1 Introduction

We study the Gross-Pitaevskii equation,

$$\begin{aligned} i \Psi _{t}+\Psi _{x x} + \Psi \left( 1-|\Psi|^{2}\right) =0 \qquad x \in {\mathbb {R}}, \end{aligned}$$
(1)

where \(\Psi: {\mathbb {R}} \times {\mathbb {R}} \rightarrow {\mathbb {C}}\) satisfies the boundary condition

$$\begin{aligned}|\Psi| \rightarrow 1 \qquad \text{ as }|x| \rightarrow \infty. \end{aligned}$$

Equation (1) appears in various fields in physics, including superfluidity and Bose-Einstein condensation ([1, 3, 4, 20,21,22]), and it describes the dark soliton in nonlinear optics ([23, 24]). Under the nonzero boundary condition, (1) has a nontrivial dynamics, in contrast with the zero boundary condition case, where the dynamics is essentially dispersion and scattering.

The energy functional

$$\begin{aligned} E(\Psi )=\int _{{\mathbb {R}}}| \Psi _{x}|^{2}+\frac{1}{2}\left(|\Psi|^{2} - 1 \right) ^{2} d x \end{aligned}$$

is a conserved quantity of (1), where \(V(|\Psi|^{2})=\frac{1}{2}(|\Psi|^{2}-1)^{2}\) is the potential.

The momentum \(P(\Psi )\) is also conserved. Section 2 will give \(P(\Psi )\) a rigorous definition.

We consider the traveling wave solution of (1): \(\Psi (x, t)=\phi (x+v t)\), where v is velocity. It satisfies

$$\begin{aligned} i v \phi _{x} + \phi _{x x}+ \phi \left( 1-|\phi|^{2}\right) =0 \qquad \text{ in } {\mathbb {R}}. \end{aligned}$$
(2)

We only focus on the case \(v \ge 0\), because a function \(\phi\) solves (2) for some v is equivalent to \(\phi (-x)\) solves it with velocity \(-v\).

Equation (2) is integrable, by the ordinary differential equation technique (see [6]). If \(v \ge \sqrt{2}\), \(\phi =1\) (modulo complex number of magnitude 1). We set \(v_{s} = \sqrt{2}\) (called the sound speed). For \(0 \le v<\sqrt{2}\), the solution is either 1, or

$$\begin{aligned} b_{v}=\sqrt{\frac{2-v^{2}}{2}} \tanh \left( \frac{\sqrt{2-v^{2}}}{2} x\right) -i \frac{v}{\sqrt{2}}, \end{aligned}$$
(3)

modulo unit length complex number and translation. For \(v \ne 0,\)\(b_{v}\) are called dark solitons and they do not vanish on \({\mathbb {R}}\). In the case \(v=0\), \(b_{0}=0\) at \(x=0\). \(b_{0}\) is called the black soliton.      

We consider orbital stability of the solution (3). Two ways are used to tackle the orbital stability problem: the first one is concentration-compactness argument in [11], and the other one is Grillakis-Shatah-Strauss theory ( [18, 19]). Our goal is to establish orbital stability using [11] for all speed \(|v| < \sqrt{2}\), under a general class of perturbations in the energy space.

The overall strategy is to implement (3) as minimizers of E at fixed P, where v serves as the Lagrange multiplier. Then using [11], we get the orbital stability result.

We introduce some function spaces. Let \(\phi \in H_{\text{ loc }}^{1}({\mathbb {R}})\) and \(\Omega \subset {\mathbb {R}}\) be an open set, we define

$$\begin{aligned} E_{\Omega }(\phi )=\int _{\Omega }| \phi ^{\prime }|^{2}+ V(|\phi|^{2}) d x \end{aligned}$$

to be the Ginzburg-Landau energy of \(\phi\) in \(\Omega\). When \(\Omega = {\mathbb {R}}\), we use \(E(\phi )\) rather than \(E_{{\mathbb {R}}}(\phi )\).

