10-3\beta$,
where $\alpha+\beta =p+1$. In particular, when $F(q,r)$ depends
only on $q$ we have instability of $\Omega_{\varphi_c}$ for $p>9$.
We observe that via our approach of instability, we do not need to
use the function $d$ and hence the existence of a smooth curve of
solitary-wave solutions depending on $c$ is not necessary in our
analysis.
\subsection{Cauchy problem and existence of solitary-wave
solutions for equation (1.1)} %2.2
\vskip.2cm
In this subsection, we establish an assumption about the
initial-value problem to (1.1) and also give some results
concerning to the existence of solitary-wave solutions, as well
as, about regularities and asymptotic properties of these
solutions.
Here we need to make an assumption more than an affirmation about
the well-posedness problem to equation (1.1) in the space
$H^2(\mathbb{R})$, that is, about existence, uniqueness, persistence
property and continuous dependence of the solution upon the
initial data. This assumption is naturally induced by the
difficulties which appear when we work with a general non-linear
term as $G(u,u_x,u_{xx})$ in (1.1). In fact, when $F(u,u_x)$
depends only on $u$ is possible to obtain a local well-posedness
(and global also) theory in $H^2(\mathbb{R})$ (see Kato [15], Saut
[26]). For more general nonlinearities, for example
$F(u,u_x)=uu_x^2-u^3$ (so that
$G(u,u_x,u_{xx})=-3u^2-u_x^2-2uu_{xx}$), Ponce in [25] has shown a
well-posedness theory in $H^s(\mathbb{R})$ with $s\geq 4$. Finally in
[17], Kenig \& Ponce \& Vega have shown that the following
higher-order nonlinear dispersive equation
\[
\partial_t u+\partial_x^{2j+1}u+P(u, \partial_x u,\dots, \partial_x^{2j}u)=0,
\quad x,t\in\mathbb{R},\;\;j \in\mathbb{N},
\]
where $P(\cdot)$ is a polynomial having no constant or linear terms, is locally well posed in the
weighted Sobolev spaces $H^s(\mathbb{R})\cap L^2(|x|^mdx)$ for some $s$ (in general sufficiently large)
and $m\in \mathbb{N}$.
\begin{assumption} \label{a21.}\rm
For any $u_0 \in H^2(\mathbb{R})$ there
exist $T=T(\|u_0\|_{H^2})>0$
and a unique solution $u(t)\equiv U(t)u_0\in C([-T,T]; H^2(\mathbb{R}))$ of (1.1) with $u(x,0)=u_0(x)$. Moreover, for $T_1~~0$ and small
enough that we can get some $\delta$ small such that
\[
K(\mathcal{W}(\tau)\varphi)=K(\varphi)+\int_0^\tau \langle K'(\mathcal{W}(\xi)\varphi),B_\psi (\mathcal{W}(\xi)\varphi)\rangle
\,d\xi=\mu-\delta<\mu\,. \tag{3.21}
\]
\begin{lemma} \label{lm3.7}
Suppose $\varphi$, $\psi$ and $S$ satisfy the
same hypotheses as those established in Lemma 3.6 and $\psi$ is
chosen such that (3.11) is true with $0<\theta/(1+\theta)<1$.
Define
\[
\mathbb{D}=\{v\in H^2(\mathbb{R}) |\; S(v)~~~~0$
such that $u(t)=U(t)u_0\in H^2(\mathbb{R})$ and satisfies (1.1) for
all $t\in [0,T_0)$. So from (1.5) we have $S(U(t)u_0)=S(u_0)<
S(\varphi)$. Therefore from this last relation and the property of
minimization of $S$ on $\mathcal{F}=\{v\in H^2(\mathbb{R})|\; K(v)= \mu \}$
by $\varphi$, we have that for all $t\in [0,T_0)$,
$K(U(t)u_0)\neq\mu$. Finally, since $t\to K(U(t)u_0)$ is
continuous on $[0,T_0)$ we obtain that $ K(U(t)u_0)<\mu$ for all $
t\in [0,T_0)$. This shows property (ii).
