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

Abstract

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 $s>\frac{1}{2}$, 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 $s>\frac{3}{2}$ in the deep-water limit and $s\ge 2$ 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 $k\ge 2$. Furthermore, this work gives the first convergence results of generalised ILW solutions on the circle with regularity $s\ge \frac{3}{4}$. 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.

arXiv comment41 pages

References in corpus (6)

• A para-differential renormalization technique for nonlinear dispersive equations
• Improvement of the energy method for strongly non resonant dispersive equations and applications
• Non-chiral Intermediate Long Wave equation and inter-edge effects in narrow quantum Hall systems
• Low regularity well-posedness for generalized Benjamin-Ono equations on the circle
• Global well-posedness and limit behavior for the modified finite-depth-fluid equation
• Deep-water and shallow-water limits of statistical equilibria for the intermediate long wave equation

Cited by in corpus (2)

• On the deep-water and shallow-water limits of the intermediate long wave equation from a statistical viewpoint
• Unconditional deep-water limit of the intermediate long wave equation in low-regularity