Orbital stability of solitary waves for generalized derivative nonlinear Schr\"odinger equations in the endpoint case

TL;DR

This paper proves the orbital stability of solitary waves for a generalized derivative nonlinear Schrödinger equation specifically at the endpoint case where the frequency equals the critical value, extending previous stability results.

Contribution

It establishes the stability of solitary waves at the critical endpoint case, which was previously unresolved in the stability theory for this equation.

Findings

Proves stability of solitary waves at the endpoint case.

Extends the stability theory to the critical parameter regime.

Provides rigorous mathematical analysis for the endpoint stability.

Abstract

We consider the following generalized derivative nonlinear Schr\"odinger equation \begin{equation*} i\partial_tu+\partial^2_xu+i|u|^{2\sigma}\partial_xu=0,\ (t,x)\in\mathbb R\times\mathbb R \end{equation*} when $\sigma\in(0,1)$. The equation has a two-parameter family of solitary waves $$u_{\omega,c}(t,x)=\Phi_{\omega,c}(x)e^{i\omega t+\frac{ic}2x-\frac i{2\sigma+2}\int_0^x\Phi_{\omega,c}(y)^{2\sigma}dy},$$ with $(\omega,c)$ satisfying $\omega>c^2/4$, or $\omega=c^2/4$ and $c>0$. The stability theory in the frequency region $\omega>c^2/4$ was studied previously. In this paper, we prove the stability of the solitary wave solutions in the endpoint case $\omega=c^2/4$ and $c>0$.

Equations

$$i{\partial }_{t}u+{\partial }_x^{2}u+i|u{|}^{2\sigma }{\partial }_{x}u=0,(t,x)\in \mathbb{R}\times \mathbb{R}$$

$$u_{\omega ,c}(t,x)={\Phi }_{\omega ,c}(x)e^{i\omega t+\frac{ic}{2}x-\frac{i}{2\sigma +2}{\int }_0^x{\Phi }_{\omega ,c}(y{)}^{2\sigma }dy},$$

$$i{\partial }_{t}v+{\partial }_x^{2}v+i{\partial }_x(|v{|}^{2}v)=0,(t,x)\in \mathbb{R}\times \mathbb{R}$$

$$u(t,x)=v(t,x)e^{\frac{i}{2}{\int }_{-\infty }^x|v(t,x){|}^{2}dx},$$

$$i{\partial }_{t}u+{\partial }_x^{2}u+i|u{|}^2{\partial }_{x}u=0,(t,x)\in \mathbb{R}\times \mathbb{R}.$$

$$\{\begin{array}{l}i{\partial }_{t}u+{\partial }_x^{2}u+i|u{|}^{2\sigma }{\partial }_{x}u=0,(t,x)\in \mathbb{R}\times \mathbb{R}\\ u(0,x)=u_0(x)\end{array}$$

$$u_{\gamma }(t,x)={\gamma }^{\frac{1}{2\sigma }}u({\gamma }^{2}t,\gamma x),\gamma >0,$$

$$M(u)=\Vert u{\Vert }_{L^2}^2=M(u_0),$$

$$E(u)=\Vert {\partial }_{x}u{\Vert }_{L^2}^2-\frac{1}{\sigma +1}Im{\int }_{\mathbb{R}}|u{|}^{2\sigma }u\overline{{\partial }_{x}u}dx=E(u_0),$$

$$P(u)=Re{\int }_{\mathbb{R}}i{\partial }_{x}u\bar{u}dx=Im{\int }_{\mathbb{R}}u\overline{{\partial }_{x}u}dx=P(u_0).$$

$$u_{\omega ,c}(t,x)=e^{i\omega t}{\varphi }_{\omega ,c}(x-ct),$$

$${\varphi }_{\omega ,c}(x)={\Phi }_{\omega ,c}(x)e^{\frac{ic}{2}x-\frac{i}{2\sigma +2}{\int }_0^x{\Phi }_{\omega ,c}(y{)}^{2\sigma }dy}$$