$(iii)$\; Let $u_0\in \mathbb{B}$. Then from the conservation
quantities in (1.5) by the flow of equation (1.1), we have that
\begin{align*}
\frac12 \int_{-\infty}^{\infty}[\partial_x^2 U(t)u_0]^2 -b
[\partial_x U(t)u_0]^2 +c [U(t)u_0]^2dx
&=S(U(t)u_0)+K(U(t)u_0)\\
&~~~~0$ such that $\|\partial_x U(t)u_0\|\leq
M(\varphi,\mu)$ and therefore $\|U(t)u_0\|_{H^2}^2\leq
M(\varphi,\mu)$ for all $t\in [0,T_0)$. If $b>0$ then from
restriction $c>b^2/4$ we get
\[
\begin{split} &\frac12\Big(1-\frac{b}{2\sqrt c}\Big
)\;\int_{-\infty}^{\infty} [\partial_x^2 U(t)u_0]^2+c
[U(t)u_0]^2dx\\
&\leq \frac12 \int_{-\infty}^{\infty}[\partial_x^2 U(t)u_0]^2 -b
[\partial_x U(t)u_0]^2 +c [U(t)u_0]^2dx~~~~0$ and
for all $j\in\mathbb{N}$ in a finite time. In fact, let $\epsilon_3>0$
determined in Lemma 3.6 and define
\[
T_j=\sup \{\tau >0 : U(t)u_0^j\in
V(\Omega_{\varphi},\epsilon_3),\quad\text{ for all } t\in (0,
\tau)\}.
\]
Then, it follows from Lemma 3.6 that for all $j\in \mathbb{N}$ and
$t\in (0,T_j)$ there exists $s=s_j(t)\in (-\sigma_3,\sigma_3)$
satisfying $ S(\varphi)\leq S(U(t)u_0^j)+ P(U(t)u_0^j)s=S(u_0^j)+
P(U(t)u_0^j)s$. Now, since $u_0^j\in \mathbb{D}$ then $t\to
U(t)u_0^j\in \mathcal{P}$ for $t\in (0,T_j)$, so we have that
\[
-P(U(t)u_0^j)\geq
\frac{S(\varphi)-S(u_0^j)}{\sigma_3}=\eta_j>0,\quad \text{for
all} \quad t\in (0,T_j). \tag{3.25}
\]
Now suppose that for some $j$, $T_j=+\infty$. Then from the
properties obtained by the flow $\tau\to \mathcal{W}(\tau)\varphi$ in
(3.14) and considering that (3.11) is true, we obtain from (3.24)
and (3.25) that $\mathcal{A}_\psi(U(t)u_0^j))\geq t \eta_j+ \mathcal{A}_\psi(u_0^j)$ for all $t\in(0,+\infty)$. Then from (3.11) it
follows
\[
K_1\geq\frac{t\eta_j+ \mathcal{A}_\psi(u_0^j)}{1+t^{\theta/(1+\theta)}}\quad \text {for
all }t\in (0,+\infty),
\]
which is a contradiction . Therefore $T_j<+\infty$, which means
that $u(t)=U(t)u_0^j$ eventually leaves
$V(\Omega_{\varphi},\epsilon_3)$. This proves the Theorem.
\hfill$\square$
\section{Instability of the Orbit $\Omega_\varphi$} %sec. 4
In this section we give conditions to assure inequality (3.12) and
so to obtain the instability of the $\varphi$-orbit,
$\Omega_\varphi$, with respect to equation (1.1). For that, we
start showing the following Lemma.