We use the notation \({\dot{H}}^{1}({\mathbb {R}})=\left\{ \phi \in L_{\text{ loc } }^{1}({\mathbb {R}}) ~\big|~ \phi ^{\prime } \in L^{2}({\mathbb {R}})\right\}\). Define the energy space

$$\begin{aligned} \begin{aligned} {\mathcal {E}}&=\left\{ \phi \in {\dot{H}}^{1}({\mathbb {R}}) ~\big|~| \phi|^{2}-1 \in L^{2}({\mathbb {R}})\right\} \\&=\left\{ \phi \in {\dot{H}}^{1}({\mathbb {R}}) ~\big|~ E(\phi )<\infty \right\}. \end{aligned} \end{aligned}$$

Denote the distance \(({\mathcal {E}}, d_{{\mathcal {E}}} )\) as

$$\begin{aligned} \begin{aligned} d_{{\mathcal {E}}}\left( \phi _{1}, \phi _{2}\right) = \left\||\phi _{1}| -|\phi _{2}| \right\| _{L^{2}({\mathbb {R}})} + \left\| \phi _{1}^{\prime } - \phi _{2}^{\prime }\right\| _{L^{2}({\mathbb {R}})} + \left\| \phi _{1}-\phi _{2}\right\| _{L^{2}+L^{\infty }({\mathbb {R}})}. \end{aligned} \end{aligned}$$
(4)

\(({\mathcal {E}}, d_{{\mathcal {E}}} )\) is a complete metric space (this can be proved following section 1 in [15], pp. 132–133).

Denote the semi-distance \(d_{0}\) on \({\mathcal {E}}\) as

$$\begin{aligned} \begin{aligned} d_{0}(\phi _{1}, \phi _{2}) = \left\||\phi _{1}| -|\phi _{2}| \right\| _{L^{2}({\mathbb {R}})} + \left\| \phi _{1}^{\prime } - \phi _{2}^{\prime }\right\| _{L^{2}({\mathbb {R}})}. \end{aligned} \end{aligned}$$
(5)

The following theorem is established in ( [6, 7]). Our main aim of this article is to provide an alternative proof of this well-known theorem.

Theorem 1.1

( [6, 7]) For \(0 < q \le \pi\), let

$$\begin{aligned} \begin{aligned} \displaystyle E_{\min }(q) = \inf _{\phi \in {\mathcal {E}}} \{ E( \phi ) ~\big|~ P( \phi ) = q \}. \end{aligned} \end{aligned}$$

Then any minimizing sequence \((\phi _{n})_{n \ge 1} \subset {\mathcal {E}}\) verifying \(E(\phi _{n}) \rightarrow E_{\min }(q)\) under the constraint \(P(\phi ) \rightarrow q\) has a convergent subsequence, under the semi-distance \(d_{0}\) (up to translations).

\(U_{q} = \{\phi \in {\mathcal {E}} ~\big|~ E(\phi ) = E_{\min }(q), ~P(\phi )=q \}\) has a unique element \(b_{v(q)}\) (up to translations and rotations), where v(q) is the unique speed v such that \(P(b_{v})=q\). The set \(U_{q}\) is orbitally stable, with respect to the semi-distance \(d_{0}\).

Theorem 1.1 is a summary of Theorem 4.1, Proposition 4.6 and Theorem 5.5.

The orbital stability of dark solitons \(v=0\), under the distance (see Lemma 10 in [12], p. 1338, and [25])

$$\begin{aligned} \begin{aligned} d\left( \phi _{1}, \phi _{2}\right) = \left| \phi _{1}(0)-\phi _{2}(0)\right| + \left\||\phi _{1}|-|\phi _{2}|\right\| _{L^{2}({\mathbb {R}})} + \left\| \phi _{1}^{\prime } - \phi _{2}^{\prime } \right\| _{L^{2}({\mathbb {R}})}, \end{aligned} \end{aligned}$$
(6)

was proved in [25]. The proof exploits the hydrodynamical form of (1), which is a Hamiltonian system and Grillakis-Shatah-Strauss theorem is applied.

This method is not valid for the case \(v=0\), since \(b_0\) vanishes at \(x=0\). Orbital stability for black soliton \((v=0)\) for distance

$$\begin{aligned} \begin{aligned} d_{A}\left( \phi _{1}, \phi _{2}\right) = \left\| \phi _{1}-\phi _{2}\right\| _{L^{\infty }[-A, A]} + \left\||\phi _{1}|-|\phi _{2}|\right\| _{L^{2}({\mathbb {R}})} + \left\| \phi _{1}^{\prime } - \phi _{2}^{\prime } \right\| _{L^{2}({\mathbb {R}})} \end{aligned} \end{aligned}$$
(7)

was established in [7] relying on variational arguments, given any \(A>0\). The orbital stability of \(b_{v}\) (\(|v| < \sqrt{2}\)) with the distance (7) has been proved in ( [6, 7]).

Using Lemma 2.2, it can be shown that the semi-distance \(d_{0}\), the distance defined in (6) and (7) are equivalent, so we state Theorem 1.1 using the semi-distance \(d_{0}\).

