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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
DOI: https://doi.org/10.1090/S0002-9947-1987-0879569-7
MathSciNet review: 879569
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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.


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Article copyright: © Copyright 1987 American Mathematical Society