TY - JOUR
T1 - On the singular nature of shallow-water convergence of the intermediate long wave equation on the real line
AU - Chapouto, Andreia
AU - Harrop-Griffiths, Benjamin
AU - Li, Guopeng
AU - Oh, Tadahiro
N1 - Publisher Copyright:
© The Author(s) 2026. Published by Oxford University Press. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0/), which permits unrestricted reuse, distribution, and reproduction in any medium, provided the original work is properly cited.
PY - 2026/9
Y1 - 2026/9
N2 - 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.
AB - 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.
UR - https://www.scopus.com/pages/publications/105050442454
U2 - 10.1093/imrn/rnag195
DO - 10.1093/imrn/rnag195
M3 - Article
AN - SCOPUS:105050442454
SN - 1073-7928
VL - 2026
JO - International Mathematics Research Notices
JF - International Mathematics Research Notices
IS - 18
M1 - rnag195
ER -