$${\Phi }_{\omega ,c}(x)=\{\begin{array}{ll}{\left(\frac{(\sigma +1)(4\omega -c^2)}{2\sqrt{\omega }cosh(\sigma \sqrt{4\omega -c^2}x-c}\right)}^{\frac{1}{2\sigma }},& if\omega >\frac{c^2}{4},\\ {\left(\frac{2(\sigma +1)c}{{\sigma }^2(cx{)}^2+1}\right)}^{\frac{1}{2\sigma }},& if\omega =\frac{c^2}{4}andc>0.\end{array}$$

$$-{\partial }_x^2\Phi +(\omega -\frac{c^2}{4})\Phi +\frac{c}{2}|\Phi {|}^{2\sigma }\Phi -\frac{2\sigma +1}{(2\sigma +2{)}^2}|\Phi {|}^{4\sigma }\Phi ,x\in \mathbb{R},$$

$$-{\partial }_x^2\varphi +\omega \varphi +ic{\partial }_x\varphi -i|\varphi {|}^{2\sigma }{\partial }_x\varphi =0x\in \mathbb{R}.$$

$$-{\partial }_x^2\varphi +\frac{c^2}{4}\varphi +ic{\partial }_x\varphi -i|\varphi {|}^{2\sigma }{\partial }_x\varphi =0,x\in \mathbb{R}.$$

$$\Vert u_0-{\varphi }_c{\Vert }_{H^1}\le \delta ,$$

$$\Vert u(t)-e^{i\theta (t)}\varphi (\cdot -y(t)){\Vert }_{H^1}\le \epsilon .$$

$$S_c(u)(t)=$$

$$=$$

$$K_c(u)(t)=\Vert {\partial }_{x}u{\Vert }_{L^2}^2+cIm{\int }_{\mathbb{R}}u\overline{{\partial }_{x}u}dx+\frac{c^2}{4}\Vert u{\Vert }_{L^2}^2-Im{\int }_{\mathbb{R}}|u{|}^{2\sigma }u\overline{{\partial }_{x}u}dx.$$

$$L_c(u)(t)=\Vert {\partial }_{x}u{\Vert }_{L^2}^2+cIm{\int }_{\mathbb{R}}u\overline{{\partial }_{x}u}dx+\frac{c^2}{4}\Vert u{\Vert }_{L^2}^2$$

$$N(u)(t)=Im{\int }_{\mathbb{R}}|u{|}^{2\sigma }u\overline{{\partial }_{x}u}dx,$$

$$S_c(u)=L_c(u)-\frac{1}{\sigma +1}N(u),K_c(u)=L_c(u)-N(u).$$

$$-{\partial }_x^2\Psi +\frac{c}{2}|\Psi {|}^{2\sigma }\Psi -\frac{2\sigma +1}{(2\sigma +2{)}^2}|\Psi {|}^{4\sigma }\Psi =0,x\in \mathbb{R},$$

$$\Psi (x)=e^{i\theta }{\Phi }_c(x-x_0)$$

$${\Phi }_c(x)={\left(\frac{2(\sigma +1)c}{{\sigma }^2(cx{)}^2+1}\right)}^{\frac{1}{2\sigma }}.$$

$$\psi (x)=e^{i\theta }{\varphi }_c(x-x_0).$$

$$d(c)=\inf \{S_c(u):u\in H^1(\mathbb{R})\setminus \{0\},K_c(u)=0\}.$$

$${\tilde{L}}_c(f):=L_c(e^{\frac{c}{2}ix}f),{\tilde{N}}_c(f):=N(e^{\frac{c}{2}ix}f),$$

$${\tilde{L}}_c(f)=\Vert {\partial }_{x}f{\Vert }_{L^2}^2,{\tilde{N}}_c(f)=-\frac{c}{2}\Vert f{\Vert }_{L^{2\sigma +2}}^{2\sigma +2}+Im{\int }_{\mathbb{R}}|f{|}^{2\sigma }f\overline{{\partial }_{x}f}dx,$$

