Comment on "Orbital stability of solitary wave solutions for an interaction equation of short and long dispersive waves"

TL;DR

This paper critically examines a previous study on the stability of solitary waves in a fluid dynamics system, identifying a flaw in a key lemma that affects the overall proof of stability.

Contribution

It provides a detailed critique of the original proof, highlighting a specific false claim in Lemma 2.4 that impacts the validity of the main stability result.

Findings

Identifies a false claim in Lemma 2.4 of the original paper

Shows the flaw affects the proof of Lemma 2.7 and Theorem 2.1

Questions the validity of the original stability proof

Abstract

J. Angulo and J. F. Montenegro (J. Differential Equations 174 (2001), no. 1, 181-199) published a paper about nonlinear stability of solitary waves for an interaction system between a long internal wave and a short surface wave in a two layer fluid considering that the fluid depth of the lower layer is sufficiently large in comparison with the wavelength of the internal wave. In this note, we show that in a critical step during the proof of Lemma 2.4 in the above mentioned paper, there is a claim used by the authors which fails to be true. Lemma 2.4 is crucial for the proof of Lemma 2.7, and for the proof of stability in Theorem 2.1 in the paper before mentioned.