\begin{lemma} \label{lm4.1}
Let $p\geq 2$ and suppose that $F(q,r)$,
homogeneous of degree $p+1$, satisfies relation (1.2). Consider
$\varphi$ a solution of (1.4) obtained via Theorem 2.2-(ii) with
$\lambda=\mu$. Then for
$\psi(x)=\int_{-\infty}^x[\varphi(y)+2y\varphi'(y)]dy$
we get that
\[
\begin{split} &\langle S''(\varphi)B_\psi(\varphi),B_\psi(\varphi)\rangle\\
&= 8b\int_{-\infty}^{\infty}\;(\varphi')^2\,dx+
(9-p)(p-1)\int_{-\infty}^{\infty}\;F(\varphi, \varphi')\,dx \\
&\quad +4\int_{-\infty}^{\infty}\;[4\varphi'F_r(\varphi, \varphi')-\varphi\varphi'F_{qr}(\varphi, \varphi')-
2(\varphi')^2F_{rr}(\varphi, \varphi')]\,dx.
\end{split}\tag{4.1}
\]
\end{lemma}
\noindent\textbf{Proof}\quad
Since $\langle\psi', \varphi\rangle=0$ it follows
immediately from (3.9) that $B_\psi(\varphi)=\psi'$ and therefore
we need only to estimate the quantity $\langle
S''(\varphi)\psi',\psi'\rangle $. In fact, it denotes by $\mathcal{L}=S''(\varphi)$ the linear operator
\[
\begin{split}
\mathcal{L}=&\partial_x^4+b\;\partial_x^2+c-F_{qq}(\varphi, \varphi')+\varphi'F_{qqr}(\varphi, \varphi')
+\varphi'F_{qrr}(\varphi, \varphi')\partial_x\\
&+\varphi''F_{qrr}(\varphi, \varphi')+\varphi''F_{rrr}(\varphi, \varphi')\partial_x+
F_{rr}(\varphi, \varphi')\partial_x^2.
\end{split}
\tag{4.2}
\]
Initially, from the homogeneity of $F$ we obtain the relations
\[
\begin{gathered}
\varphi F_{q}(\varphi, \varphi')+\varphi' F_{r}(\varphi, \varphi')=
(p+1)F(\varphi, \varphi'),\\
\varphi^2 F_{qq}(\varphi, \varphi')+2\varphi\varphi' F_{qr}(\varphi, \varphi')
+(\varphi')^2 F_{rr}(\varphi, \varphi')=
p(p+1)F(\varphi, \varphi').
\end{gathered}
\tag{4.3}
\]
Now, using (1.2) and (1.4), we have from (4.2)
\[
\begin{gathered}
\mathcal{L} \varphi=F_{q}-\varphi'F_{qr}-
\varphi F_{qq}+ \varphi(F_{qr})_x+
\varphi'(F_{rr})_x\\
\mathcal{L}
(x\varphi')=-2b\varphi''-4c\varphi+4F_{q}-4\varphi'F_{qr}-2\varphi''F_{rr}
+(\varphi')^2 F_{qrr}+ \varphi'\varphi''F_{rrr}.
\end{gathered}
\tag{4.4}
\]
So, using the first equation in (4.4), integration by parts and (4.3) we get
that
\[
\begin{split} \langle\mathcal{L} \varphi,\varphi\rangle
&=\int_{-\infty}^{\infty}\; [\varphi F_{q}-
\varphi\varphi'F_{qr}-\varphi^2F_{qq}+\varphi^2 (F_{qr})_x
+\varphi \varphi'(F_{rr})_x]\,dx\\
&=\int_{-\infty}^{\infty}\; [\varphi F_{q}-
\varphi\varphi'F_{qr}-\varphi \varphi''F_{rr}-p(p+1)F]\,dx\\
&=\int_{-\infty}^{\infty}\;[ \varphi
F_{q}+\varphi'F_{r}-p(p+1)F]\,dx
=(p+1)(1-p)\int_{-\infty}^{\infty}\;F\,dx.
