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

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.