= Whitham modulation equation
{c}
Fix the phase convention $k=\Theta_X$, $\omega=-\Theta_T$. If the averaged action is $\int\overline L(k,\omega,\ldots)\,dX\,dT$, a compactly supported phase variation gives $\delta k=\partial_X\delta\Theta$, $\delta\omega=-\partial_T\delta\Theta$. <Integration by parts> then yields
$$
\partial_X\overline L_k-\partial_T\overline L_\omega=0.
$$
Defining the <wave-action density> $W=\overline L_\omega$ and its flux $J=-\overline L_k$ puts the same equation in the conservative form $W_T+J_X=0$. This sign follows from the declared negative time derivative of phase. It must be combined with $k_T+\omega_X=0$ and any profile-parameter equations to close a modulation problem.
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