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

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.