A motivation of this work is that, previous work, e.g. ( [6, 7, 12]), treated the cases \(0<v<\sqrt{2}\) and \(v=0\) separately, while our proof strategy deals with the two cases in a unified framework.

[16] proved orbital stability of black soliton, under a very restricted class of perturbations. See [12] for a detailed study of the stability problem of the traveling waves for the nonlinear Schrödinger equation, under the distance (6). Generalizations of the orbital stability to variations of the 1-dimensional Gross-Pitaevskii equation (with non local terms and general nonlinearities) was shown in ( [5, 14]). The asymptotic stability was shown in [8].

In space dimension \(N \ge 2\), the constraint minimization procedure is used in [13] to obtain a class of orbitally stable traveling waves, for general nonlinearity (including the Gross-Pitaevskii equation).

We then comment on the proof methods. We rely on the ideas in [13]. An important quantity called modified Ginzburg-Landau energy is indispensable in analyzing the traveling waves in space dimension \(\ge 2\) ( [13, 27]). However, for the one dimensional equation (1), we don’t need this modified Ginzburg-Landau energy because the Ginzburg-Landau energy E itself can be used to control \(\Vert|\phi| - 1 \Vert _{L^ {\infty } ({\mathbb {R}} ) },\) see Lemma 2.2.

We use the concentration-compactness principle (similar to [13]) to prove that \(b_{v(q)}\) is minimizer (modulo translations and rotations) for \(E_{\min }(q)\). If “vanishing” holds, we have that \(\left\| \left| \phi _{n}\right| - 1\right\| _{L^{\infty }} \rightarrow 0\), provided \(\left( \phi _{n}\right) _{n \ge 1}\) is a vanishing minimizing sequence. Then from Lemma 3.2 (ii) we get \(E(\phi _{n}) \ge v \left| P ( \phi _{n} ) \right|\) for all \(v \in (0, v_{s} ),\) Taking limit \(v \uparrow v_{s}\) we obtain \(E_{\text{ min }}(q) \ge v_{s} q\), which contradicts the upper bound \(E_{\min }(q)<v_{s} q\) (see Lemma 3.3).

If we have “dichotomy”, then we show that \(E_{\text{ min }}(q)=E_{\text{ min }}(q_{1} ) + E_{\text{ min }}(q - q_{1}),\)\(q_{1} \in (0, q)\), which contradicts with \(E_{\min }\) is strictly subadditive (see Lemma 3.5).

Hence, we have concentration since vanishing and dichotomy are excluded.

1.1 Outline

Section 2 gives the rigorous definition of momentum. Section 3 contains some properties of \(E_{\min }\). Section 4 shows the precompactness of the minimizing sequence. Section 5 presents the orbital stability result. Finally, in the Appendix A, we give a technical result: a splitting lemma, which is used to ruling out dichotomy in the proof of Theorem 4.1.

2 The Definition of Momentum in 1D

To solve (2) via a variational approach, we need a reasonable definition of momentum. In dimension \(N \ge 3\), a definition of the momentum for all functions in the energy space has been given in [27]. The definition of momentum in dimension 2 is given in [13]. In dimension 1, a definition called untwisted momentum for any function in \({\mathcal {E}}\) has been provided by [7]. We propose an alternative definition in 1D, generalizing a strategy given in [27] for dimension \(\ge 3\), and show that this definition is equivalent to the one given in [7]. We will use this alternative definition in the following sections.

We now give some observations of why we need to give a definition of momentum. The momentum should be defined as

$$\begin{aligned} P(\phi )=\int _{{\mathbb {R}}}\left\langle i \phi ^{\prime }, \phi -1\right\rangle d x. \end{aligned}$$

provided \(\phi -1 \in H^{1}({\mathbb {R}})\). But there are functions \(\phi -1 \in {\mathcal {E}} \setminus H^{1}({\mathbb {R}})\) satisfying \(\langle i \phi ^{\prime }, \phi -1\rangle \notin L^{1}({\mathbb {R}})\).

If \(\phi \in {\mathcal {E}}\) has a lifting \(\phi =\rho e^{i \theta }\), and \(\lim _{x \rightarrow \infty } \phi, ~\lim _{x \rightarrow -\infty } \phi\) exist, a computation gives

$$\begin{aligned} \int _{{\mathbb {R}}}\left\langle i \phi ^{\prime }, \phi -1\right\rangle d x=-\int _{{\mathbb {R}}}\left( \rho ^{2}-1\right) \theta ^{\prime } d x + [ {{\text {Im}}} (\phi ) - \theta ]|_{-\infty } ^{\infty }. \end{aligned}$$

But there exists \(\phi \in {\mathcal {E}}\) such that \(\phi\) can not be liftted. Also, \(\lim _{x \rightarrow \infty } \phi (x)\), \(\lim _{x \rightarrow -\infty } \phi (x)\) may not exist.