\end{split}
\tag{4.5}
\]
Now, from the first equation in (4.4), some integrations by parts and
the first equation of (4.3),
we estimate the following term
\[
\begin{split} \langle \mathcal{L} \varphi,x\varphi'\rangle
&=\int_{-\infty}^{\infty}\;[ -\varphi F_{q}-
2x\varphi(F_{r})_x-2x\varphi'(F_{q})_x-\varphi\varphi' F_{qr}-(\varphi')^2F_{rr}]\,dx\\
&=\int_{-\infty}^{\infty}\;[ -\varphi F_{q}+2x(F)_x+2(\varphi
F_{q}+\varphi' F_{r})
-\varphi\varphi' F_{qr}-(\varphi')^2F_{rr}]\,dx\\
&=(p-1)\int_{-\infty}^{\infty}\;F\,dx+
\int_{-\infty}^{\infty}\;[\varphi' F_{r}-\varphi\varphi'
F_{qr}-(\varphi')^2F_{rr} ]\,dx.
\end{split}
\tag{4.6}
\]
We are going to estimate the term $\langle \mathcal{L}
(x\varphi'),x\varphi'\rangle $. Initially from the second equation in (4.4)
and integration by parts we get
\[
\begin{split}
&\langle \mathcal{L} (x\varphi'),x\varphi'\rangle \\
&=\int_{-\infty}^{\infty}\;[b(\varphi')^2+4x\varphi'F_{q}-4x(\varphi')^2F_{qr}-
2x\varphi'\varphi''F_{rr}+x(\varphi')^2(F_{rr})_x+2c\varphi^2]\,dx\\
&=\int_{-\infty}^{\infty}\;[b(\varphi')^2+4x\varphi'F_{q}-4x\varphi'(F_{r})_x-(\varphi')^2F_{rr}+2c\varphi^2]\,dx\\
&=\int_{-\infty}^{\infty}\;[b(\varphi')^2-4F+4\varphi'F_{r}-(\varphi')^2F_{rr}+2c\varphi^2]\,dx.
\end{split}
\tag{4.7}
\]
We need now an expression for
$\int_{-\infty}^{\infty}2c\varphi^2\,dx$. In fact, multiplying
(1.4) by $x\varphi'$, integrating by parts several times and using
the relation
$$\int_{-\infty}^{\infty}\;x\varphi'\partial_x^4
\varphi\,dx= \frac{3}{2}
\int_{-\infty}^{\infty}\;(\varphi'')^2\,dx
$$
we have
\[
\int_{-\infty}^{\infty}\;\Big
[\frac32(\varphi'')^2-\frac{b}{2}(\varphi')^2+F-\varphi'
F_{r}-\frac{c}{2}\varphi^2\Big ]\,dx=0. \tag{4.8}
\]
Moreover, from (1.4) we obtain immediately that
\[
\int_{-\infty}^{\infty}\;(\varphi'')^2\,dx=\int_{-\infty}^{\infty}\;[b(\varphi')^2+(p+1)F-c\varphi^2]\,dx.
\tag{4.9}
\]
So, from (4.8) and (4.9) we get the main relation
\[
\int_{-\infty}^{\infty}\;\Big [b(\varphi')^2+\frac32 \varphi
F_q+\frac12 \varphi' F_r +F\Big ]\,dx=
\int_{-\infty}^{\infty}\;2c\varphi^2\,dx. \tag{4.10}
\]
Then, from (4.7) and (4.10) we obtain
\[
\langle \mathcal{L}
(x\varphi'),x\varphi'\rangle=\int_{-\infty}^{\infty}\;\Big
[2b(\varphi')^2 + \frac{3(p-1)}{2}F \Big
]\,dx+\int_{-\infty}^{\infty}\;[3\varphi' F_r -(\varphi')^2
F_{rr}]\,dx. \tag{4.11}
\]
Finally, from (4.5), (4.6) and (4.11) we get (4.1). This completes
the Proof. \hfill$\square$
With Theorem 3.5 and Lemma 4.1 we are ready to establish our
Theorem of instability for solitary-wave solutions associated to
the fifth-order equation (1.1).
