= Energy–momentum slope of a solitary wave
{title2=$dE/dp_x=v$}
With $E=\frac12\int[|\nabla\psi|^2+\frac12(1-|\psi|^2)^2]$ and the <renormalized momentum of a condensate>, a <Gross–Pitaevskii solitary wave> is a critical point of $E-vp_x$. Along a differentiable solution family, fixed bulk normalization and vanishing variation boundary terms give $dE/d\lambda=v\,dp_x/d\lambda$. Therefore $dE/dp_x=v$ wherever $dp_x/d\lambda\ne0$. At a cusp or turning point the parametrized identity is the appropriate statement; one must not assume a globally single-valued dispersion branch.
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