Lemma 2.1

Let \(\phi \in {\mathcal {E}}\) satisfies \(0< c_1 \le|\phi| <\infty\) on \({\mathbb {R}}\) for a constant \(c_1\). Then we can write \(\phi =\rho e^{i \theta }\) with \(\rho -1 \in H^{1}({\mathbb {R}}), ~\theta \in {\dot{H}}^{1}({\mathbb {R}}),\)

$$\begin{aligned} \begin{aligned} \left\langle i \phi ^{\prime }, \phi -1\right\rangle =\frac{d}{d x}( {\text {Im}} (\phi ) - \theta ) - \left( \rho ^{2}-1\right) \theta ^{\prime } \text{ a.e. } \text{ on } {\mathbb {R}}. \end{aligned} \end{aligned}$$
(8)

In addition, \(\smallint _{\mathbb {R}}\mid(\rho^2-1){\theta}^{\prime}\mid{dx}{\leq} \frac{1}{\sqrt{2} c_1} E(\phi )\).

Proof

Since \(\phi \in H_{\text {loc}}^{1}({\mathbb {R}})\), the existence of \(\rho, ~\theta \in H_{\text {loc}}^{1}({\mathbb {R}})\) such that \(\phi =\rho e^{i \theta }\) a.e. can be obtained using Theorem 1 in ( [10], p. 37). Direct calculation shows

$$\begin{aligned} \begin{aligned} \left| \phi ^{\prime }\right| ^{2}=\left| \rho ^{\prime }\right| ^{2}+\rho ^{2}\left| \theta ^{\prime }\right| ^{2}. \end{aligned} \end{aligned}$$
(9)

Since \(\rho =|\phi| \ge c_1\) and \(\phi ^{\prime } \in L^{2}({\mathbb {R}})\), it follows that \(\rho ^{\prime }, \theta ^{\prime } \in L^{2}({\mathbb {R}})\). We have \(\rho ^{2}-1 \in L^{2}({\mathbb {R}})\) because \(\phi \in {\mathcal {E}}\). Since \(|\rho -1|=\frac{\left| \rho ^{2}-1\right| }{\rho +1} \le \left| \rho ^{2}-1\right|\), then \(\rho - 1 \in L^{2}({\mathbb {R}})\).

A short computation yields

$$\begin{aligned} \begin{aligned}&\left\langle i \phi ^{\prime }, \phi -1\right\rangle =\left\langle i \phi ^{\prime },-1\right\rangle -\rho ^{2} \theta ^{\prime } = \frac{d}{d x}({\text {Im}} (\phi ) - \theta )-\left( \rho ^{2}-1\right) \theta ^{\prime }. \end{aligned} \end{aligned}$$

Using 9, we have \(|\theta ^{\prime }| \le \frac{1}{\rho }|\phi ^{\prime }| \le \frac{1}{c_1}|\phi ^{\prime }|\), and

$$\begin{aligned}\int_{\mathbb {R}} \mid(\rho^2-1)\theta^\prime{\mid}dx&{\leq} {\parallel}(\rho^2-1){\parallel}_{L^2}{\parallel}\theta^\prime{\parallel}_{L^2}{\leq}\frac{1}{c_1}{\parallel}(\rho^2-1){\parallel}_{L^2}{\parallel}\phi^\prime{\parallel}_{L^2}\\ &{\leq}\frac{1}{\sqrt{2}c_1}(\frac{1}{2}{\parallel}(\rho^2-1){\parallel}_{L^2}^2+{\parallel}\phi^{\prime}{\parallel}_{L^2}^2)=\frac{1}{\sqrt{2}c_1}E(\phi).\end{aligned}$$

\(\hfill{\square}\)

We use the notation

$$\begin{aligned} X^{1}({\mathbb {R}})=\left\{ \phi \in L^{\infty }({\mathbb {R}}) ~\big|~ \phi ^{\prime } \in L^{2}({\mathbb {R}})\right\}. \end{aligned}$$

Lemma 2.2

We have \({\mathcal {E}} \subset L^{\infty }({\mathbb {R}})\). There exists a universal constant C such that

$$\begin{aligned} \Vert \phi \Vert _{L^{\infty }({\mathbb {R}})} \le C(1+\sqrt{ E(\phi ) } ). \end{aligned}$$

