On the phase shift in the Kuzmak—Whitham ansatz for nonlinear waves

We consider one-phase (formal) asymptotic solutions in the Kuzmak-Whitham form for evolutionary nonlinear equations. In this case, the leading asymptotic expansion term has the form X(S(x, t)/ɛ + ϕ(x, t), A(x, t), x, t) + O(ɛ), where ɛ is a small parameter and the phase S(x, t) and slowly changing parameters A(x, t) are to be found from the system of averaged Whitham equations. The equation for the phase shift ϕ(x, t) is appearing 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. We formulate the general conjecture (checked for some examples), which essentially follows from papers by R. Haberman and collaborators, is that if one incorporates the phase shift ϕ(x, t) into the phase and adjust A by setting S → (x, t, ɛ) = S + ɛϕ + O(h2), A → Ã(x, t, ɛ) = A + ɛa + O(ɛ²), then the new functions (x, t, ɛ) and Ã(x, t, ɛ) become solutions of the Cauchy problem for the same Whitham system but with modified initial conditions. These functions completely determine the leading asymptotic term in the Whitham method.