Shallow-water convergence of the intermediate long wave equation in $L^2$

TL;DR

This paper proves that the intermediate long wave (ILW) equation converges to the Korteweg-de Vries (KdV) equation in the shallow-water limit within the $L^2$ framework, completing the understanding of ILW's behavior on both the real line and circle.

Contribution

It establishes the $L^2$-convergence of scaled ILW solutions to KdV in the shallow-water limit, using integrability and normal form methods, extending prior deep-water results.

Findings

Convergence of ILW to KdV in $L^2$ on both geometries.

Uniform equicontinuity of ILW solutions in $L^2$.

Effective control of high and low frequency parts of solutions.

Abstract

We continue our study on the convergence issue of the intermediate long wave equation (ILW) on both the real line and the circle. In particular, we establish convergence of the scaled ILW dynamics to that of the Korteweg-de Vries equation (KdV) in the shallow-water limit at the $L^2$-level. Together with the recent work by the first three authors and D. Pilod (2024) on the deep-water convergence in $L^2$, this work completes the well-posedness and convergence study of ILW on both geometries within the $L^2$-framework. Our proof equally applies to both geometries and is based on the following two ingredients: the complete integrability of ILW and the normal form method. More precisely, by making use of the Lax pair structure and the perturbation determinant for ILW, recently introduced by Harrop-Griffths, Killip, and Vi\c{s}an (2025), we first establish weakly uniform (in small depth parameters) equicontinuity in $L^2$ of solutions to the scaled ILW, providing a control on the high frequency part of solutions. Then, we treat the low frequency part by implementing a perturbative argument based on an infinite iteration of normal form reductions for KdV.