Moreover,

$$\begin{aligned} \begin{aligned}|\phi|^{2}-1 \in H^{1}({\mathbb {R}}), \quad \forall \phi \in {\mathcal {E}}. \end{aligned} \end{aligned}$$
(10)

Proof

Let \(\chi _{1} \in C_{0}^{\infty }({\mathbb {C}})\) with \(0 \le \chi _{1} \le 1\), \(\chi _{1}(x)=1\) for \(|x| \le 2\), and \(\chi _{1}(x) = 0\) for \(|x| \ge 3\). Let us decompose

$$\begin{aligned} \phi =\phi _{1}+\phi _{2}, \quad \phi _{1}=\chi _{1}(\phi ) \phi, \quad \phi _{2}=\left( 1-\chi _{1}(\phi )\right) \phi. \end{aligned}$$

Using Lemma 1.5 in ( [15], p. 132), we have

$$\begin{aligned} \left\| \phi _{1}\right\| _{X^{1}({\mathbb {R}})}+\left\| \phi _{2}\right\| _{H^{1}({\mathbb {R}})} \le C_{1}+C_{2} \sqrt{E(\phi )}. \end{aligned}$$

By Sobolev inequality in 1D ([9], pp. 212–213),

$$\begin{aligned} \begin{aligned} \Vert \phi \Vert _{L^{\infty }({\mathbb {R}})}&\le \left\| \phi _{1}\right\| _{L^{\infty }({\mathbb {R}})}+\left\| \phi _{2}\right\| _{L^{\infty }({\mathbb {R}})} \le \left\| \phi _{1}\right\| _{X^{1}({\mathbb {R}})} + C \left\| \phi _{2}\right\| _{H^{1}({\mathbb {R}})} \\&\le C(1+\sqrt{E(\phi )}). \end{aligned} \end{aligned}$$

Since \(\left(|\phi|^{2}-1\right) ^{\prime } = 2\left\langle \phi, \phi ^{\prime }\right\rangle\), we have

$$\begin{aligned} \begin{aligned} \Vert (|\phi|^{2}-1 )^{\prime } \Vert _{L^{2}({\mathbb {R}})}&=2\left\| \left\langle \phi, \phi ^{\prime }\right\rangle \right\| _{L^{2}({\mathbb {R}})} \le 2\left( \int _{{\mathbb {R}}}|\phi|^{2}|\phi ^{\prime }|^{2} d x \right) ^{\frac{1}{2}} \\&\le 2\Vert \phi \Vert _{L^{\infty }({\mathbb {R}})}\left\| \phi ^{\prime }\right\| _{L^{2}({\mathbb {R}})} \le C(1+\sqrt{E(\phi )}) \sqrt{E(\phi )}<\infty, \end{aligned} \end{aligned}$$

thus, \(\left(|\phi|^{2}-1\right) ^{\prime } \in L^{2}({\mathbb {R}})\). Combining with the fact that \(|\phi|^{2} - 1 \in L^{2}({\mathbb {R}})\), we have \(|\phi|^{2}-1 \in H^{1}({\mathbb {R}})\). \(\hfill{\square}\)

Remark 2.3

[7] uses the energy space

$$\begin{aligned} \chi ^{1} = \left\{ \gamma \in L^{\infty }({\mathbb {R}}) ~\big|~ 1-| \gamma|^{2} \in L^{2}({\mathbb {R}}) \text { and } \gamma ^{\prime } \in L^{2}({\mathbb {R}}) \right\}. \end{aligned}$$

Using Lemma 2.2, we see that \({\mathcal {E}} = \chi ^{1}.\)

Lemma 2.4

Let \(\chi \in C_{c}^{\infty }({\mathbb {C}}, {\mathbb {R}})\) satisfies \(\chi =1\) on \(\{x ~\big|~||x| - 1| < \frac{1}{4} \}\), \(0 \le \chi \le 1\) and \({\text {supp}}(\chi ) \subset \{x ~\big|~||x| - 1| < \frac{1}{2} \}\). For any \(\phi \in {\mathcal {E}}\), denote \(\phi _{1}-1=\chi (\phi )(\phi -1)\) and \(\phi _{2}-1=(1-\chi (\phi ))(\phi -1)\). Then \(\phi _{1} \in {\mathcal {E}}\), \(\phi _{2}-1 \in H^{1}\left( {\mathbb {R}}\right)\) and we have the following:

$$\begin{aligned} \begin{aligned}|\phi _{i}^{\prime }| \le C|\phi ^{\prime }| \quad \quad i=1, 2, \text { with C depends only on } \chi; \end{aligned} \end{aligned}$$
(11)
$$\begin{aligned} \begin{aligned} \left\| \phi _{2}-1\right\| _{L^{2}({\mathbb {R}})} \le C_{1}\left\||\phi|^{2}-1\right\| _{L^{2}({\mathbb {R}})} \text { and } \\ \left\| \left( 1-\chi ^{2}(\phi )\right) (\phi -1)\right\| _{L^{2}({\mathbb {R}})} \le C_{2}\left\||\phi|^{2}-1\right\| _{L^{2}({\mathbb {R}})}; \end{aligned} \end{aligned}$$
(12)
$$\begin{aligned} \begin{aligned} \int _{{\mathbb {R}}}(\left| \phi _{1}\right| ^{2}-1)^{2} d x \le C_{3} \int _{{\mathbb {R}}}\left(|\phi|^{2}-1\right) ^{2} d x; \end{aligned} \end{aligned}$$
(13)
$$\begin{aligned} \begin{aligned} \int _{{\mathbb {R}}}(\left| \phi _{2}\right| ^{2}-1)^{2} d x \le C_{3} \int _{{\mathbb {R}}}\left(|\phi|^{2}-1\right) ^{2} d x. \end{aligned} \end{aligned}$$
(14)

Let \(\phi _{1}=\rho e^{i \theta }\) be the lifting of \(\phi _{1}\), provided by Lemma 2.1. Then

$$\begin{aligned} \begin{aligned}&\left\langle i \phi ^{\prime }, \phi -1\right\rangle =\left( 1 - \chi ^{2}(\phi )\right) \left\langle i \phi ^{\prime }, \phi -1\right\rangle -\left( \rho ^{2}-1\right) \theta ^{\prime }+\frac{d}{d x}\left( {\text {Im}} (\phi _{1}) - \theta \right). \end{aligned} \end{aligned}$$
(15)

Proof

Since \(\left| \phi _{i}\right| \le|\phi -1|+1\) we have \(\phi _{i} \in L^{\infty }({\mathbb {R}})\) for \(i=1, 2\) by Lemma 2.2. It can be shown that \(\phi _{i} \in H_{\text {loc}}^{1}({\mathbb {R}})\) (see Lemma C1 in [10], p. 66) and we have

$$\begin{aligned} \begin{aligned} \phi _{1}^{\prime }=\left( \partial _{1} \chi (\phi ) \frac{d({\text {Re}}(\phi ))}{d x} + \partial _{2} \chi (\phi ) \frac{d\left( {\text {Im}} (\phi )\right) }{d x}\right) (\phi -1) + \chi (\phi ) \phi ^{\prime }. \end{aligned} \end{aligned}$$
(16)

For \(\phi _{2}\) we have a similar formula. Since \(\partial _{i} \chi (\phi )(\phi -1)\) are bounded, \(i=1, 2\), we have (11).

Since \(||\phi|-1| \ge \frac{1}{4}\) on the support of \((1 - \chi (\phi )) \phi\), there exists \(C_{1}>0\) such that

$$\begin{aligned} \left\| \phi _{2}-1\right\| _{L^{2}({\mathbb {R}})} = \left\| (1-\chi (\phi ))(\phi -1) \right\| _{L^{2}({\mathbb {R}})} \le \left\||\phi|+1 \right\| _{L^{2}({\mathbb {R}})} \le C_{1}\left\||\phi|^{2}-1\right\| _{L^{2}({\mathbb {R}})}. \end{aligned}$$

Thus we get the first part in (12). Similarly we have the second part.        

Since \(\phi _{1}(x)=\phi (x)\) when \(||\phi|-1| \le \frac{1}{4}\), so

$$\begin{aligned} \begin{aligned} \int _{\{||\phi|-1| \le \frac{1}{4} \}} (\left| \phi _{1}\right| ^{2}-1 )^{2} d x = \int _{\{|| \phi|-1| \le \frac{1}{4}\}}\left(|\phi|^{2}-1\right) ^{2} d x. \end{aligned} \end{aligned}$$

There exists \(C_{3}>0\) such that

$$\begin{aligned} (\left| \phi _{1}\right| ^{2}-1 )^{2} \le C_{3}\left(|\phi|^{2}-1\right) ^{2} \end{aligned}$$

if \(||\phi| - 1| \ge \frac{1}{4}\). Thus

$$\begin{aligned} \int _{\{|| \phi|-1|> \frac{1}{4} \}} (\left| \phi _{1}\right| ^{2}-1 )^{2} d x \le C_{3} \int _{\left\{|| \phi|-1|>\frac{1}{4}\right\} }\left(|\phi|^{2}-1\right) ^{2} d x. \end{aligned}$$

This implies (13). (14) is similar.