\begin{theorem}[Instability for $\Omega_{\varphi}$] \label{thm4.2}
Let $p\geq 2$ and suppose that $F(q,r)$, homogeneous of degree $p+1$,
satisfies relation (1.2). Consider $\varphi$ a solution of (1.4)
obtained via Theorem 2.2-(ii) with $\lambda=\mu$. Then if
$F(\varphi, \varphi')$ is homogeneous in $\varphi'$ of degree
$\beta$, $ \beta\in[0,p+1]$, then the conditions
\begin{gather*}
b=0\quad\text{and}\quad \beta>\frac{9-p}{2},\quad\text{or}\\
b<0\quad\text{and}\quad \beta\geq \frac{9-p}{2},\quad\text{or}\\
\beta>\frac{9-p}{2},\; b>0\quad\text{and}\quad b\quad\text{small},
\end{gather*}
imply that the $\varphi$-orbit, $\Omega_{\varphi}$, is unstable by
the flow of (1.1).
\end{theorem}
\noindent\textbf{Proof}\quad From Lemma 3.4 we have that (3.11) is
true since we have $0<\theta/(1+\theta)<1$, and from Lemma 2.3 we
obtain the properties of regularity on
$\psi(x)=\int_{-\infty}^x[\varphi(y)+2y\varphi'(y)]dy$ required by
Theorem 3.5. So, we only need to verify condition (3.12) and
therefore from Lemma 4.1 we need to know when expression (4.1) is
negative. In fact, let $F(\varphi, \varphi')$ be homogeneous in
$\varphi'$ of degree $\beta$, $\beta\in[0,\ p+1]$. Since $F$
satisfies the relations $\varphi'F_{r}=\beta F$,
$(\varphi')^2F_{rr}=\beta(\beta-1) F$, $\varphi F_{q}=\alpha F$
and $\varphi^2F_{qq}=\alpha(\alpha-1) F$, where
$\alpha+\beta=p+1$, we have initially from (4.1) and the second
equation in (4.3) that
\[
4\int_{-\infty}^{\infty}\;[4\varphi'F_r-\varphi\varphi'F_{qr}-
2(\varphi')^2F_{rr}]\,dx=4\beta(5-\beta-p)\int_{-\infty}^{\infty}\;F\,dx
\]
so from (4.1) we get
\[
\begin{aligned}
&\langle S''(\varphi)B_\psi(\varphi),B_\psi(\varphi)\rangle\\
&=\Big[(1-p)(p-9)+4(5-p)\beta-4\beta^2\Big]
\int_{-\infty}^{\infty}F(\varphi,\varphi')\,dx+
8b\int_{-\infty}^{\infty}\;(\varphi')^2\,dx,
\end{aligned} \tag{4.12}
\]
which is negative if either $b=0$ and $\beta>\frac{9-p}{2}$ or
$b<0$ and $\beta\geq \frac{9-p}{2}$.
Now if we consider $\beta>\frac{9-p}{2}$ and $b>0$ then it follows
from condition $c>b^2/4$, properties obtained for $\varphi$ and
Theorem 2.2, that
\[
\begin{split} b\int_{-\infty}^{\infty}\;(\varphi')^2\,dx&\leq
b\int_{-\infty}^{\infty}\;[(\varphi'')^2+ \varphi^2]\,dx\leq
\frac{b\quad\text{max}\{1,c\}}{c}
\int_{-\infty}^{\infty}\;[(\varphi'')^2+c
\varphi^2]\,dx\\
&\leq
\frac{4b\quad\text{max}\{1,c\}}{\sqrt{c}(2\sqrt{c}-b)}\;I_{c,b}(\varphi)=
\frac{4b\quad\text{max}\{1,c\}}{\sqrt{c}(2\sqrt{c}-b)}\;\mu^{\frac{2}{p+1}}M_c(1)\\
&=\frac{4b\quad\text{max}\{1,c\}}{\sqrt{c}(2\sqrt{c}-b)}\;
\Big(\frac{1}{p+1}\Big)^{\frac{2}{p-1}}[M_c
(1)]^{\frac{p+1}{p-1}}.
