BOLETÍN ELECTRÓNICO CIENTÍFICO
DEL NODO BRASILERO
DE INVESTIGADORES COLOMBIANOS
Número 2(Artículo 20), 2000

TÍTULO
ORBITAL STABILITY OF SOLITARY WAVE SOLUTIONS FOR AN INTERACTION EQUATION OF SHORT AND LONG DISPERSIVE WAVES

TIPO:

AUTOR: Jaime Angulo Pava1angulo@ime.unicamp.br & José Fabio B. Montenegro2

IDIOMA: Inglés

DIRECCIÓN PARA CONTACTO

1IMECC, Universidade Estadual de Campinas,
Campinas, CEP 13083-970, São Paulo, Brazil
2Departamento de Matemática, Campus do Pici, Universidade Federal do Ceará,
Fortaleza, CEP 60455-000, Ceará, Brazil

ENTIDADES QUE FINANCIARON LA INVESTIGACIÓN: The second author is supported by CNPq, a Brazilian government agency that supports the development of science and technology.

PALABRAS CLAVE:

Abstract:

We study the existence and orbital stability of solitary wave solutions for an interaction equation between a long internal wave and a short surface wave in a two layer fluid. If the short wave term is denoted by $u=u(x,t):\Bbb R\times \Bbb R\to \Bbb C$ and the long wave term by $v=v(x,t):\Bbb R\times \Bbb R\to \Bbb R$, the phenomena of interaction is described by the following equation ( Funakoshi and Oikawa (J. Phys. Soc. Japan, $\bold {52}$ (1983), 1982-1995),

\begin{displaymath} \cases iu_t+u_{xx}=\alpha vu,\\ v_t+\gamma \Cal Hv_{xx}=\beta (\vert u\vert^2)_x, \endcases \end{displaymath}

where $\alpha, \beta >0$, $\gamma \in \Bbb R$ and $\Cal H$ is the Hilbert transform. Via the Implicit Function Theorem we show the existence de smooth real solutions $\phi$ and $\psi$ of the system

\begin{displaymath} \cases \phi''-\sigma\phi=\alpha \,\psi\phi\\ \gamma\Cal H\psi'-c\psi=\beta \phi^2, \endcases \end{displaymath}

where $\sigma, c>0$ and $\gamma$ in some neighbourhood of zero. Moreover, using perturbation theory of closed operators on Hilbert spaces we show that the functions $\Phi(\xi)=e^{ic\xi/2}\phi(\xi)$ and $\Psi(\xi)=\psi(\xi)$ are solutions orbitally stable in $H^1(\Bbb R)\times H^{\frac12}(\Bbb R)$, at least when $\gamma$ is negative near zero .

1. INTRODUCTION

In this work we continue our study on an interaction equation between a long internal wave and a short surface wave in a two layer fluid when the fluid depth of the lower layer is sufficiently large in comparison with the wavelength of the internal wave. The fluids are assumed with different densities, inviscid and incompressible, and their motions to be two-dimensional and irrotational. If the short wave term is denoted by $u=u(x,t):\Bbb R\times \Bbb R\to \Bbb C$ and the long wave term by $v=v(x,t):\Bbb R\times \Bbb R\to \Bbb R$, the phenomena of interaction is described by the following nonlinear coupled system (see Funakoshi and Oikawa [14]),

\begin{displaymath} \cases iu_t+u_{xx}=\alpha vu,\\ v_t+\gamma Dv_x=\beta (\... ..._x,\\ u(x,0)=u_0(x),\,\,v(x,0)=v_0(x), \endcases \tag 1.1 \end{displaymath}

where $\alpha, \beta$ are positive constants, $\gamma \in \Bbb R$ and $D=\Cal H\partial_x$, is a linear differential operator representing the dispersion of the internal wave. Here $\Cal H$ denotes the Hilbert transform defined by

\begin{displaymath} \Cal H f(x)=\, p.v. \,\frac1{\pi}\,\int\,\frac{f(y)}{y-x}\,dy \end{displaymath}

therefore, $D$ is the multiplier with Fourier operator defined as $\widehat {Dv}(\xi)=\vert\xi\vert\widehat v(\xi)$. Here the circumflex over a function denotes the function Fourier transform. Several results for Eq. (1.1) have been obtained. Funakoshi and Oikawa ([14]) computed numerical solitary wave solutions. Angulo and Montenegro ([4]) proved the existence of solitary wave solutions via the concentration compactness method (Lions [18], [19]), as well as, the evenness and analyticity of these solutions. Bekiranov, Ogawa and Ponce ([6]) proved the local well-posedness theory in $H^s(\Bbb R)\times H^{s-\frac12}(\Bbb R)$. More precisely, if $\vert\gamma\vert<1$ and $s\geqq 0$ then for any $(u_0,v_0)\in H^s(\Bbb R)\times H^{s-\frac12}(\Bbb R)$ there exists $T>0$ such that the initial value problem (1.1) admits a unique solution $(u(t),v(t))\in C([0,T);H^s(\Bbb R))\times C([0,T);H^{s-\frac12}(\Bbb R))$. Moreover, for $T>0$ the map $(u_0,v_0)\to (u(t),v(t))$ is Lipschitz continuous from $H^s(\Bbb R)\times H^{s-\frac12}(\Bbb R)$ to $C([0,T);H^s(\Bbb R))\times C([0,T);H^{s-\frac12}(\Bbb R))$. For the case $\vert\gamma\vert=1$, they obtain the same results as above, but for $s>0$. We note that as consequence of the relations (1.2) and (1.3) below, $T$ can be chosen arbitrarily large if $\gamma \leqq 0$ and $s=1$. For any $s\geqq 0$, we have that the solution $u$ preserves its $L^2(\Bbb R)$-norm, i.e., if

\begin{displaymath} H(u)=\int_{\Bbb R}\,\vert u(x)\vert^2\,dx \tag 1.2 \end{displaymath}

then for any $0<t<T$, $H(u(t))=H(u_0)$. Moreover, we have the conservation laws of momentum and energy:

\begin{displaymath} \split &G(u,v)\equiv Im\,\int_{\Bbb R}\,u(x)\overline{u_x(... ..._{\Bbb R}\,\vert D^{1/2}v(x)\vert^2\;dx, \endsplit \tag 1.3 \end{displaymath}

where $D^{1/2}$ is the multiplier with Fourier operator defined as $\widehat {D^{1/2}v}(\xi)=\vert\xi\vert^{\frac12}\widehat v(\xi)$. The purpose of this paper is to consider the existence and orbital stability of solitary wave solutions for Eq. (1.1) of the form

\begin{displaymath} \cases u(x,t)=e^{i\omega t}\varphi(x-ct),\\ v(x,t)=\psi(x-ct), \endcases \tag 1.4 \end{displaymath}

where we have that, $\varphi:\Bbb R\to \Bbb C$, $\psi:\Bbb R\to \Bbb R$, are smooth functions such that for each $n\in \Bbb N$, $\vert\varphi^{(n)}(\xi)\vert\to 0$, and $\psi^{(n)}(\xi )\to 0$, as $\vert\xi\vert\to \infty $, $c>0 $ and $\omega\in \Bbb R$. Thus, substituting $(u,v)$ as above in (1.1) we obtain the coupled system of equations

\begin{displaymath} \cases \varphi''-\omega\varphi-ic\varphi'=\alpha \,\psi\va... ...H\psi'-c\psi=\beta \,\vert\varphi\vert^2 \endcases \tag 1.5 \end{displaymath}

