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 (FORMULA PRESENTED) in the limit of small dispersion, i.e., 0. When the unperturbed equation (FORMULA PRESENTED) 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 0. b. We show that in the neighborhood of a point (x, t) the solution u(x, t,) 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/). 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. © 1987 American Mathematical Society.
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