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

Andreia Chapouto, Guopeng Li, Tadahiro Oh, Tengfei Zhao

TL;DR

This work proves that the scaled ILW equation converges to the KdV equation in the shallow-water limit at the $L^2$ level on both $\mathbb{R}$ and $\mathbb{T}$. It combines the complete integrability framework, via the perturbation determinant $\alpha(\kappa;u)$ and the ILW Lax pair, with an infinite-iteration normal-form approach to compare dispersive phases to those of KdV, controlling high frequencies through equicontinuity and low frequencies via perturbative expansions encoded by ordered trees. The result completes the $L^2$ well-posedness and convergence theory for ILW on both geometries, bridging the ILW-BO deep-water convergence with the ILW-KdV shallow-water limit and providing a unified, cross-geometry methodology. The techniques have potential implications for invariant measures and long-time behavior in low-regularity regimes, and they underscore the utility of combining integrable structures with perturbative normal-form analyses in dispersive PDEs.

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 Vişan (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.

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

TL;DR

This work proves that the scaled ILW equation converges to the KdV equation in the shallow-water limit at the level on both and . It combines the complete integrability framework, via the perturbation determinant and the ILW Lax pair, with an infinite-iteration normal-form approach to compare dispersive phases to those of KdV, controlling high frequencies through equicontinuity and low frequencies via perturbative expansions encoded by ordered trees. The result completes the well-posedness and convergence theory for ILW on both geometries, bridging the ILW-BO deep-water convergence with the ILW-KdV shallow-water limit and providing a unified, cross-geometry methodology. The techniques have potential implications for invariant measures and long-time behavior in low-regularity regimes, and they underscore the utility of combining integrable structures with perturbative normal-form analyses in dispersive PDEs.

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 -level. Together with the recent work by the first three authors and D. Pilod (2024) on the deep-water convergence in , this work completes the well-posedness and convergence study of ILW on both geometries within the -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 Vişan (2025), we first establish weakly uniform (in small depth parameters) equicontinuity in 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.

Paper Structure

This paper contains 15 sections, 8 theorems, 232 equations.

Table of Contents

  1. Introduction
  2. Intermediate long wave equation
  3. Deep-water regime
  4. Shallow-water regime: main result
  5. Preliminaries
  6. Notations
  7. On the basic multiplier
  8. Shallow-water convergence in $L^2$
  9. High frequency analysis: weakly uniform equicontinuity
  10. Low frequency analysis: normal form method
  11. Perturbative approach: first step
  12. General step: ordered trees
  13. Proof of Proposition \ref{['PROP:low']}
  14. Proof of Proposition \ref{['PROP:NN']} on the circle
  15. Proof of Proposition \ref{['PROP:NN']}: on the real line

Key Result

Theorem A

Let $\mathcal{M} = \mathbb{R}$ or $\mathbb{T}$ and $s\ge 0$. Given $u_0 \in H^s(\mathcal{M})$, let $u$ be the global solution to the BO equation BO1 with $u|_{t = 0} = u_0$ constructed in MP. Let $\{u_{0, \delta}\}_{1 \le \delta < \infty}\subset H^s(\mathcal{M})$ such that $u_{0, \delta}$ converges

Theorems & Definitions (23)

  • Theorem A
  • Definition 1.1
  • Theorem 1.2
  • Definition 1.3
  • Theorem B
  • Proposition 1.4
  • Remark 1.5
  • Remark 1.6
  • Remark 1.7
  • Lemma 2.1
  • ...and 13 more