Quartic-time transverse phase scaling ($Y=\epsilon y,T={\epsilon }^{4}t$)

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