Full text

[TABLE]

where $\sigma\in(0,1)$. The equation has a two-parameter family of solitary waves

[TABLE]

with $(\omega,c)$ satisfying $\omega>c^{2}/4$, or $\omega=c^{2}/4$ and $c>0$. The stability theory in the frequency region $\omega>c^{2}/4$ was studied previously. In this paper, we prove the stability of the solitary wave solutions in the endpoint case $\omega=c^{2}/4$ and $c>0$.

MSC: 35A15, 35B35, 35Q55

Keywords: Generalized Derivative Nonlinear Schrödinger equation; Orbital stability; Endpoint case; Variational method

1. Introduction

The derivative nonlinear Schrödinger (DNLS) equation

[TABLE]

appears in the long wave-length approximation of Alfvén waves propagating in plasma [1, 2, 3]. Applying the gauge transformation

[TABLE]

the equation (1.1) has the Hamiltonian form

[TABLE]

The Cauchy problem for (1.2) has been studied by many researchers. The locally well-posedness theory in the energy space $H^{1}(\mathbb{R})$ was studied in [16, 17, 18, 28]. Local well-posedness in low-regularity spaces $H^{s}(\mathbb{R}))$, $s\geq\frac{1}{2}$ was investigated by Takaoka [26] using the Fourier restricted method. Biagioni and Linares [4] proved that when $s<\frac{1}{2}$, the solution map from $H^{s}(\mathbb{R})$ to $C([-T,T]:H^{s}(\mathbb{R})),T>0$ for (1.2) is not locally uniformly continuous.

The problem of global well-posedness has attracted the attention of a number of authors. Hayashi and Ozawa [18, 24] proved the global existence in $H^{1}(\mathbb{R})$ with $\|u_{0}\|_{L^{2}}^{2}<2\pi$. Wu [29, 30] showed it holds for initial data $u_{0}$ having the mass $\|u_{0}\|_{L^{2}}^{2}$ less than threshold $4\pi$. For the initial data with low regularity, Colliander, Keel, Staffilani, Takaoka, and Tao [6] proved that the $H^{s}$-solution is global if $\|u_{0}\|_{L^{2}}^{2}<2\pi$ when $s>1/2$ by the I-method (see also [5, 26]). Miao, Wu and Xu [23] showed that $H^{\frac{1}{2}}$-solution is global if $\|u_{0}\|_{L^{2}}^{2}<2\pi$. Guo and Wu [13] improved this result to obtain that $H^{\frac{1}{2}}$-solution is global if $\|u_{0}\|_{L^{2}}^{2}<4\pi$. Despite the amount of studies devoted to (1.2), existence of blowing up solutions remains an open problem.

It is known that (1.2) has a two-parameter family of the solitary waves $u_{\omega,c}(t,x)=e^{i\omega t}\phi_{\omega,c}(x-ct)$, where $(\omega,c)$ satisfies $\omega>c^{2}/4$ or $\omega=c^{2}/4$ and $c>0$. Boling Guo and Yaping Wu [12] proved that the solitary waves $u_{\omega,c}$ are orbitally stable when $\omega>c^{2}/4$ and $c<0$ by the abstract theory of Grillakis, Shatah, and Strauss [10, 11] and the spectral analysis of the linearized operators. Colin and Ohta [7] proved that the solitary waves $u_{\omega,c}$ are orbitally stable when $\omega>c^{2}/4$ by characterizing the solitary waves from the view point of a variational structure. The case of $\omega=c^{2}/4$ and $c>0$ is treated by Kwon and Wu [19]. Recently, the stability of the multi-solitons is studied by Miao, Tang, and Xu [22] and Le Coz and Wu [20].

