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We then demonstrate that the solutions of ILW converge to those of the Benjamin--Ono equation in $H^s(\\mathbb{T})$ in the infinite-depth limit.\n  Our methods do not rely on the complete integrability of ILW, but rather treat ILW as a perturbation of the Benjamin--Ono equation by a linear term of order zero. 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The previous record for well-posedness was $s\\geq 0$, and the system is known to be ill-posed for $s<-\\frac12$. We then demonstrate that the solutions of ILW converge to those of the Benjamin--Ono equation in $H^s(\\mathbb{T})$ in the infinite-depth limit.\n  Our methods do not rely on the complete integrability of ILW, but rather treat ILW as a perturbation of the Benjamin--Ono equation by a linear term of order zero. 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