with $\lq\lq ' ''=\frac{d}{d\xi}$ and $\xi=x-ct$. Now, if we consider $\varphi(\xi)=e^{ic\xi/2}\phi(\xi)$, for $\phi$ real-valued, and replace it in (1.5) finally we obtain the pseudo-differential system

\begin{displaymath} \cases \phi''-\sigma\phi=\alpha \,\psi\phi\\ \gamma\Cal H\psi'-c\psi=\beta \phi^2, \endcases \tag 1.6 \end{displaymath}

where $\sigma=\omega-\frac{c^2}{4}$. We shall show the existence of smooth real solutions $\phi$, $\psi$ of (1.6) for $\sigma>0$ and $\gamma$ in some neighbourhood of zero via the Implicit Function Theorem. Moreover, adapting a method developed by Albert, Bona and Henry ([1]), Benjamin ([7]) and Weinstein ([23], [24]), together with some results from perturbation theory of closed operators on Hilbert spaces (Kato [16]), we shall prove that these solutions are orbitally stable in $H^1(\Bbb R)\times H^{\frac12}(\Bbb R)$ at least when $\gamma$ is negative near zero. In comparison with Eq. (1.1), we consider the following system

\begin{displaymath} \cases iu_t+u_{xx}=\alpha vu\\ v_t=\beta (\vert u\vert^2)_x, \endcases \tag 1.7 \end{displaymath}

i.e., $\gamma=0$ in (1.1). This model is the most typical in the theory of wave interaction and occurs when the fluid depth is sufficiently small in comparison with the wavelength of the internal wave. Eq. (1.7) also has been considered under various settings, see for example, Benney ([8],[9]), Bekiranov, Ogawa and Ponce ([5]), Grimshaw ([15]), Laurençot ([17]), Ma ([20]) and Tsutsumi and Hatano ([22]). In the particular case of the existence and stability theory of solitary wave solutions, the results are more definitive (see [17]), in the sense that solitary waves for (1.7) ($\gamma=0$ and $\sigma>0$ in (1.6)) are unique (up to translations) and may be computed explicitly as

\begin{displaymath} \cases \phi_0(\xi)=\sqrt{\frac{2c\sigma}{\alpha\beta}}\,\t... ...si_0(\xi)=-\frac{\beta}{c}\phi_0^2(\xi), \endcases \tag 1.8 \end{displaymath}

however, formulas of the form given in (1.8) have no known counterpart for equations of type (1.6) with $\gamma\neq 0$. The plan of this paper is as follows. In section 2, we prove the existence and stability of solitary-wave solutions corresponding to values of $\gamma$ negative near zero. In the Appendix, we briefly review some results about perturbation theory of closed linear operators on Hilbert spaces necessary in the development of our work.
Notations. Throughout this paper we will denote by $\widehat f$ the Fourier transform of $f$, defined as $ \widehat f(\xi)=\int_{\Bbb R}\;f(x)e^{-i\xi x}\;dx$. $\vert f\vert _{L^p}$ denotes the $L^p(\Bbb R)$ norm of $f$, $1\leqq p\leqq \infty$. In particular, $\vert\cdot\vert _{L^2}=\Vert\cdot\Vert$ and $\vert\cdot\vert _{L^{\infty}}=\vert\cdot\vert _\infty$. We denote by $H^s(\Bbb R)$ the Sobolev space of all $f$ (tempered distributions) for which the norm $\Vert f\Vert _s^2=\int_{\Bbb R}\;(1+\vert\xi\vert^2)^{s}\vert\widehat f(\xi)\vert^2\;d\xi$ is finite. The product norm in $H^s(\Bbb R)\times H^{r}(\Bbb R)$ is denoted by $\Vert\cdot\Vert _{s\times r}$. $D^s=(-\partial_x^2)^{s/2}$ is the Riesz potential of order $-s$, and is defined by $\widehat{D^sf}(\xi)=\vert\xi\vert^s\widehat f(\xi)$. $B(X;Y)$ denote the space of all bounded linear operators from $X$ into $Y$. If $X=Y$, $B(X;Y)=B(X)$. If $S$ is any closed operator on $L^2(\Bbb R)$, we denote its spectrum by $\Sigma (S)$.

2. EXISTENCE AND STABILITY OF SOLITARY WAVE SOLUTIONS

For any $(c,\omega)\in \Bbb R^+\times \Bbb R$ define the functions $\Phi(\xi)=e^{ic\xi/2}\phi(\xi)$ and $\Psi(\xi)=\psi(\xi)$, where $(\phi,\psi)$ is a solution of (1.6). Then we said that the solitary-wave solution $(U_s(x,t),V_s(x,t))=(e^{i\omega t}\Phi(x-ct), \Psi(x-ct))$ is orbitally stable in $H^1(\Bbb R)\times H^{\frac12}(\Bbb R)$ if for every $\epsilon >0$, there exists $\delta (\epsilon) >0$ such that when $(u_0, v_0)\in H^1(\Bbb R)\times H^{\frac12}(\Bbb R)$ and satisfies both $\Vert u_0-\Phi\Vert _1<\delta $ and $\Vert v_0-\Psi\Vert _\frac12<\delta $, and $(u,v)$ is the solution of (1.1) corresponding to $(u_0,v_0)$, then $(u,v)\in C([0,\infty);H^1(\Bbb R))\times C([0,\infty); H^{\frac12}(\Bbb R))$ and

\begin{displaymath} \split \;\underset{\theta\in [0,2\pi)}\to{\underset{x_0\in... ...t+x_0,t)-\Psi\Vert _{\frac12} <\epsilon, \endsplit \tag 2.1 \end{displaymath}

for all $t\in [0,\infty)$. The main result to be proved in this section is that stable solitary-wave solutions of (1.1) exist when $\sigma>0$ and $\gamma$ is negative and near 0.
Theorem 2.1 Let $\alpha, \beta >0$, and $\sigma=\omega-\frac{c^2}{4}>0$, for $c>0 $ and $\omega\in \Bbb R$ fixed numbers. Then there exists $\gamma_0>0$ such that for each $\gamma\in (-\gamma_0, 0)$, Eq. (1.6) has a solution $(\phi_\gamma,\psi_\gamma)$. Moreover, for $\Phi_\gamma(\xi)=e^{ic\xi/2}\phi_\gamma(\xi)$ and $\Psi_\gamma(\xi)=\psi_\gamma(\xi)$, we have that $(e^{i\omega t}\Phi(x-ct), \Psi(x-ct))$ is a stable solitary-wave solution of (1.1). The proof of Theorem 2.1 will proceed by adapting a method developed by Benjamin ([7]), Albert, Bona, and Henry ([1]) and Weinstein ([24]), together with some results on spectral theory of closed linear operators on Hilbert spaces. We begin our study proving the existence of solitary-wave solutions of (1.1). Since our argument is based on the Implicit Function Theorem, we need to establish several facts on the structure spectral of the self-adjoint operator on $L^2(\Bbb R)$,

\begin{displaymath} \Cal L_0= -\frac{d^2}{d\xi^2} +\sigma- \frac{3\alpha \beta}{c}{\phi_0}^2, \tag 2.2 \end{displaymath}

