---
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. 

    [*MSC:*]{} 35B35; 35Q51; 35Q35

    [*Keywords:*]{} Nonlinear stability; Solitary waves; Hilbert transform; Dispersive waves
author:
- 'Borys Alvarez-Samaniego'
title: 'Comment on “Orbital Stability of Solitary Wave Solutions for an Interaction Equation of Short and Long Dispersive Waves"'
---

[^1] [^2]

Department of Mathematics, Statistics, and Computer Science

University of Illinois at Chicago - UIC

SEO 322, m/c 249, 851 S Morgan St, Chicago, IL, 60607, USA

Throughout this manuscript we keep the terminology and notations of [@am:am] as much as possible.  We start (see [@am:am Lemma 2.4, p. 188]) by remarking that applying the dominated convergence theorem, we get $$\Big{\|}\Big{[}\frac{1}{c}-\mathcal{K}_{\gamma}^{-1}\Big{]} (\phi_0 f) 
 \Big{\|}^2 = \frac{1}{c^2} \int_{\mathbb{R}} |\xi|^2
 |\widehat{K_{\mu}}(\xi) \widehat{\phi_0 f}(\xi)|^2 d\xi 
 \rightarrow 0,$$ as $\gamma \rightarrow 0^-$, where $\mu=\frac{-c}{\gamma}$. Now, by using the last expression, (2.4) and (2.5) in [@am:am], we obtain $\|(\mathcal{M}_{\gamma}-\mathcal{M}_0)f\|\rightarrow 0$ as $\gamma \rightarrow 0^{-}$, for all $f \in L^2(\mathbb{R})$.

We note that since $\widehat{K_{\mu}}(\xi)=\frac{1}{|\xi|+\mu}$, it follows that the correct expression for the kernel $K_{\mu}$ is given by $$K_{\mu}(x)  =\left\{
  \begin{array}
  [c]{r}
  \sqrt{\frac{2}{\pi}} \int_0^{+\infty}\frac{\cos(\xi x)}
  {\xi+\mu} d\xi =
  \sqrt{\frac{2}{\pi}} \int_0^{+\infty}\frac{\tau e^{-x\tau}}
  {\mu^2+\tau^2}d\tau,\text{ }x>0,\\
  K_{\mu}(-x),\text{ }x<0,
  \end{array}
\right.$$ where the Fourier transform of $g \in L^1(\mathbb{R})$ is defined as $\hat g(\xi)=\frac{1}{\sqrt{2\pi}}\int_{\mathbb{R}} 
g(x)e^{-i\xi x}dx$. It is not difficult to see, by using the dominated convergence theorem, that $$K_{\mu}'(x)  =\left\{
  \begin{array}
  [c]{r}
  -\sqrt{\frac{2}{\pi}} \int_0^{+\infty}\frac{\tau^2 e^{-x\tau}}
  {\mu^2+\tau^2}d\tau,\text{ }x>0,\\
  -K_{\mu}'(-x),\text{ }x<0. 
  \end{array}
\right.$$ Now, it is easy to verify that $K_{\mu}'\notin L^1(\mathbb{R})$. Therefore, it is not possible to use inequality (2.6) in [@am:am] to conclude that $\|\mathcal{M}_{\gamma}-\mathcal{M}_0\|_{B(L^2)}\rightarrow 0$ when $\gamma \rightarrow 0^{-}$, as it was claimed in [@am:am].  Hence, claim (b) in [@am:am Lemma 2.4, p. 187], namely $\lim_{\gamma \rightarrow 0^-}\hat \delta(\mathcal{L}_{\gamma}, 
\mathcal{L}_0)=0$, remains unproven. 

Finally, we remark that the proof of [@am:am Lemma 2.7] and the proof of stability in [@am:am Theorem 2.1] make use of the spectral structure of the operator $\mathcal{L}_{\gamma}$ for $\gamma$ negative and near zero.  These spectral properties were obtained in [@am:am Lemma 2.4] by using the fact that $\lim_{\gamma \rightarrow 0^-}\hat \delta(\mathcal{L}_{\gamma},\mathcal{L}_0)=0$, and [@am:am Theorem A.2] (which follows from [@k:k Theorem IV.3.16]).  In consequence, the part regarding the stability of solitary-wave solutions of system (1.1) in [@am:am Theorem 2.1] remains still opened in [@am:am]. 

[**[Acknowledgements.]{}**]{}  The author wants to express his gratitude to Professor Jerry L. Bona (UIC) for his hospitality, and many stimulating conversations during his stay at UIC. 

[99]{} J. Angulo, J. F. Montenegro, *Orbital stability of solitary wave solutions for an interaction equation of short and long dispersive waves*, J. Differential Equations 174 (2001) 181-199.  T. Kato, *Perturbation theory for linear operators*, Reprint of the 1980 Ed.,  Springer-Verlag, Berlin, 1995. 

[^1]: [*D*ate]{}: 07th May 2005. 

[^2]: Research supported by FAPESP/Brazil under grant No.  2003/09593-2.