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# 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

DOI: https://dx.doi.org/10.4310/CMS.2023.v21.n3.a3

#### Authors

#### Abstract

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.

#### Keywords

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