where $\phi_0$ is defined in (1.8). In fact, since $\phi_0'$ has a single zero and $\phi_0(\xi)\to 0$ as $\vert\xi\vert\to \infty $, we obtain from the Sturm-Liouville theory (see, [13], [10]) and perturbation theory of closed linear operators on Hilbert spaces (see, Kato [16]) that $\Cal L_0$ has one simple negative eigenvalue $\lambda_0$ with eigenfunction $\varphi_0>0$, a simple eigenvalue at zero with eigenfunction $\phi_0'$, the essential spectrum is the interval $[\sigma,\infty)$, and the remainder of the spectrum of $\Cal L_0$ consist of isolated eigenvalues of finite multiplicity. Moreover, there exists $\eta_0>0$ such that for $\beta_0\in \Sigma(\Cal L_0)-\{\lambda_0, 0\}$, we have that $\beta_0>\eta_0$. For $r\geqq 0$, let $H_e^r(\Bbb R)$ denote the closed subspace of all even functions in $H^r(\Bbb R)$. In order to prove Theorem 2.1 we have first,
Lemma 2.2 Let $\alpha, \beta >0$, and $\sigma=\omega-\frac{c^2}{4}>0$, for $c>0 $ and $\omega\in \Bbb R$ fixed numbers. Then there exists $\gamma_1>0$ such that for every $\gamma\in (-\gamma_1, \gamma_1)$, Eq. (1.6) has a solution $(\phi_\gamma,\psi_\gamma)\in H_e^{2}(\Bbb R)\times H_e^{1}(\Bbb R)$, and the correspondence $\gamma\to (\phi_\gamma,\psi_\gamma)$ defines a continuous map from $(-\gamma_1, \gamma_1)$ to $ H_e^{2}(\Bbb R)\times H_e^{1}(\Bbb R)$. In particular, for $\gamma<0$ and $\gamma \to 0$, $(\phi_\gamma(x),\psi_\gamma(x))$ converges to $(\phi_0(x),\psi_0(x))$, uniformly for $x\in \Bbb R$, where $\phi_0$ and $\psi_0$ are defined as in (1.8). Proof Without loss of generality take $\alpha=1$ and $\beta=1/2$. Let $X_e=H_e^{2}(\Bbb R)\times H_e^{1}(\Bbb R)$ and define a map $G:\Bbb R\times X_e \to L_e^{2}(\Bbb R)\times L_e^{2}(\Bbb R)$ by

\begin{displaymath} G(\gamma,\phi, \psi)=(-\phi''+\sigma\phi+\phi\psi,-\gamma D\psi+c\psi+\phi^2). \end{displaymath}

A calculation shows that the Fréchet derivative $G_{(\phi,\psi)}=\frac{\partial G(\gamma,\phi, \psi)}{\partial(\phi,\psi)}$ exists on $\Bbb R\times X_e$ and is defined as a map from $\Bbb R\times X_e$ to $B(X_e;L_e^{2}(\Bbb R)\times L_e^{2}(\Bbb R))$ by

\begin{displaymath} G_{(\phi,\psi)}(\gamma,\phi, \psi)=\left(\matrix -\frac{d^2... ...igma+ \psi & \phi\\ \phi & -\gamma D+c \endmatrix\right). \end{displaymath}

From (1.8) it follows that for $ \Phi_0=(\phi_0,\psi_0)$, $G(0,\Phi_0)=\vec 0^t$ and that the operator $\Cal S_0=G_{(\phi,\psi)}(0, \Phi_0)$ has a one-dimensional nullspace $\Cal N(\Cal S_0)$ in $L^{2}(\Bbb R)\times L^{2}(\Bbb R)$. In fact, it follows immediately from (1.8) that $\Cal S_0({\Phi_0'}^t)={\vec 0}^t$, and hence $\Phi_0'\in \Cal N (\Cal S_0)$. Now, we consider $\Psi=(f,g)$ such that $\Cal S_0({\Psi}^t)={\vec 0}^t$, then $f$ satisfies the differential equation

\begin{displaymath} -f'' +\sigma f-\frac{3}{2c}{\phi_0}^2 f=0. \end{displaymath}

Thus, $f\in \Cal N(\Cal L_0)$ and therefore $f=\eta_1 {\phi_0}'$ for some $\eta_1\in \Bbb R$. Since $g=-\frac{1}{c}\phi_0 f$ it follows that $\Psi=\eta_1 \Phi_0'$ and therefore $\Cal N(\Cal S_0)$ is generated by $\Phi_0'$. Since $\Phi_0'\notin X_e$, it follows that $S_0:X_e \to L_e^{2}(\Bbb R)\times L_e^{2}(\Bbb R)$ is invertible. Finally, since $G$ and $G_{(\phi,\psi)}$ are continuous maps on their domains, we have from the Implicit Function Theorem that there exist a number $\gamma_1>0$ and a continuous map $\gamma\to (\phi_\gamma,\psi_\gamma)$ from $(-\gamma_1, \gamma_1)$ to $X_e$ such that $G(\gamma,\phi_\gamma, \psi_\gamma)=0$ for all $\gamma\in (-\gamma_1, \gamma_1)$. This shows the Lemma. $\blacksquare$
Remark 2.3. Notice that for $\gamma$ near $0$, $\phi_\gamma$ obtained in Lemma 2.2 is strictly positive. In fact, since $\psi(\xi)\to 0$ as $\vert\xi\vert\to \infty $ there exists $M>0$ such that for each $x\geqq M$, $\sigma+\alpha\psi(x)>\frac{\sigma}{2}$, thus

\begin{displaymath} -\phi_\gamma(x)\phi_\gamma'(x)\geqq \frac{\sigma}{2}\int_x^... ...ty \phi_\gamma^2(y)dy+\int_x^\infty (\phi_\gamma'(y))^2 dy>0. \end{displaymath}

Therefore, $\phi_\gamma(x)\ne 0$ for each $x\geqq M$. Now, since $\phi_0$ is strictly positive and $\phi_\gamma\to \phi_0$ as $\gamma \to 0$ uniformly in $[0, M]$, we have that for $\gamma$ near zero $\phi_\gamma(x)>0$ for $x\in [0, M]$. Thus from the continuity of $\phi_\gamma$ we obtain the affirmation. The existence of the desired family of solitary-wave solutions of Eq. (1.6) established in Theorem 2.1 has now been demonstrated, and it remains to prove that these solitary waves are stable at least when $\gamma$ is negative and near zero. In order to show this, we need to establish some results on the spectrum of a operator associated with the solitary wave $(\phi_\gamma,\psi_\gamma)$, it will necessary to use some results from perturbation theory of closed linear operator on Hilbert spaces ( ch. IV-V of Kato [16]). The linear, self-adjoint, closed, and unbounded operator on $L^2(\Bbb R)$ of our interest here is given by

\begin{displaymath} \Cal L_\gamma= -\frac{d^2}{d\xi^2} +\sigma +\alpha\psi_\gam... ...\phi_\gamma\circ\Cal K_\gamma^{-1}\circ\phi_\gamma, \tag 2.3 \end{displaymath}

