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

Author:
Stephanos Venakides

Journal:
Trans. Amer. Math. Soc. **301** (1987), 189-226

MSC:
Primary 35Q20; Secondary 35B25, 35L67

DOI:
https://doi.org/10.1090/S0002-9947-1987-0879569-7

MathSciNet review:
879569

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Abstract | References | Similar Articles | Additional Information

Abstract: We study the initial value problem for the Korteweg-de Vries equation

When the unperturbed equation

In our study:

a. We compute the weak limit of the solution of (i) for periodic initial data as .

b. We show that in the neighborhood of a point the solution 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 .

c. We compute the number of phases and the wave parameters associated with each phase of the approximating solution as functions of and .

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.

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DOI:
https://doi.org/10.1090/S0002-9947-1987-0879569-7

Article copyright:
© Copyright 1987
American Mathematical Society