Periodic KdV is locally well-posed in H^s for s > -2/3 with small data, proved without using complete integrability.
Babin, Alexei A
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On Local Well-posedness of the Periodic Korteweg-de Vries Equation Below $H^{-\frac{1}{2}}(\mathbb{T})$
Periodic KdV is locally well-posed in H^s for s > -2/3 with small data, proved without using complete integrability.