Quartic-time transverse phase scaling
= Quartic-time transverse phase scaling
{title2=$Y=\varepsilon y,\quad T=\varepsilon^4t$}
At a transverse phase-diffusion threshold the coefficient of the second spatial derivative is of order $\varepsilon^2$. Its product with two long-wave derivatives therefore has order $\varepsilon^4$, the same order as the stabilizing fourth derivative. This balance determines the slow time $T=\varepsilon^4t$ and retains both terms in the leading <phase modulation> equation.