On the singular nature of shallow-water convergence of the intermediate long wave equation on the real line

We investigate regularity properties of the solution map for the intermediate long wave equation (ILW) on the real line. More precisely, we study the scaled ILW that was shown to converge to the Korteweg–de Vries equation (KdV) in (Formula presented) in the shallow-water limit in a recent work by the first, third, and fourth authors with T. Zhao (2025). By decomposing the dynamics into the low-frequency part and the residual part, we show that, when the depth parameter is sufficiently small, the solution map for the low-frequency part is analytic in (Formula presented), while the solution map for the residual part fails to be (Formula presented). Moreover, we establish shallow-water convergence in (Formula presented) of the low-frequency dynamics to KdV. This explains the mechanism of the regularity gain of the solution map in the shallow-water limit.