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

Author:

Li GuopengORCID

Abstract

Abstract In this paper, we study the low regularity convergence problem for the intermediate long wave equation (ILW), with respect to the depth parameter δ > 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 Hs -Sobolev space for s > 1 2 , to those of BO in the deep-water limit (as δ ), and to those of KdV in the shallow-water limit (as δ → 0). This improves previous convergence results by Abdelouhab et al (1989 Physica D 40 360–92), which required s > 3 2 in the deep-water limit and s 2 in the shallow-water limit. Moreover, the convergence results also apply to the generalised ILW equation, i.e. with nonlinearity x ( u k ) for k 2 . Furthermore, this work gives the first convergence results of generalised ILW solutions on the circle with regularity s 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.

Funder

EPSRC Centre for Doctoral Training in Mathematical Analysis and its Applications:Maxwell Institute Graduate School in Analysis & Applications - MIGSAA

Publisher

IOP Publishing

Reference65 articles.

Cited by 1 articles. 订阅此论文施引文献 订阅此论文施引文献,注册后可以免费订阅5篇论文的施引文献,订阅后可以查看论文全部施引文献

1. Intermediate long wave equation in negative Sobolev spaces;Proceedings of the American Mathematical Society, Series B;2024-09-12

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