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Asymptotic Analysis of a Perturbed Periodic Solution for the KdV Equation

We consider the solution of the Korteweg–de Vries (KdV) equation with periodic initial value where C, A, k, μ, and β are constants. The solution is shown to be uniformly bounded for all small ɛ, and a formal expansion is constructed for the solution via the method of multiple scales. By using the energy method, we show that for any given number T > 0, the difference between the true solution v(x, t; ɛ) and the Nth partial sum of the asymptotic series is bounded by ɛN+1 multiplied by a constant depending on T and N, for all −∞ < x < ∞, 0 ≤t≤T/ɛ, and 0 ≤ɛ≤ɛ0.