where $(\phi_\gamma,\psi_\gamma)$ is a solution of (1.6) for $\gamma \leqq 0$ obtained in Lemma 2.2, the operator $\Cal K_\gamma^{-1}$, defined as $\widehat {\Cal K_\gamma^{-1}f}(\xi)=\frac{1}{-\gamma \vert\xi\vert+c}\widehat f(\xi)$, is the inverse operator of $\Cal K_\gamma\in B(H^s(\Bbb R); H^{s-1}(\Bbb R))$, $s\geqq 1$, defined by $\Cal K_\gamma= -\gamma D+c$, and $\phi_\gamma\circ\Cal K_\gamma^{-1}\circ\phi_\gamma$ is given by $[\phi_\gamma\circ\Cal K_\gamma^{-1}\circ\phi_\gamma] (f)=\phi_\gamma\Cal K_\gamma^{-1}(\phi_\gamma f)$. When $\gamma=0$ in (2.3) we obtain from (1.8) the operator $\Cal L_0$ defined in (2.2). The following Lemma shows that for $\gamma$ negative and near zero, the spectrum of $\Cal L_\gamma$ is similar to $\Cal L_0$.
Lemma 2.4 There exists a constant $\gamma_2>0$ such that if $\gamma\in (-\gamma_2, 0)$, then the self-adjoint operator $\Cal L_\gamma$ on $L^2(\Bbb R)$, defined in (2.3) with domain $H^2(\Bbb R)$, has the following properties:
(1) $\Cal L_\gamma$ has simple negative eigenvalue $\lambda_\gamma$ with eigenfunction $\varphi_\gamma$, such that $\int_\Bbb R \varphi_\gamma\phi_\gamma dx >0$.
(2) $\Cal L_\gamma$ has a simple eigenvalue at zero with eigenfunction $\phi_\gamma'$, and
(3) There is $\eta_\gamma>0$ such that for $\beta_\gamma\in \Sigma(\Cal L_\gamma)-\{\lambda_\gamma, 0\}$, we have that $\beta_\gamma>\eta_\gamma$. Moreover, the essential spectrum of $\Cal L_\gamma$ is the interval $[\sigma,\infty)$, $\lambda_\gamma\to \lambda_0$ as $\gamma\to 0^-$, and $\varphi_\gamma\to \varphi_0$ as $\gamma\to 0^-$ in $L^2(\Bbb R)$-norm, where $\varphi_0$ is the eigenfunction of the operator $\Cal L_0$ associated to the eigenvalue $\lambda_0$. Proof Without loss of generality take $\alpha=1$ and $\beta=1/2$. First we state the following facts:
(a) $ \underset{\gamma\to 0^-}\to{\lim}\;\phi_\gamma(x)=\phi_0(x),\;\text{and}\;\und... ...-}\to{\lim}\;\psi_\gamma(x)=\psi_0(x),\;\;\;\text{uniformly for}\;\;x\in \Bbb R$,
(b) $\underset{\gamma\to 0^-}\to{\lim}\;\widehat \delta(\Cal L_\gamma,\Cal L_0)=0$,
where $\widehat \delta$ is a metric on $C(L^2(\Bbb R))$, the space of closed operators on $L^2(\Bbb R)$ ( see Appendix). Part (a) follows from Lemma 2.2. To prove part (b), first write $\Cal L_\gamma= -\frac{d^2}{d\xi^2} + \Cal M_\gamma$ and $\Cal L_0= -\frac{d^2}{d\xi^2} + \Cal M_0$, where $\Cal M_\gamma= \sigma +\psi_\gamma-\phi_\gamma\circ\Cal K_\gamma^{-1}\circ\phi_\gamma$ and $\Cal M_0=\sigma-\frac{3}{2c}{\phi_0}^2$. Next, note that $\Cal M_\gamma-\Cal M_0\in B(L^2(\Bbb R))$, with operator norm tending to $0$ as $\gamma\to 0^-$. In fact, first one has for $f\in L^2(\Bbb R)$ that

\begin{displaymath} \Vert(\Cal M_\gamma-\Cal M_0)f\Vert\leqq \vert\psi_\gamma-\... ...-\phi_\gamma\Cal K_\gamma^{-1}(\phi_\gamma f)\Vert. \tag 2.4 \end{displaymath}

Next, we estimate the second norm on the right hand side of (2.4). From the relations

\begin{displaymath} \frac1c \phi_0^2f-\phi_\gamma\Cal K_\gamma^{-1}(\phi_\gamma... ...hi_0 f)-\phi_\gamma\Cal K_\gamma^{-1}((\phi_\gamma-\phi_0) f) \end{displaymath}

and $\Vert\Cal K_\gamma^{-1}(g)\Vert\leqq \frac1c \Vert g\Vert$ we have that,

\begin{displaymath} \split \Vert\frac1c \phi_0^2f-\phi_\gamma\Cal K_\gamma^{-1... ...ma-\phi_0\vert _{_{\infty}}\Vert f\Vert. \endsplit \tag 2.5 \end{displaymath}

Now, for the kernel $K_\mu$ defined by

\begin{displaymath} K_\mu(x)=\frac1\pi\, \int_0^\infty\,\frac{e^{-x\tau}}{\mu^2+\tau^2}\,d\tau,\qquad\text{for}\;x>0, \end{displaymath}

and $K_\mu(x)=K_\mu(-x)$ for $x<0$, we have that $K_\mu '\in L^1(\Bbb R)$. Moreover, if $\mu=\frac{-c}{\gamma}$ then $\widehat {K_\mu}(\xi)=\frac{1}{\vert\xi\vert+\mu}$. Thus, it follows from the Plancherel Theorem and Young's inequality that

\begin{displaymath} \split \Vert[\frac1c -\Cal K_\gamma^{-1}](\phi_0 f)\Vert^2... ...ert K_\mu '\vert _{L^1}^2\Vert f\Vert^2. \endsplit \tag 2.6 \end{displaymath}

Therefore, from the equality $ \vert K_\mu '\vert _{L^1}=\frac1\mu$, item (a) above, and (2.5), (2.6), it follows from (2.4) that $\Vert\Cal M_\gamma-\Cal M_0\Vert _{_{B(L^2)}}\to 0$ as $\gamma\to 0^-$. Hence, from Theorem A.1 in the Appendix we have that

\begin{displaymath} \split \widehat \delta(\Cal L_\gamma,\Cal L_0)&=\widehat \... ...2)\;\Vert\Cal M_\gamma-\Cal M_0\Vert _{_{B(L^2)}}, \endsplit \end{displaymath}

and since $\Vert\Cal M_\gamma\Vert _{_{B(L^2)}}$ is uniformly bounded for $\gamma$, we have immediately the advertised result in (b). We now turn to the proof of the Lemma. That the essential spectrum of $\Cal L_\gamma$ is $[\sigma,\infty)$ it follows from the fact that the operator $\phi_\gamma\circ\Cal K_\gamma^{-1}\circ\phi_\gamma$ is relatively compact with respect to the operator $\Cal J=-\frac{d^2}{d\xi^2} +\sigma +\psi_\gamma $, because $\vert\phi_\gamma(\xi)\vert\to 0$ as $\vert\xi\vert\to \infty $, $\Cal K_\gamma^{-1}$ is a bounded operator, and the essential spectrum of $\Cal J$ is $[\sigma,\infty)$ (see [2], [16]). Now, from Theorem A.1 in the Appendix and following similar arguments to those of the proof of Theorem 5 of Albert, Bona and Henry ([1]) we have that there exists a positive constant $\gamma_2$ such that for each $\gamma\in (-\gamma_2, 0)$, the spectrum of $\Cal L_\gamma$ have the properties established in the Lemma. Finally, since $\int_\Bbb R \varphi_0\phi_0 dx >0$ we have from Lemma 2.2 and Theorem A.1 that for $\gamma$ small $\int_\Bbb R \varphi_\gamma\phi_\gamma dx >0$. $\blacksquare$
Remark 2.5. We note that another proof of the existence of unique negative eigenvalue (simple) and that $0$ is a nondegenerate eigenvalue, can be given using the min-max principle. In fact, for $f\in D(\Cal L_\gamma)$, we have that

\begin{displaymath} \split <\Cal L_\gamma f&,f>= <\Cal L_0 f,f> -\frac{\gamma}... ...gamma f\Cal K_\gamma^{-1}(\phi_\gamma f)\Big ]\;dx \endsplit \end{displaymath}