Liu, Simpson, and Sulem [21] introduced an extension of (1.2) with general power nonlinearity, which is the so-called generalized derivative nonlinear Schrödinger equation:

[TABLE]

where $\sigma>0$. The equation of (1.3) is invariant under the scaling transformation

[TABLE]

which implies that its critical Sobolev exponent is $s_{c}=\frac{1}{2}-\frac{1}{2\sigma}$. Hayashi and Ozawa [15] proved local well-posedness in $H^{1}(\mathbb{R})$ when $\sigma\geq 1$ and showed that the following quantities are conserved:

[TABLE]

[TABLE]

[TABLE]

Moreover, they proved global well-posedness for small initial data. They also constructed global solutions for any initial data in the $L^{2}$-subcritial case $1/2\leq\sigma<1$. Recently, Fukaya, Hayashi, Inui [9] and Miao, Tang, xu [22] investigate the global well-posedness for (1.3) in the case $\sigma>1$ by variational argument.

Similar to the equation (1.2), by [21], (1.3) has a two-parameter family of solitary waves

[TABLE]

with $(\omega,c)$ satisfying $\omega>c^{2}/4$, or $\omega=c^{2}/4$ and $c>0$,

[TABLE]

and

[TABLE]

Moreover, $\Phi_{\omega,c}$ is the positive even solution of

[TABLE]

and the complex-valued function $\phi_{\omega,c}$ satisfies

[TABLE]

Liu, Simpson and Sulem [21] proved that when $1<\sigma<2$, for some $z_{0}=z_{0}(\sigma)\in(0,1)$, if $-2\sqrt{\omega}<c<2z_{0}\sqrt{\omega}$, the solitary waves are orbitally stable and if $2z_{0}\sqrt{\omega}<c<2\sqrt{\omega}$, they are orbitally unstable. They also showed that the solitary waves for all $\omega>c^{2}/4$ are orbitally unstable when $\sigma\geq 2$ and orbitally stable when $0<\sigma<1$. In [8], it is proved that the solitary waves are orbitally unstable if $c=2z_{0}\sqrt{\omega}$ when $3/2<\sigma<2$. Tang and Xu [27] investigated stability of the two sum of solitary waves for (1.3). We also refer to [14, 25] for some lower regularity results.

In this present work, we consider the stability of solitary wave solutions of (1.3) in the endpoint case $\omega=c^{2}/4$ when $\sigma\in(0,1)$. For simplicity, we denote $\phi_{c}=\phi_{c^{2}/4,c}$, which solves

[TABLE]

More precisely, we prove that when $\omega=c^{2}/4$, the solitary waves (1.7) are orbitally stable in the sense of the following.

Theorem 1.1**.**

For any $\varepsilon>0$, there exists some $\delta=\delta(\varepsilon)$ such that if

[TABLE]

then there exist $\theta(t)\in[0,2\pi),y(t)\in\mathbb{R}$ such that the solution $u(t)$ of the equation (1.3) satisfies that, for any $t\in\mathbb{R}$,

[TABLE]

We use a variational method to prove Theorem 1.1, but it is not standard and the difficulty is from the “zero mass” property of (1.10) in this endpoint case.

The rest of the paper is organized as follows. In section 2, we give a variational characterization of solitary wave solutions. Then by a variational argument, we prove Theorem 1.1 in section 3.

2. variational characterization

In this section, we give a variational characterization of the solitary wave solution $\phi_{c}$ of (1.10) defined by (1.7). Note that it is not standard because of the “zero mass” of the equation (1.10). Our approach is inspired by Kwon, Wu [19]. In addition to (1.4), (1.5) and (1.6), we define several other variational functionals as follows:

[TABLE]

[TABLE]

We also denote for convenience that

[TABLE]

and

[TABLE]

which imply that

[TABLE]

Using a standard argument as in Berestycki and Lion [4], we can obtain the uniqueness result as follows.