Since \(\partial _{1} \chi (\phi ) \frac{d({\text {Re}}(\phi ))}{d x}+\partial _{2} \chi (\phi ) \frac{d\left( {\text {Im}} (\phi )\right) }{d x} \in {\mathbb {R}}\), using (16) to get

$$\begin{aligned} \left\langle i \phi _{1}^{\prime }, \phi _{1}-1\right\rangle =\chi ^{2}(\phi )\left\langle i \phi ^{\prime }, \phi -1\right\rangle. \end{aligned}$$

From Lemma 2.1,

$$\begin{aligned} \begin{aligned} \left\langle i \phi _{1}^{\prime }, \phi _{1}-1\right\rangle = \chi ^{2}(\phi )\left\langle i \phi ^{\prime }, \phi -1\right\rangle =\frac{d}{d x}\left( {\text {Im}} (\phi _{1})-\theta \right) -\left( \rho ^{2}-1\right) \theta ^{\prime }, \end{aligned} \end{aligned}$$
(17)

hence,

$$\begin{aligned} \begin{aligned} \left\langle i \phi ^{\prime }, \phi -1\right\rangle =&\left( 1-\chi ^{2}(\phi )\right) \left\langle i \phi ^{\prime }, \phi -1\right\rangle -\left( \rho ^{2}-1\right) \theta ^{\prime }+\frac{d}{d x}\left( {\text {Im}} (\phi _{1})-\theta \right) \end{aligned} \end{aligned}$$

and this gives (15). \(\hfill{\square}\)

Consider the Banach space \({\mathcal {Y}} =\{u^{\prime } ~\big|~ u \in {\dot{H}}^{1}({\mathbb {R}})\}\) (see [27], p. 122). Defining the norm as \(\left\| u^{\prime }\right\| _{{\mathcal {Y}}}=\Vert u\Vert _{{\dot{H}}^{1}({\mathbb {R}})}=\left\| u^{\prime }\right\| _{L^{2}({\mathbb {R}})}\).

For any \(\phi \in {\mathcal {E}}\), from (15), Lemma 2.1 and Lemma 2.4, we see that \(\left\langle i \phi ^{\prime }, \phi -1\right\rangle \in L^{1}({\mathbb {R}}) + {\mathcal {Y}}\). It motivates us to give:

Definition 2.5

For any \(\phi \in {\mathcal {E}}\), let \(\chi, ~\phi _{1}, ~\phi _{2}, ~\rho, ~\theta\) are as in Lemma 2.4, the momentum of \(\phi\) is

$$\begin{aligned} \begin{aligned} P(\phi )=\int _{{\mathbb {R}}}\left( 1-\chi ^{2}(\phi )\right) \left\langle i \phi ^{\prime }, \phi -1\right\rangle -\left( \rho ^{2}-1\right) \theta ^{\prime } d x. \end{aligned} \end{aligned}$$
(18)

The above formula is independent of the choice of the \(\chi\).

If \(\phi \in {\mathcal {E}}\) can be lifted, that is, \(\phi =\rho e^{i \theta }\) with \(\rho-1 \in H^1(\mathbb{R})\) and \(\theta \in {\dot{H}}^{1}(\mathbb{R})\), then from lemma 2.1 and Definition 2.5 we have

$$\begin{aligned} P(\phi )=-\int _{{\mathbb {R}}}\left( \rho ^2-1\right) \theta ^{\prime } d x. \end{aligned}$$
(19)

Remark 2.6

We have \(|\phi|^{2}-1 \in H^{1}({\mathbb {R}})\) by Lemma 2.2, then necessarily \(\lim _{x \rightarrow \infty }|\phi (x)|=\lim _{x \rightarrow -\infty }|\phi (x)|=1\), and then \(\lim _{x \rightarrow \pm \infty }\left( \phi _{1}-\phi \right) = \lim _{x \rightarrow \pm \infty }(\chi (\phi )(\phi -1)+1-\phi )=0\). From (15) and (18), we have