where the last inequality is due to that $D\Cal K_\gamma^{-1}$ is a positive operator and $\gamma<0$. Thus, if $\Vert f\Vert=1$ and $f\perp \varphi_0$, $f\perp \phi_0'$ then from the spectral structure of $\Cal L_0$ and item (a) in the proof of Lemma 2.4 it follows that for $\gamma$ near zero $<\Cal L_\gamma f,f>\;>\eta_0/2$. Therefore, from min-max principle ([21]) we have the advertised result. As the stability considered here is with respect to form, i.e., up to translation in space and phase, it is propitious to introduce the following map on $H^1(\Bbb R)$ (see [#!7!#], [#!11!#], [#!12!#] and [#!24!#]), namely, consider $(\phi_\gamma,\psi_\gamma)$ a solution of (1.6) obtained in Lemma 2.2 for $\gamma<0$. If $(u_0,v_0)\in H^{1}(\Bbb R)\times H^{\frac12}(\Bbb R)$ and $(u,v)$ is the solution to (1.1) corresponding to these initial data (see [6]), we define for all $t\geqq 0$

\begin{displaymath} \rho_\sigma(u(\cdot,t), \phi_\gamma)^2\equiv\;\;\underset{\... ...(T_c u)(\cdot+x_0,t)- \phi_\gamma(\cdot)\Vert^2\}, \tag 2.7 \end{displaymath}

where we denote by $T_c$ the bounded linear operator from $L^2_{\text{loc}}(\Bbb R^+;H^{1}(\Bbb R))$ to itself defined by $ (T_c u)(x,t)=e^{-ic(x-ct)/2}u(x,t)$. The following Lemma stating essential properties of the map $\rho_\sigma$ ( analog of Lemma 1 in [11]), will show that there exist maps $\theta=\theta(t)$ and $x_0=x_0(t)$ minimizing the function

\begin{displaymath} \Omega_t(x_0,\theta)= \Vert e^{i\theta}(T_c u)'(\cdot+x_0,t... ...t e^{i\theta}(T_c) u(\cdot+x_0,t)- \phi_\gamma(\cdot)\Vert^2. \end{displaymath}


Lemma 2.6 We consider $(u_0,v_0)\in H^{1}(\Bbb R)\times H^{\frac12}(\Bbb R)$ such that $\Vert u_0\Vert=\Vert\phi_\gamma\Vert$ and $(u,v)$ the solution to (1.1) corresponding to this initial data. Suppose that for some $t_0\in [0,\infty)$ and some $(\widetilde {x_0},\widetilde \theta)\in \Bbb R \times [0,2\pi)$, it is the case that

\begin{displaymath} \Omega_{t_0}(\widetilde {x_0},\widetilde \theta)< \sigma \Vert\phi_\gamma\Vert^2. \tag 2.8 \end{displaymath}

Then, it follows that

\begin{displaymath} \text{Inf} \; \{\Omega_{t_0}(x_0,\theta)\vert\;(x_0,\theta)\in \Bbb R\times[0,2\pi)\} \tag 2.9 \end{displaymath}

is attained at least once in $\Bbb R\times[0,2\pi)$. Proof It is immediate that $\Omega_{t_0}(x_0,\theta)$ is a continuous function of $(x_0,\theta)$ on $\Bbb R\times[0,2\pi)$. Moreover, for any $\theta \in [0,2\pi)$, we have

\begin{displaymath} \underset{\vert x_0\vert\to \infty}\to{\text{lim}}\,\Omega_... ..._\gamma \Vert^2 + 2\sigma \Vert\phi_\gamma\Vert^2. \tag 2.10 \end{displaymath}

The hypothesis (2.8), continuity of $\Omega_{t_0}$, and (2.10) imply the result. $\blacksquare$ Next, it is established that the infimum in (2.9) is attained at points $(x_0,\theta)$ at least for $t_0$ in some interval of the form $[0,T]$. To this end, it is sufficient to obtain condition (2.8) in such an interval. Let $\epsilon >0$ be such that $ \epsilon^2 < \;\sigma \Vert\phi_\gamma\Vert^2/2(1+\omega)$. The solitary-wave solution $(U_s(x,t), V_s(x,t))=(e^{i\omega t}e^{ic(x-ct)/2}\phi_\gamma(x-ct),\psi_\gamma(x-ct))$ is globally defined. Hence from continuous dependence theory for (1.1) with $\vert\gamma\vert<1$, established in Bekiranov, Ogawa and Ponce ([6]), we deduced that for $T>0$ there exists a $\delta>0$ such that if $\Vert(u_0, v_0)-(e^{icx/2}\phi_\gamma,\psi_\gamma)\Vert _{_{1\times \frac12}}<\delta$, then the solution $(u,v)$ of (1.1) corresponding to $(u_0,v_0)$ exists at least for $0\leqq t \leqq T$. Moreover

\begin{displaymath} \Vert(u(\cdot,t),v(\cdot,t))-(U_s(\cdot,t), V_s(\cdot,t))\Vert _{_{1\times \frac12}}< \epsilon, \end{displaymath}

for all $ t\in [0,T]$. Now, since

\begin{displaymath} \split \Vert e^{-i\omega t}\frac{d}{dy}(T_c u)(\cdot+ct,t)... ...dot,t)\Vert^2\\ &< 2(1+\frac{c^2}{4})\epsilon ^2 \endsplit \end{displaymath}

it follows that $\Omega_{t}(ct,-\omega t)< 2(1+\omega)\epsilon^2$, and therefore we have (2.8) because of the value of $\epsilon$ just specified above. Thus, the infimum (2.9) is taken on at values $ (x_0(t),\theta (t))$ throughout the time-interval $[0,T]$. We now turn to the proof of Theorem 2.1. Proof of Theorem 2.1, (Stability) We consider the functions $\Phi_\gamma(\xi)=e^{ic\xi/2}\phi_\gamma(\xi)$ and $\Psi_\gamma(\xi)=\psi_\gamma(\xi)$, where $(\phi_\gamma,\psi_\gamma)$ is the solution of (1.6) obtained in Lemma 2.2 and $\gamma$ is negative close to zero such that Remark 2.3, Lemma 2.4 are true. Moreover, we consider $(u_0,v_0)\in H^{1}(\Bbb R)\times H^{\frac12}(\Bbb R)$ such that $\Vert u_0\Vert=\Vert\phi_\gamma\Vert$ and $(u,v)$ the solution to (1.1) corresponding to this initial data. The proof of stability is based on the continouos functional $L$ defined on $H^{1}(\Bbb R)\times H^{\frac12}(\Bbb R)$ by

\begin{displaymath} L(u,v)= E(u,v)+ c\; G(u,v) + \omega\; H(u,v) \end{displaymath}

where $E, G, H$ are defined by (1.2) and (1.3). We observe that for $(u,v)$ solution of (1.1), $L(u(t),v(t))=L(u_0,v_0)$ at any time $t\in [0,\infty)$. To prove (2.1), write a renormalized version of $(u,v)$, namely,

\begin{displaymath} \split \xi (x,t)&=e^{i\theta}(T_c u)(x+x_0,t)-\phi_{\gamma... ...eta (x,t)&= v(x+x_0,t)-\psi_{\gamma}(x) \endsplit \tag 2.11 \end{displaymath}

where $\theta=\theta(t)$ and $x_0=x_0(t)$ are chosen such that the infimum (2.9) is take on at this values, at least for $ t\in [0,T]$. Thus, if $p(x,t)=\text{Re}(\xi (x,t))$ and $q(x,t)=\text{Im}(\xi (x,t))$ the result of Lemma 2.6 together with (1.6) provide us with compatibility relations on $p$ and $q$, namely

