The zero dispersion limit of the Korteweg-de Vries equation with periodic initial data

Abstract:

We study the initial value problem for the Korteweg-de Vries equation \[ ({\text {i}})\quad {u_t} - 6u{u_x} + {\varepsilon ^2}{u_{xxx}} = 0\] in the limit of small dispersion, i.e., $\varepsilon \to 0$. When the unperturbed equation \[ ({\text {ii}})\quad {u_t} - 6u{u_x} = 0\] develops a shock, rapid oscillations arise in the solution of the perturbed equation (i) In our study: a. We compute the weak limit of the solution of (i) for periodic initial data as $\varepsilon \to 0$. b. We show that in the neighborhood of a point $(x, t)$ the solution $u(x, t, \varepsilon )$ can be approximated either by a constant or by a periodic or by a quasiperiodic solution of equation (i). In the latter case the associated wavenumbers and frequencies are of order $O(1/\varepsilon )$. c. We compute the number of phases and the wave parameters associated with each phase of the approximating solution as functions of $x$ and $t$. d. We explain the mechanism of the generation of oscillatory phases. Our computations in a and c are subject to the solution of the Lax-Levermore evolution equations (7.7). Our results in b-d rest on a plausible averaging assumption.