Remark on the phase shift in the Kuzmak–Whitham ansatz

We consider one-phase (formal) asymptotic solutions in the Kuzmak–Whitham form for the nonlinear Klein–Gordon equation and for the Korteweg–de Vries equation. In this case, the leading asymptotic expansion term has the form $X(S(x,t)/h+\Phi(x,t),I(x,t),x,t)+O(h)$, where $h\ll1$ is a small parameter and the phase $S(x,t)$ and slowly changing parameters $I(x,t)$ are to be found from the system of “averaged” Whitham equations. We obtain the equations for the phase shift $\Phi(x,t)$ by studying the second-order correction to the leading term. The corresponding procedure for finding the phase shift is then nonuniform with respect to the transition to a linear (and weakly nonlinear) case. Our observation, which essentially follows from papers by Haberman and collaborators, is that if we incorporate the phase shift $\Phi$ into the phase and adjust the parameter $\tilde{I}$ by setting $\widetilde{S}=S+h\Phi+O(h^2)$, $\tilde{I}=I+hI_1+O(h^2)$, then the functions $\widetilde{S}(x,t,h)$ and $\tilde{I}(x,t,h)$ become solutions of the Cauchy problem for the same Whitham system but with modified initial conditions. These functions completely determine the leading asymptotic term, which is $X(\widetilde{S}(x,t,h)/h,\tilde{I}(x,t,h),x,t)+O(h)$.