\begin{displaymath} \int_{\Bbb R} q(x,t)\phi_{\gamma}(x) \psi_{\gamma}(x)dx =0, \tag 2.12 \end{displaymath}

and

\begin{displaymath} \int_{\Bbb R} p(x,t)(\phi_{\gamma}(x) \psi_{\gamma}(x))'dx=0 \tag 2.13 \end{displaymath}

for all $ t\in [0,T]$. Now, using the representation (2.11) and (1.6) we have

\begin{displaymath} \split \Delta L&(t):= L(u(t),v(t))- L(\Phi_{\gamma}, \Psi_... ...l K}_{\gamma}^{-1/2} (p^2+q^2)\Big]dx \endsplit \tag 2.14 \end{displaymath}

where $\Cal L_\gamma$ is defined as in (2.3), $\Cal L_\gamma^+$ is given by

\begin{displaymath} \Cal L_\gamma^+ = -\frac{d^2}{d \xi^2}+\sigma + \alpha \psi_{\gamma}, \tag 2.15 \end{displaymath}

and ${\Cal K}_{\gamma}^{1/2}$, ${\Cal K}_{\gamma}^{-1/2}$ are the positive roots of ${\Cal K}_{\gamma}$ and ${\Cal K}_{\gamma}^{-1}$ respectively. Now, we need to find a lower bound for $\Delta {L}(t)$. The first step will be to obtain a suitable lower bound of the last term on the right-hand side of (2.14). In fact, since ${\Cal K}_{\gamma}^{-1/2}$ is a bounded operator on $L^2(\Bbb R)$, $\phi_\gamma$ is uniformly bounded, and from the continuous embedding of $H^1(\Bbb R)$ in $L^4(\Bbb R)$ and in $L^{\infty}(\Bbb R)$, we have that

\begin{displaymath} -\frac{\alpha \beta}{2}\int_{\Bbb R} \Big [\vert{\Cal K}_{\... ...eq -C_1\Vert\xi \Vert _1^3-C_2\Vert\xi \Vert _1^4, \tag 2.16 \end{displaymath}

where $C_1$ and $C_2$ are positive constants. The estimates for $<\Cal L_\gamma p,p>$ and $<\Cal L_\gamma^+ q,q>$ are obtained in the next two Lemmas.
Lemma 2.7 Consider $\gamma<0$ and near zero such that Lemma 2.4 is true. If $\Cal L_\gamma$ is defined as in (2.3), then

\begin{displaymath} \split &(a) \;\;\inf \{ <\Cal L_\gamma f,f>\vert\;\; \Vert... ...<f, (\phi_{\gamma}\psi_{\gamma})'>=0\}\equiv D>0, \endsplit \end{displaymath}

Proof Proof of part (a). From Lemma 2.4, $\Cal L_\gamma$ has exactly one negative eigenvalue $\lambda_\gamma$ (simple) with eigenfunction $\varphi_\gamma$, such that $\int_{\Bbb R}\varphi_\gamma\phi_\gamma >0$ for $\gamma$ near zero. Moreover, since zero is not an eigenvalue for $\Cal L_0$ in $L^2_e(\Bbb R)$, we have that $\Cal L_\gamma$ when considered as an operator on $L^2_e(\Bbb R)$, it does not have zero in its spectrum for values of $\gamma$ sufficiently close to zero, because $\underset{\gamma\to 0^-}\to{\lim}\;\widehat \delta(\Cal L_\gamma,\Cal L_0)=0$. Thus $\Cal L_\gamma^{-1}$ is well-defined on $L^2_e(\Bbb R)$, and from Theorem IV. 2.25 of Kato ([16]) and Lemma 2.2, it follows that the function $\chi_\gamma=\Cal L_\gamma^{-1}(\phi_\gamma)$ depends continuously on $\gamma \leqq 0$ in the $L^2(\Bbb R)$-norm. Now, since $\Cal L_0(-\frac{d}{d\sigma}\phi_0)=\phi_0$ and the nullspace of $\Cal L_0$ is spanned by $\phi_0'$, then $<\chi_0,\phi_0>=<-\frac{d}{d\sigma}\phi_0,\phi_0> \;<0$. Therefore by continuity $<\chi_\gamma,\phi_\gamma>\;<0$ for $\gamma$ near zero. Thus, as consequence of the last reasoning and Theorem A.3 in the Appendix we have that $D_0=0$ for $\gamma$ in some neighbourhood of zero. Proof of part (b). Because of part (a), we know that $D\geq 0$. Suppose that $D=0$ and let $\{f_j\}$ be a sequence of $H^1(\Bbb R)$-functions with $\Vert f_j\Vert=1$, $<f_j,\phi_{\gamma}>=0,$ $<f_j,(\phi_{\gamma}\psi_{\gamma})'>=0,$ and $<\Cal L_\gamma f_j,f_j>\to 0$ as $j\to \infty$. Then, for any $\delta>0$, there is a natural number $j_0$ such that for $j>j_0$,

\begin{displaymath} \split 0<\sigma \leq \int_{\Bbb R}\vert f'_j\vert^2dx+\sig... ...rt\phi_{\gamma}\vert^2_{\infty}+\delta. \endsplit \tag 2.17 \end{displaymath}

Then (2.17) implies that there is a subsequence of $\{f_j\}$, which we denote again by $\{f_j\}$, and a function $f^*\in H^1(\Bbb R)$ such that $f_j \rightharpoonup f^*$ weakly in $H^1(\Bbb R)$ and uniformly for each compact subset of $\Bbb R$. So $f^*$ satisfies the conditions $f^*\perp \phi_{\gamma}$ and $f^*\perp (\phi_{\gamma}\psi_{\gamma})'$. Moreover, from the just mentioned properties of the sequence $\{f_j\}$, the decay of $\phi_{\gamma}(\xi)$ to zero as $\vert\xi\vert\to \infty $, and that $\Cal K_{\gamma}^{-1}$ is a bounded operator, we have

\begin{displaymath} \split \int_{\Bbb R} \phi_{\gamma}\vert f_j\vert^2dx &\to ... ...mma}f^* {\Cal K}_{\gamma}^{-1}(\phi_{\gamma}f^*)dx \endsplit \end{displaymath}

as $j\to \infty$. Taking the limit in (2.17) as $j\to \infty$ yields

\begin{displaymath} 0<\sigma \leq -\alpha \int_{\Bbb R} \psi_{\gamma}\vert f^*\... ...gamma}f^* {\Cal K}_{\gamma}^{-1}(\phi_{\gamma}f^*)dx +\delta. \end{displaymath}

Since $\delta>0$ is arbitrary and $\psi_\gamma <0$, it must be the case that $f^*\neq 0$. It is now show that the infimum is achieved. Indeed, weak convergence is lower semi-continuous, so

\begin{displaymath} 0\leq <\Cal L_\gamma f^*,f^*>\leq \liminf_{j\to \infty}\;<\Cal L_\gamma f_j,f_j>=0 \end{displaymath}

Now, define $g^*=\frac{f^*}{ \Vert f^*\Vert}$. Then, $\Vert g^*\Vert=1$, $g^*\perp \phi_{\gamma}$, $g^*\perp(\phi_{\gamma}\psi_{\gamma})'$, and $<\Cal L_\gamma g^*,g^*>=0$. A consequence of the last reasoning is that there exist non-trivial critical points $(g^*,\lambda , \theta , \eta )$ for the Lagrange problem,

\begin{displaymath} \cases \Cal L_\gamma f=\lambda f+\theta \phi_{\gamma}+ \e... ...\;<f,(\phi_{\gamma}\psi_{\gamma})'>=0 . \endcases \tag 2.18 \end{displaymath}