\end{split}
\tag{4.13}
\]
So, we get from (4.12) and (4.13) that
\[
\begin{split}
\langle S''(\varphi)B_\psi(\varphi),B_\psi(\varphi)\rangle
&\leq
\Big[[(9-p)(p-1)+4(5-p)\beta-4\beta^2]\Big(\frac{1}{p+1}\Big)^{\frac{p+1}{p-1}}\\
&\quad+
\frac{32b\quad\text{max}\{1,c\}}{\sqrt{c}(2\sqrt{c}-b)}\Big(\frac{1}{p+1}\Big)^{\frac{2}{p-1}}\Big]
[M_c(1)]^{\frac{p+1}{p-1}}\\
&\equiv K_0(b)[M_c(1)]^{\frac{p+1}{p-1}}.
\end{split}
\]
Therefore, if we choose $b$ small such that $K_0(b)<0$ then we
obtain from the last inequality that $\langle
S''(\varphi)B_\psi(\varphi),B_\psi(\varphi)\rangle<0$. This
finishes the proof. \hfill$\square$
\subsection*{Acknowledgments} This paper was finished while
the author was a visitor to the Department of Mathematics of the
University of Texas at Austin. The author also wants to express
his gratitude to Professor Jerry Bona for his interest in this
work.
\begin{thebibliography}{00} \frenchspacing
\bibitem{a1} C.J. Amick and J.F. Toland,
\textit{Homoclinic orbits in the dynamic phase-space analogy of an elastic strut},
European J. Appl. Math., Vol. 3(2), (1992), 97-114.
\bibitem{a2} J. Angulo,
\textit{On the instability of solitary waves solutions of the generalized
Benjamin equation},
To appear in Advances in Differential Equations, 2003.
\bibitem{b1} T. B. Benjamin,
\textit{A new kind of solitary waves},
J. Fluid Mechanics, Vol. 254 (1992), 401--411
\
\bibitem{b2} T. B. Benjamin,
\textit{Solitary and periodic waves of a new kind},
Phil. Trans. Roy. Soc. London Ser. A, Vol. 354 (1996), 1775--1806.
\bibitem{b3} D. J. Benney,
\textit{A general theory for interactions between short and long waves},
Stud. Appl. Math., Vol. 56 (1977), 81--94.
\bibitem{b4} J. L. Bona, P. E. Souganidis and W. A. Strauss,
\textit{Stability and instability of solitary waves of Korteweg-
de Vries type},
Proc. Royal. Soc. London Ser. A, Vol. 411 (1987), 395--412.
\bibitem{b5} T. J. Bridges and G. Derks,
\textit{Linear instability of solitary wave solutions of the Kawahara equation and its
generalization},
SIAM J. Math. Anal., Vol. 33 (2002), 1356--1378.
\bibitem{c1} A. R. Champneys and M. D. Groves,
\textit{A global investigation of solitary-wave solutions to a
two-parameter model for water waves},
Fluid Mech., Vol. 342 (1996), 199--229.
\bibitem{c2} W. Craig and, M.D. Groves,
\textit{Hamiltonian long-wave approxiamtions to the water-wave problem},
Wave Motion, Vol. 19 (1994), 367--389.
\bibitem{e1} A. Erd\'elyi, W. Magnus, F. Oberhettinger and F. Tricomi,
\textit{Tables of integral transform}, Vol. v. 2, McGraw-Hill, New York, 1954.
\bibitem{g1} J. M. Gon\c calves Ribeiro,
\textit{Instability of symmetric stationary states for some nonlinear
Schr\"odinger equations with an external magnetic field},
Ann. Inst. H. Poincar\'e; Phys. Th\'eor. Vol. 54 (1992), 403--433.
