Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



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
MathSciNet review: 879569
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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

$\displaystyle ({\text{i}})\quad {u_t} - 6u{u_x} + {\varepsilon ^2}{u_{xxx}} = 0$

in the limit of small dispersion, i.e., $ \varepsilon \to 0$.

When the unperturbed equation

$\displaystyle ({\text{ii}})\quad {u_t} - 6u{u_x} = 0$

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.

References [Enhancements On Off] (What's this?)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 35Q20, 35B25, 35L67

Retrieve articles in all journals with MSC: 35Q20, 35B25, 35L67

Additional Information

Article copyright: © Copyright 1987 American Mathematical Society