Deep-water and shallow-water limits of the intermediate long wave equation

In this paper, we study the low regularity convergence problem for the intermediate long wave equation (ILW), with respect to the depth parameter $\delta>0$, on the real line and the circle. As a natural bridge between the Korteweg-de Vries (KdV) and the Benjamin-Ono (BO) equations, the ILW equation is of physical interest. We prove that the solutions of ILW converge in the $H^s$-Sobolev space for $s>\frac12$, to those of BO in the deep-water limit (as $\delta\to\infty$), and to those of KdV in the shallow-water limit (as $\delta\to 0$). This improves previous convergence results by Abdelouhab, Bona, Felland, and Saut (1989), which required $s>\frac32$ in the deep-water limit and $s\geq2$ in the shallow-water limit. Moreover, the convergence results also apply to the generalised ILW equation, i.e.~with nonlinearity $\partial_x (u^k)$ for $k\geq 2$. Furthermore, this work gives the first convergence results of generalised ILW solutions on the circle with regularity $s\geq \frac34$. Overall, this study provides mathematical insights for the behaviour of the ILW equation and its solutions in different water depths, and has implications for predicting and modelling wave behaviour in various environments.