\bibitem{g2} K.A. Gorshkov, L.A. Ostrovsky and V.V. Papko,
\textit{Hamiltonian and non-Hamiltonian models for water waves},
Lecture Notes in Physics, Vol. 195, Springer, Berlin (1984), 273--290.
\bibitem{g3} M. Grillakis, J. Shatah and W. Strauss,
\textit{Stability theory of solitary waves in the presence of
symmetry I.}, J. Funct. Anal. Vol. 74 (1987), 160--197.
\bibitem{h1} J. Hunter, J. Scheurle\textit{Existence of
perturbed solitary wave solutions to a model equation for water
waves}, Physica D Vol. 32 (1988), 253--268.\
\bibitem{k1} T. Kato,
\textit{Quasilinear equations of evolution, with applications to
Partial Differential Equations},
Lectures Notes in Math., Vol. 448 (1975) 25--70.
\bibitem{k2} T. Kawahara,
\textit{Oscillatory solitary waves in dispersive media},
J. Phys. Soc. Jpn., Vol. 33 (1972) 260-264.
\bibitem{k3} C. Kenig, G. Ponce and L. Vega,
\textit{Higher-order nonlinear dispersive equations},
Proc. Amer. Math. Soc., Vol. 122 (1), 1994, 157-166.
\bibitem{k4} S. Kichenassamy,
\textit{Existence of solitary waves
for water-wave models},
Nonlinearity, Vol. 10(1), 1997, 133-151.
\bibitem{k5} S. Kichenassamy and P. Olver,
\textit{Existence and nonexistence of solitary wave solutions
to higher-order model evolution equations},
SIAM J. Math. Anal., Vol. 23 (1992), 1141-1166.
\bibitem{l1} S. P. Levandosky,
\textit{A stability analysis of fifth-order water wave models},
Physica D, Vol. 125 (1999), 222-240.
\bibitem{l2} P.L. Lions,
\textit{The concentration-compactness principle in the calculus of variations.
The locally compact case, part 1}, Ann. Inst. H. Poincar\'e,
Anal. Non lin\'eare, Vol. 1 (1984), 109--145.
\bibitem{l3} P. L. Lions,
\textit{The concentration-compactness principle in the calculus of variations.
The locally compact case, part 2}, Ann. Inst. H. Poincar\'e,
Anal. Non lin\'eare Vol. 4 (1984), 223--283.
\bibitem{m1} P. J. McKenna and W. Walter,
\textit{Traveling waves in a suspension bridge},
SIAM, J. Appl. Math., Vol. 50 (1990), 703--715
\bibitem{o1} P. J. Olver,
\textit{Hamiltonian and non-Hamiltonian models for water waves},
Lecture Notes in Physics, Vol. 195, Springer, Berlin, 1984, 273--290.
\bibitem{p1} G. Ponce,
\textit{Lax pairs and higher order models for water waves},
J. Differential Equations, Vol. 102(2), 1993, 360--381.
\bibitem{s1} J. C. Saut,
\textit{Quelques g\'en\'eralisations de l'\'equation de
Korteweg-de Vries, II},
J. differential Equations, Vol. 33 (1979), 320--335.
\bibitem{s2} J. Shatah and W. A. Strauss,
\textit{Instability of nonlinear bound states},
Comm. Math. Phys., Vol. 100 (1985), 173--190.
\bibitem{s3} P. E. Souganidis and W. A. Strauss,
\textit{Instability of a class of dispersive solitary waves},
Proc. Royal. Soc. Eding., Vol. 114A (1990), 195--212.
\bibitem{w1} M. Weinstein,
\textit{Existence and dynamic stability of solitary wave solutions of
equations in long wave propagation}, Comm. P.D.E., Vol. 12 (1987), 1133--1173
\
\bibitem{z1} J. Zufiria,
\textit{Symmetric breaking in periodic and solitary gravity-capillary
waves on water of finite depth}, J. Fluid Mech., Vol. 184 (1987),
183--206.
\
\end{thebibliography}
\end{document}
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