Differential and Integral Equations

A remark on the Cauchy problem for the Korteweg-de Vries equation on a periodic domain

Bing Yu Zhang

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Abstract

The Cauchy problem of the Korteweg-de Vries equation on a periodic domain $S$, a unit circle in the complex plane establishes a nonlinear map $K$ from the initial data $\phi \in H^s(S) $ to the solution $u(x,t) \in C([-T,T];H^s(S) )$ for $s\geq 0$ (cf. [4] and [10]). Based on Bourgain's new approach [4] to periodic solutions of the KdV equation, it is shown that the nonlinear map $K$ is analytic from $H^s(S) $ to $C([-T,T];H^s(S) ) $ in the following sense. For any $\phi \in H^s(S) $, there exists a $\delta > 0 $ such that if $h\in H^s(S) $ with $\|h\| _{H^s(S) } \leq \delta $ and $\int _S h(x)dx = 0$, then $K (\phi + h) $ has the following Taylor series expansion: $$ K(\phi +h ) =\sum ^{\infty }_{n=0} \frac{K^{(n)} (\phi ) [h^n ] }{n!}, $$ where the series converges in the space $C([-T,T];H^s(S) ) $ uniformly for $\|h\| _{H^s (S)} \leq \delta $ and $K^{(n)} (\phi ) $ is the n-th order Fr\'{e}chet derivatives of $K$ at $\phi $. As a consequence, the periodic solution $u(x,t)$ of the KdV equation can be obtained by solving a series of linear problems since each term in the above Taylor series is a solution of a linearized KdV equation.

Article information

Source
Differential Integral Equations, Volume 8, Number 5 (1995), 1191-1204.

Dates
First available in Project Euclid: 20 May 2013

Permanent link to this document
https://projecteuclid.org/euclid.die/1369056051

Mathematical Reviews number (MathSciNet)
MR1325553

Zentralblatt MATH identifier
0826.35114

Subjects
Primary: 35Q53: KdV-like equations (Korteweg-de Vries) [See also 37K10]
Secondary: 35B65: Smoothness and regularity of solutions 35C10: Series solutions

Citation

Zhang, Bing Yu. A remark on the Cauchy problem for the Korteweg-de Vries equation on a periodic domain. Differential Integral Equations 8 (1995), no. 5, 1191--1204. https://projecteuclid.org/euclid.die/1369056051


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