Lemma 2.1**.**

If $\Psi\in H^{1}(\mathbb{R})\setminus\{0\}$ is a solution of

[TABLE]

then there exists some $(\theta,x_{0})$ such that

[TABLE]

with

[TABLE]

The difficulty is that we have no $L^{2}$-control from $L_{c}(u)$. A counterpart result is as follows.

Lemma 2.2**.**

If $\psi\in H^{1}(\mathbb{R})\setminus\{0\}$ is a solution of (1.10) then there exists some $(\theta,x_{0})$ such that

[TABLE]

We consider the following minimization problem:

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Letting

[TABLE]

then

[TABLE]

and

[TABLE]

[TABLE]

Hence, equivalently, we have

[TABLE]

We need the following result to give the characterization of $d(c)$.

Lemma 2.3**.**

Assume that $f\in H^{1}(\mathbb{R})\setminus\{0\}$ satisfies

[TABLE]

then $\tilde{K}_{c}(f)\geq 0.$

Proof.

We argue by contradiction to assume that there exists some $f\in H^{1}(\mathbb{R})\setminus\{0\}$ such that $\tilde{K}_{c}(f)<0$. Then, there exists some $\gamma\in(0,1)$ such that $\tilde{K}_{c}(f_{\gamma})=0$, with $f_{\gamma}(x)=\gamma^{\frac{1}{\sigma}}f(\gamma x)$. Indeed, since

[TABLE]

$\tilde{K}_{c}(f)<0$ implies that we may choose

[TABLE]

to get $\tilde{K}_{c}(f_{\gamma})=0$. Therefore, by definition of $d(c)$ in (2.7), $\tilde{S}_{c}(f_{\gamma})\geq d(c)$, which gives then

[TABLE]

or $\gamma^{\frac{2}{\sigma}+1}\|\partial_{x}f\|_{L^{2}}^{2}\geq\frac{\sigma+1}{\sigma}d(c).$ Since $\gamma<1$, this contradicts the assumption $\|\partial_{x}f\|_{L^{2}(\mathbb{R})}^{2}\leq\frac{\sigma+1}{\sigma}d(c)$ and we conclude the proof. ∎

Now we give the characterization of $d(c)$.

Lemma 2.4**.**

It holds that

[TABLE]

with $\phi_{c}$ defined by (1.7) with $w=c^{2}/4$.

Proof.

We first claim that

[TABLE]

Indeed, considering

[TABLE]

Now, if $d(c)=0$, then we obtain some minimizing sequence $\{v_{n}\}\subset H^{1}(\mathbb{R})\setminus\{0\}$, such that

[TABLE]

Thus, we have $\|\partial_{x}v_{n}\|_{L^{2}}^{2}\rightarrow 0$ and

[TABLE]

that is

[TABLE]

Note that

[TABLE]

with some $\theta\in(1,2\sigma+2)$ and some constants $A>0$. From (2.8),(2.9) and $\|\partial_{x}v_{n}\|_{L^{2}}^{2}\rightarrow 0$ , we obtain that

[TABLE]

which gives by interpolation that

[TABLE]

Using (2.10), we obtain that

[TABLE]

which gives $v_{n}=0$. This is a contradiction and gives the claim $d(c)>0$.

Next, let $\{v_{n}\}\subset H^{1}(\mathbb{R})\setminus\{0\}$ be the minimizing sequence such that as $n\rightarrow\infty$,

[TABLE]

and

[TABLE]

There holds then $\|\partial_{x}v_{n}\|_{L^{2}}^{2}\rightarrow\frac{\sigma+1}{\sigma}d(c)$. Moreover, by a similar argument of (2.11), there exists some absolute constant $C>0$ such that

[TABLE]

Now we apply the profile decomposition to the uniformly bounded sequence $\{v_{n}\}$ in $\dot{H}^{1}(\mathbb{R})\bigcap L^{2\sigma+2}(\mathbb{R})$ to obtain that there exist some sequences $\{V^{j}\}_{j=1}^{\infty}$ and $\{x_{n}^{j}\}_{n,j=1}^{\infty}$ such that, up to some supsequence, for each $L\geq 1$,

