 Whitham modulation equation

Fix the phase convention $k={\Theta }_X$, $\omega =-{\Theta }_T$. If the averaged action is $\int \overline{L}(k,\omega ,\dots )dXdT$, a compactly supported phase variation gives $\delta k={\partial }_X\delta \Theta$, $\delta \omega =-{\partial }_T\delta \Theta$. Integration by parts then yieldsDefining 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.