Communications in Mathematical Sciences

Volume 21 (2023)

Number 3

Some models for the interaction of long and short waves in dispersive media. Part II: Well-posedness

Pages: 641 – 669



Chuangye Liu (School of Mathematics and Statistics and Hubei Key Laboratory of Mathematical Sciences, Central China Normal University, Wuhan, P.R. China)

Nghiem V. Nguyen (Department of Mathematics and Statistics, Utah State University, Logan, Ut., U.S.A.)


The (in)validity of a system coupling the cubic, nonlinear Schrödinger equation (NLS) and the Korteweg-de Vries equation (KdV) commonly known as the NLS-KdV system for studying the interaction of long and short waves in dispersive media was discussed in part I of this work [N.V. Nguyen and C. Liu, Water Waves, 2:327–359, 2020]. It was shown that the NLS-KdV system can never be obtained from the full Euler equations formulated in the study of water waves, nor even the linear Schrödinger–Korteweg–de Vries system where the two equations in the system appear at the same scale in the asymptotic expansion for the temporal and spatial variables. A few alternative models were then proposed for describing the interaction of long and short waves.

In this second installment, the Cauchy problems associated with the alternative models introduced in Part I are analyzed. It is shown that all of these models are locally well-posed in some Sobolev spaces. Moreover, they are also globally well-posed in those spaces for a range of suitable parameters.


Euler equations, linear Schrödinger equation, NLS-equation, KdV-equation, BBM-equation, NLS-KdV system, abcd-system

2010 Mathematics Subject Classification

35A35, 35M30, 35Q31, 35Q35, 76B15

Received 30 April 2021

Received revised 23 May 2022

Accepted 8 July 2022

Published 27 February 2023