[TABLE]

with

[TABLE]

and

[TABLE]

Moreover,

[TABLE]

[TABLE]

and

[TABLE]

Thus by Lemma 2.3, $\tilde{K}_{c}(V^{j})=K_{c}(e^{\frac{c}{2}ix}V^{j})\geq 0$ or $V^{j}=0$. Since also $\tilde{K}_{c}(R_{n}^{L})\geq 0$, from (2.16) and $\tilde{K}(v_{n})=0$, we get that for any $j=1,2,\cdots,L$ there must hold that $\tilde{K}_{c}(V^{j})=0,$ which means that $\tilde{S}_{c}(V^{j})\geq d(c)$ or $V^{j}=0$. Now from (2.14), there exists only one $j$, say $j=1$, such that $\tilde{S}_{c}(V^{1})=d(c)$ and $V^{j}=0$ for $j=2,\cdots,L$. Hence, $V^{1}$ is the function such that

[TABLE]

Then there exists some Lagrange constant $\rho$ such that

[TABLE]

which implies

[TABLE]

That is to say

[TABLE]

which implies that $\rho=0$. Therefore, we have that

[TABLE]

and then $e^{\frac{c}{2}ix}V^{1}(x)$ solves the equation (1.10). Hence by Lemma 2.2,

[TABLE]

It follows that

[TABLE]

∎

Finally in this section, we prove the following lemma, which is useful to show our main result.

Lemma 2.5**.**

111Due to a private discussion with Cui Ning.

There hold that

[TABLE]

Proof.

First of all, by the definition of (1.7) and (1.8) and straight calculation, it follows that

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Next, letting $\phi:=\phi_{1}$, it follows that $\phi_{c}(x)=c^{\frac{1}{2\sigma}}\phi(cx)$,

[TABLE]

Hence,

[TABLE]

Note that

[TABLE]

and recall

[TABLE]

We have then

[TABLE]

Finally, we are sufficed to show the integration

[TABLE]

In fact, by straight calculation,

[TABLE]

which implies that if $\sigma<2$,

[TABLE]

or

[TABLE]

Since $\sigma<1$, we obtain that

[TABLE]

As a result, we conclude Lemma 2.5 by (2.18) and (2.19). ∎

3. Proof of Theorem 1.1

In this section, we prove Theorem 1.1. The proof is based on the variational characterization of solitary wave solutions in section 2. Using the notations defined in the above section, we set

[TABLE]

[TABLE]

Lemma 3.1**.**

The sets $\mathcal{A}^{+}$ and $\mathcal{A}^{-}$ are invariant under the flow of (1.3), i.e., if $u_{0}\in\mathcal{A}^{+}$ (resp. $\mathcal{A}^{-}$), then the solution $u(t)$ of (1.3) with $u(0)=u_{0}$ belongs to $\mathcal{A}^{+}$ (resp. $\mathcal{A}^{-}$) as long as $u(t)$ exists.

Proof.

Let $u_{0}\in\mathcal{A}^{+}$ and $I=(-T_{*},T^{*})$ be the maximal existence interval of the solution $u(t)$ of (1.3) with $u(0)=u_{0}$. By $u_{0}\neq 0$ and the conservation laws (1.5), (1.4) and (1.6), we have that $u(t)\neq 0$ for $t\in I$. By definition of $S_{c}(u)$, $S_{c}$ is also conserved, which means that $S_{c}(u(t))=S_{c}(u_{0})<d(c)$ for $t\in I$. By continuity of the function $t\mapsto L_{c}(u(t))$, we assume that there exits some $t_{0}$ such that

[TABLE]

Thus, from

[TABLE]

we get that $\tilde{K}_{c}(e^{-\frac{c}{2}ix}u(t_{0}))=K_{c}(u(t_{0}))<0$. On the other hand, by Lemma 2.3, it holds that

[TABLE]

and we get a contradiction. Hence $\mathcal{A}^{+}$ is invariant under the flow of (1.3).