Using (2.18) and $<\Cal L_\gamma g^*,g^*>=0$, it is easily seen that $\lambda =0$. Taking the inner product of (2.18) with $\phi'_{\gamma}$, we have from equality $\Cal L_\gamma \phi'_{\gamma}=0$ that

\begin{displaymath} 0=\eta \int_{\Bbb R} \phi'_{\gamma}(\phi_{\gamma}\psi_{\gamma})'dx, \tag 2.19 \end{displaymath}

but the integral in (2.19) converge to

\begin{displaymath} \int_{\Bbb R} \phi'_0(\phi_0\psi_0)'dx=\frac{-3\beta}{c}\int_{\Bbb R} \phi_0^2(\phi_0')^2dx <0 \end{displaymath}

as $\gamma \to 0$. Then, for $\gamma$ negative and near zero we have from (2.19) that $\eta =0$ and therefore

\begin{displaymath} \Cal L_\gamma g^*=\theta \phi_{\gamma}. \tag2.20 \end{displaymath}

Now, taking inner product of (2.20) with $\chi_{\gamma}=\Cal L_\gamma^{-1} \phi_{\gamma}$, we have

\begin{displaymath} 0=<g^*,\phi_{\gamma}>=<g^*,\Cal L_\gamma \chi_{\gamma}>=\theta <\phi_{\gamma},\chi_{\gamma}>, \end{displaymath}

since $<\phi_{\gamma},\chi_{\gamma}>\neq 0$, for $\gamma$ near zero, it follows that $\theta =0$ and $\Cal L_\gamma g^*=0$, therefore from Lemma 2.4 $g^*=\nu \phi_{\gamma}'$ for some $\nu \neq 0$, which is a contradiction since $\phi_{\gamma}'$ is not orthogonal to $(\phi_{\gamma} \psi_{\gamma})'$ in $L^2(\Bbb R)$, for $\gamma$ near zero. Thus, the minimum is positive and the proof of the Lemma is completed. $\blacksquare$
Lemma 2.8 Consider $\gamma<0$ near zero such that Remark 2.3 is true. If $\Cal L_\gamma^+$ is defined as in (2.15), then there is a positive value $C_0$ such that

\begin{displaymath} \inf \{ <\Cal L_\gamma^+ f,f>\vert\;\; \Vert f\Vert =1,\;\;<f,\phi_{\gamma}\psi_{\gamma}>=0\}\equiv C_0>0. \tag 2.21 \end{displaymath}

Proof Because of Remark 2.3, $\phi_\gamma$ is strictly positive for $\gamma$ in some neighbourhood of zero. Moreover, $\phi_\gamma$ satisfies that $\Cal L_\gamma^+\phi_\gamma=0$, thus it follows that $\Cal L_\gamma^+$ is a non-negative operator on $L^2(\Bbb R)$. Therefore, the infimum on the left-hand side of (2.21) is non-negative. Suppose that $C_0=0$. Following the proof of part (b) in Lemma 2.7, we have that the minimum is attained at an admissible function $g^*\neq 0$ and there is $(\lambda , \theta ) \in \Bbb R^2$ such that

\begin{displaymath} \Cal L_\gamma^+ g^*=\lambda g^* +\theta \phi_{\gamma}\psi_{\gamma}. \tag 2.22 \end{displaymath}

Thus, as a consequence of the condition $<\Cal L_\gamma^+ g^*,g^*>=0$, we have $\lambda =0$. Now, taking the inner product of (2.22) with $\phi_{\gamma}$ it is deduced that

\begin{displaymath} 0=<\Cal L_\gamma^+\phi_{\gamma},g^*>=<\phi_{\gamma},\Cal L_... ...^+ g^*>=\theta \int_{\Bbb R} \phi_{\gamma}^2\psi_{\gamma}dx \end{displaymath}

and therefore $\theta =0$, because $\phi_{\gamma}^2\psi_{\gamma}<0$ for $\gamma$ near zero. Then, since $\phi_{\gamma}$ is the ground state for $\Cal L_\gamma^+$, it follows that $g^*=\nu \phi_{\gamma}$ for some $\nu \neq 0$, which is a contradiction since $\phi_{\gamma}$ is not orthogonal to $\phi_{\gamma}\psi_{\gamma}$ in $L^2(\Bbb R)$. This completes the proof. $\blacksquare$ We now again turn to the proof of Theorem 2.1. Attention is now turned to estimating the terms $<\Cal L_\gamma p,p>$ and $<\Cal L_\gamma^+ q,q>$ in (2.14), where $p$ and $q$ satisfy the relations (2.12) and (2.13) respectively. Thus, from Lemma 2.8, there exists $C_0>0$ such that

\begin{displaymath} <\Cal L_\gamma^+ q,q>\geqq C_0\Vert q\Vert^2. \tag 2.23 \end{displaymath}

Then from the particular form of the operator $\Cal L_\gamma^+$ it follows that there is $C_1>0$ such that

\begin{displaymath} <\Cal L_\gamma^+ q,q>\geqq C_1\Vert q\Vert _1^2. \tag 2.24 \end{displaymath}

Now, suppose without loss of generality that $\Vert\phi_\gamma\Vert=1$. We write $p_{_{\perp}}= p-p_{_{\vert\vert}}$, where $p_{_{\vert\vert}}=<p,\phi_\gamma>\phi_\gamma$. Then, from (2.13) and the positivity of operator ${\Cal K}_{\gamma}^{-1}$ it follows that $<p_{_{\perp}},(\phi_{\gamma}\psi_{\gamma})'>=0$, and therefore from Lemma 2.7, it follows $ <\Cal L_\gamma p_{_{\perp}},p_{_{\perp}}>\geqq D\Vert p_{_{\perp}}\Vert^2$. Thus, from hypothesis $\Vert u_0\Vert=\Vert\phi_{\gamma}\Vert$, $ <\Cal L_\gamma \phi_{\gamma},\phi_{\gamma} >\;<0$, Cauchy-Schwartz inequality and the specific form of the operator $\Cal L_\gamma$, we obtain

\begin{displaymath} <\Cal L_\gamma p ,p>\geqq D_1\Vert p\Vert _1^2-D_2\Vert\xi\Vert _1^3-D_3\Vert\xi\Vert _1^4, \tag 2.25 \end{displaymath}

with $D_i>0$. Finally, collecting the results in (2.16),(2.24),(2.25) and substituting them in (2.14), we obtain

\begin{displaymath} \Delta L(t) \geqq d_1\Vert\xi\Vert _1^2-d_2\Vert\xi\Vert _1^3-d_3\Vert\xi\Vert _1^4, \tag 2.26 \end{displaymath}

where $d_i>0$. Therefore, from standard arguments ( Bona [11], Weinstein [24]) for $\epsilon >0$ sufficiently small there is a $\delta(\epsilon)>$ such that if $\Vert u_0-\Phi_\gamma\Vert _1<\delta(\epsilon) $ and $\Vert v_0-\Psi_\gamma\Vert _\frac12<\delta(\epsilon) $ then

\begin{displaymath} \rho_\sigma(u(t), \phi_\gamma)=\Vert\xi(t)\Vert _1<\epsilon \tag 2.27 \end{displaymath}

for $t\in [0,\infty)$. Now, it follows from (2.14) and from the above study of $\xi$ that

\begin{displaymath} \epsilon\geqq \frac{\alpha}{2\beta}\int_{\Bbb R} \Big [{\Ca... ...\gamma}p)+\beta {\Cal K}_{\gamma}^{-1/2}( p^2+q^2)\Big]^2dx. \end{displaymath}

