  We consider the nonlinear Korteweg-de Vries (KdV) equation in a bounded
interval equipped with the Dirichlet boundary condition and the Neumann
boundary condition on the right. It is known that there is a set of critical
lengths for which the solutions of the linearized system conserve the
$L^2$-norm if their initial data belong to a finite dimensional subspace $\M$.
In this paper, we show that all solutions of the nonlinear KdV system decay to
0 at least with the rate $1/ t^{1/2}$ when $\dim \M = 1$ or when $\dim \M$ is
even and a specific condition is satisfied, provided that their initial data is
sufficiently small. Our analysis is inspired by the power series expansion
approach and involves the theory of quasi-periodic functions. As a consequence,
we rediscover known results which were previously established for $\dim \M = 1$
or for the smallest critical length $L$ with $\dim \M = 2$ by a different
approach using the center manifold theory, and obtain new results. We also show
that the decay rate is not slower than $\ln (t + 2) / t$ for all critical
lengths.
