# Unconditional deep-water limit of the intermediate long wave equation in low-regularity

## TL;DR

This paper proves a mathematical connection between two wave equations in low-regularity settings.

## Contribution

The paper establishes the unconditional deep-water limit of the intermediate long wave equation to the Benjamin-Ono equation in low-regularity Sobolev spaces.

## Key findings

- Unconditional uniqueness results for ILW are proven in specific Sobolev spaces on the real line and circle.
- The ILW equation is shown to converge to the BO equation in the deep-water limit under low-regularity conditions.

## Abstract

In this paper, we establish the unconditional deep-water limit of the intermediate long wave equation (ILW) to the Benjamin-Ono equation (BO) in low-regularity Sobolev spaces on both the real line and the circle. Our main tool is new unconditional uniqueness results for ILW in \documentclass[12pt]{minimal}
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\begin{document}$$H^s$$\end{document}Hs when \documentclass[12pt]{minimal}
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\begin{document}$$s_0<s\le \frac{1}{4}$$\end{document}s0<s≤14 on the line and \documentclass[12pt]{minimal}
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\begin{document}$$s_0<s< \frac{1}{2}$$\end{document}s0<s<12 on the circle, where \documentclass[12pt]{minimal}
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\begin{document}$$s_0 = 3-\sqrt{33/4}\approx 0.1277$$\end{document}s0=3-33/4≈0.1277. Here, we adapt the strategy of Moşincat-Pilod (Pure Appl Anal 5:285–322, 2023) for BO to the setting of ILW by viewing ILW as a perturbation of BO and making use of the smoothing property of the perturbation term.