Thus, from (2.27) and the equivalence of the norms $\Vert{\Cal K}_{\gamma}^{1/2}\eta\Vert$ and $\Vert\eta\Vert _{\frac12}$ we obtain (2.1). We have thus proved that $(\Phi_\gamma,\Psi_\gamma)$ is stable relative to small perturbation which preserve the $L^2(\Bbb R)$ norm of $\Phi_\gamma$. Now we discuss the stability relative to general perturbation. First,we remark that from Lemma 2.2 we have that for

\begin{displaymath} \cases f_\gamma(x)=(\frac{1}{c^3})^{1/2}\phi_\gamma(\frac{... ...g_\gamma(x)=\frac{1}{c^2}\psi_\gamma(\frac{x}{c}), \endcases \end{displaymath}

the correspondence $\gamma\to (f_\gamma,g_\gamma)$ defines a continuous map from $(-\gamma_1, 0)$ to $ H_e^{2}(\Bbb R)\times H_e^{1}(\Bbb R)$. Moreover, $(f_\gamma,g_\gamma)$ satisfies the system

\begin{displaymath} \cases f''_\gamma -\frac{\sigma}{c^2}f_\gamma=\alpha f_\ga... ...ma Dg_\gamma-g_\gamma=\beta f_\gamma^2. \endcases \tag 2.28 \end{displaymath}

Let $(u_0,v_0)$ be such that $\Vert u_0-\Phi_\gamma\Vert _1<\delta(\epsilon) $ and $\Vert v_0-\Psi_\gamma\Vert _\frac12<\delta(\epsilon) $, then we can find $c_1>0$ and $\omega_1\in \Bbb R$ such that $\frac{\sigma_1}{c_1^2}=\frac{\sigma}{c^2}$ and $c_1\Vert f_\gamma\Vert=\Vert u_0\Vert$, where $\sigma_1= \omega_1-\frac{c_1^2}{4}$. Thus, we have that the functions

\begin{displaymath} \cases {\phi_{\gamma,}}_1(x)=(c_1^3)^{1/2}\;f_\gamma(c_1x)... ..._{\gamma,}}_1(x)=c_1^2\;g_\gamma(c_1x), \endcases \tag 2.29 \end{displaymath}

satisfy the system

\begin{displaymath} \cases {\phi''_{\gamma,}}_1 -\sigma_1 {\phi_{\gamma,}}_1=\... ...1-c_1{\psi_{\gamma,}}_1=\beta {\phi^2_{\gamma,}}_1 \endcases \end{displaymath}

and $\Vert{\phi_{\gamma,}}_1\Vert=\Vert u_0\Vert$. Moreover, from the choice of $c_1$ it follows that $\Vert\phi_\gamma-{\phi_{\gamma,}}_1\Vert _1\leqq \epsilon/2$ (see [3] pg. 17 for a similar situation). Therefore, applying the preceding theory to the case of ${\Phi_{\gamma,}}_1=e^{ic_1\xi/2}{\phi_{\gamma,}}_1(\xi)$ and ${\Psi_{\gamma,}}_1(\xi)={\psi_{\gamma,}}_1(\xi)$ we have (2.1) for $(\Phi,\Psi)=(\Phi_\gamma,\Psi_\gamma)$. Theorem 2.1 is now established $\blacksquare$

APPENDIX

In this Appendix, we state some facts from perturbation theory of closed linear operators on Hilbert spaces that we have used along this work (see Kato [16] for details). We consider $L^2(\Bbb R)\times L^2(\Bbb R)$ the Hilbert space with norm defined by $\Vert\vert(f,g)\Vert\vert=(\Vert f\Vert^2+\Vert g\Vert^2)^{1/2}$, and for any closed operator $T$ on $L^2(\Bbb R)$ with domain $D(T)$, its graph, $\Cal G(T)=\{(f,g)\in L^2(\Bbb R)\times L^2(\Bbb R)\vert f\in D(T),\; T(f)=g\}$. Then a metric $\widehat \delta$ on $C(L^2(\Bbb R))$, the space of closed operators on $L^2(\Bbb R)$, may be defined as follows: for any $S, T\in C(L^2(\Bbb R))$,

\begin{displaymath} \widehat \delta (S,T)=\Vert P_S-P_T\Vert _{_{B(L^2\times L^2)}}, \end{displaymath}

where $P_S$ and $P_T$ are the orthogonal projections on $\Cal G(S)$ and $\Cal G(T)$, and $ \Vert\cdot\Vert _{_{B(L^2\times L^2)}}$ denotes the operator norm on the space of bounded operators on $L^2(\Bbb R)\times L^2(\Bbb R)$.
Theorem A.1 Let $S, T\in C(L^2(\Bbb R))$, and suppose $A$ is a bounded operator on $L^2(\Bbb R)$ with operator norm $\Vert A\Vert _{_{B(L^2)}}$. Then

\begin{displaymath} \split a)\;&\widehat \delta (T+A,T)\leqq \Vert A\Vert _{_{... ...Vert A\Vert^2_{_{B(L^2)}})\;\widehat \delta (S,T). \endsplit \end{displaymath}


Theorem A.2 Let $T\in C(L^2(\Bbb R))$ and let $\Cal U$ denote an open subset of the complex plane whose boundary is a smooth contour $\Gamma$. Suppose that $\Sigma (T)\cap \Gamma=\emptyset$ and $\Sigma (T)\cap \Cal U$ consists of a finite number of eigenvalue of $T$, each with finite (algebraic) multiplicity. Then there exists $\delta>0$ such that if $S\in C(L^2(\Bbb R))$ and $\widehat \delta (S,T)<\delta$, then $\Sigma (S)\cap \bar {\Cal U}$ consists of a finite number of eigenvalue of finite multiplicity, the sum of their multiplicities being equal to the sum of the multiplicities of the eigenvalues of $T$ in $\Cal U$. In particular, suppose $\Sigma (T)\cap \Cal U$ consists of a single, simple eigenvalue $\lambda_0$ with eigenfunction $f_0$. If $\{S_n\}$ is a sequence in $C(L^2(\Bbb R))$ such that $\widehat \delta (S_n,T)\to 0$ as $n\to\infty$, then for $n$ large, $\Sigma (S_n)\cap \Cal U$ consists of a single simple eigenvalue $\lambda_n$, and $\lambda_n\to \lambda_0$ as $n\to\infty$. Moreover, there is an eigenfunction $f_n$ associated to $\lambda_n$ such that $f_n\to f_0$ as $n\to\infty$ in $L^2(\Bbb R)$-norm. Finally, the next result is discussed in Weinstein [23].
Theorem A.3 Let $S$ be a self-adjoint operator on $L^2(\Bbb R)$ having exactly one negative eigenvalue $\lambda$ with corresponding ground-state eigenfunction $f_{\lambda}$ and let $g\in \Cal N^{\bot}(S)$. Assume $<g,f_{\lambda}>\neq 0$ and that

\begin{displaymath} -\infty<\eta\equiv \underset{<f,g>=0}\to{\underset{\Vert f\Vert=1}\to{\text{Min}}}\,<Sf,f>. \end{displaymath}

If $<S^{-1}g,g>\;\leqq0$, then it must be the case that $\eta\geqq 0$ .

REFERENCES

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