In the same way, we see that $\mathcal{A}^{-}$ is invariant under the flow of (1.3). ∎

Lemma 3.2**.**

Let $w\in H^{1}(\mathbb{R})$. For any $\varepsilon>0$, there exists some $\delta>0$ such that if

[TABLE]

then

[TABLE]

Proof.

By contradiction, we assume that there exist $\varepsilon_{0}>0$ and some sequences $\{w_{n}\}\subset H^{1}(\mathbb{R})$ such that

[TABLE]

but

[TABLE]

On the other hand, we follow the the proof of Lemma 2.4, using the profile decomposition, to find that

[TABLE]

i.e.

[TABLE]

and $e^{\frac{c}{2}ix}V^{1}(x)=e^{i\theta}\phi_{c}(x-x_{0})$ solves the equation (1.10). Hence, we obtain that for large $n$ it holds that

[TABLE]

which is a contradiction with (3.2). ∎

Lemma 3.3**.**

For any $\varepsilon>0$, there exists some $\delta>0$ such that if $\|u_{0}-\phi_{c}\|_{H^{1}}<\delta$, then the solution $u(t)$ of (1.3) with $u(0)=u_{0}$ satisfies

[TABLE]

where $I$ is the maximal lifespan.

Proof.

For sufficiently small $\delta>0$, which will be determined later, it follows that $\|u_{0}-\phi_{c}\|_{H^{1}}<\delta$ implies

[TABLE]

where $\lambda$ will be chosen later such that $|\lambda|$ is small. Applying Taylor expansion to the function $S_{c+\lambda}(\phi_{c+\lambda})$ of $\lambda$, we obtain that

[TABLE]

where we use the formular

[TABLE]

which is negative by Lemma 2.5. Thus, combined with (3.3), we obtain that

[TABLE]

For any $\lambda$ satisfying $|\lambda|\in(0,\lambda_{0})$ with some $\lambda_{0}>0$ small enough, we may choose $\delta>0$ small such that

[TABLE]

Now we deal with $L_{c}$. Note that

[TABLE]

By Taylor expansion, we estimate

[TABLE]

which, by choosing $\lambda_{0}$ and $\delta>0$ smaller, is small than $\frac{\sigma+1}{\sigma}d(c+\lambda)$, i.e.

[TABLE]

Similarly, it must hold that for small $\lambda_{0}$ and $\delta>0$,

[TABLE]

In view of the invariant sets $\mathcal{A}^{\pm}$, these estimates, combined with (3.4), imply that for any $t\in I=(-T_{*},T^{*})$,

[TABLE]

Therefore, we can choose sufficiently small $\lambda_{0}>0$ and $\delta>0$ such that if $\|u_{0}-\phi_{c}\|_{H^{1}}<\delta$,

[TABLE]

with some constant $A>0$ large enough. Since $K_{c}=(\sigma+1)S_{c}-\sigma L_{c}$, we finally obtain that

[TABLE]

∎

Now we prove Theorem 1.1.

The Proof Theorem 1.1. By contradiction, we assume that there exists some $\varepsilon_{0}>0$ such that for any small $\delta>0$ there exists some sequence $\{t_{n}\}$ satisfying $\|u_{0}-\phi_{c}\|_{H^{1}}<\delta$, but

[TABLE]

Taking $\delta$ small enough, we get from Lemma 3.3 that

[TABLE]

where $\tilde{\delta}>0$ can be chosen sufficiently small such that, using Lemma 3.2, it holds that

[TABLE]

Moreover, by mass conversation, we have

[TABLE]

Choosing $\delta$ smaller, we obtain then

[TABLE]

Finally, there holds that

[TABLE]

which contradicts (3.6).                                                              $\Box$

**Acknowledgment ** The author would like to thank Cui Ning for useful discussions. This work was supported by the National Natural Science Foundation of China (No.11301564, 11771469).