }_Z$. Substitution in the Gross–Pitaevskii equation gives the stationary Gross–Pitaevskii solitary wave equationThe complex bulk field is fixed by $\Psi \to 1$ far from the localized disturbance. Spatial derivatives in this equation are taken in the translating frame. There is no additional uniform temporal phase here because the normalization has already made the background $\psi =1$ stationary.

Use decaying variations with the same complex bulk field one, so that integration by parts produces no variation boundary term. The functional derivatives of energy and renormalized momentum of a condensate areFor the second identity, variation of the two momentum terms and integration of the ${\partial }_z\delta {\psi }^{\ast }$ term each supply $(2i{)}^{-1}{\psi }_z$, so together they give $-i{\psi }_z$. It follows thatis exactly the stationary Gross–Pitaevskii solitary wave equation. Thus each solitary wave is a constrained energy critical point with multiplier $U$.

Let $s$ parametrize a differentiable family of these waves. Apply the critical-point identity to the variation ${\partial }_s\psi$, keeping the multiplier fixed at its value for that member of the family. Including the complex conjugate variation givesHence the energy–momentum slope of a solitary wave iswherever $dp/ds\ne 0$. At a momentum turning point the parametrized identity $dE/ds=Udp/ds$ remains the correct statement; a globally single-valued energy as a function of momentum is not needed. Differentiating $E-U(s)p$ as a composite function would include an extra $-p{U}^{\prime }$ term, so that composite derivative must not be set to zero.

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.