$$\begin{aligned} \begin{aligned} P(\phi )&=\int _{{\mathbb {R}}}\left( 1-\chi ^{2}(\phi )\right) \left\langle i \phi ^{\prime }, \phi -1\right\rangle -\left( \rho ^{2}-1\right) \theta ^{\prime } dx \\&= \int _{{\mathbb {R}}} \left\langle i \phi ^{\prime }, \phi -1\right\rangle -\frac{d}{d x}\left( {\text {Im}} (\phi _{1})-\theta \right) dx \\&=\int _{{\mathbb {R}}}\left\langle i \phi ^{\prime },-1\right\rangle -\frac{d}{d x} {\text {Im}} (\phi _{1})+\left\langle i \phi ^{\prime }, \phi \right\rangle +\theta ^{\prime } dx \\&=\lim _{x_{0} \rightarrow \infty }\left[ \left. \left( {\text {Im}} (\phi )-{\text {Im}} (\phi _1)\right) \right| _{-x_{0}} ^{x_{0}}+\int _{-x_{0}}^{x_{0}}\left\langle i \phi ^{\prime }, \phi \right\rangle dx +\left. \theta \right| _{-x_{0}} ^{x_{0}}\right] \\&=\lim _{x_{0} \rightarrow \infty }\left[ \int _{-x_{0}}^{x_{0}}\left\langle i \phi ^{\prime }, \phi \right\rangle +\left. \arg \phi \right| _{-x_{0}} ^{x_{0}}\right]. \end{aligned} \end{aligned}$$

The last formula above is an alternative definition for momentum of \(\phi\) in \({\mathcal {E}}\) and is precisely the untwisted momentum defined in ( [7], Lemma 1.8), when mod \(2\pi\).

Remark 2.7

We have

$$\begin{aligned} \begin{aligned}&P\left( b_{v}\right) =-v \sqrt{2-v^{2}}-2 \arctan \frac{v}{\sqrt{2-v^{2}}}+\pi. \\&\frac{d}{d v} P\left( b_{v}\right) =-2 \sqrt{2-v^{2}}. \end{aligned} \end{aligned}$$

\(P(b_v)\) is a diffeomorphism from \((0, \sqrt{2})\) to \((0, \pi )\). It follows from Proposition 2.6 in ( [6], p. 63) that \(E\left( b_{v}\right) =\frac{2\left( 2-v^{2}\right) ^{\frac{3}{2}}}{3}\). It can be easily shown as in ( [6], p. 64) that the map \(P \mapsto E(P)\) satisfies \(E(P)< v_{s} P\) on \((0, \pi ]\).

Corollary 2.8

For any constant \(c_{1} \in {\mathbb {C}}\) and \(\phi \in {\mathcal {E}}\) such that \(\phi + c_{1} \in {\mathcal {E}},\) we have \(P\left( \phi +c_{1}\right) =P(\phi )\).

Proof

For any \(\phi \in {\mathcal {E}}\), let \(\phi _{1}, ~\rho, ~\theta\) are given by Lemma 2.4. Then (17) gives

$$\begin{aligned} \begin{aligned} \left\langle i \phi ^{\prime }, \phi +c_{1}-1\right\rangle&= \left( 1-\chi ^{2}(\phi )\right) \left\langle i \phi ^{\prime }, \phi -1\right\rangle + \chi ^{2}(\phi )\left\langle i \phi ^{\prime }, \phi -1\right\rangle +\left\langle i \phi ^{\prime }, c_{1}\right\rangle \\&=\left( 1-\chi ^{2}(\phi )\right) \left\langle i \phi ^{\prime }, \phi -1 \right\rangle +\left\langle i \phi ^{\prime }, c_{1}\right\rangle + \frac{d}{d x}\left( {\text {Im}} (\phi _{1})-\theta \right) -\left( \rho ^{2}-1\right) \theta ^{\prime }. \end{aligned} \end{aligned}$$

Then using a calculation similar to Remark 2.6, we have

$$\begin{aligned} \begin{aligned} P\left( \phi +c_{1}\right) =&\lim _{R \rightarrow \infty } \int _{-R}^{R}\left\langle i \phi ^{\prime }, \phi +c_{1}-1\right\rangle -\frac{d}{d x}\left( {\text {Im}} (\phi _{1})\right) -\left\langle i \phi ^{\prime }, c_{1}\right\rangle +\theta ^{\prime } d x \\ =&\lim _{R \rightarrow \infty } \int _{-R}^{R}\left\langle i \phi ^{\prime }, \phi \right\rangle +\theta ^{\prime } d x \\ =&P(\phi ). \end{aligned} \end{aligned}$$

\(\hfill{\square}\)

Lemma 2.9

Let \(\phi \in {\mathcal {E}}\) and \(w \in H^{1}({\mathbb {R}})\), we have

$$\begin{aligned} \int _{{\mathbb {R}}}\left\langle i \phi ^{\prime }, w\right\rangle +\left\langle i \phi, w^{\prime }\right\rangle d x=0. \end{aligned}$$
(20)

Proof

Since \(w, \phi ^{\prime } \in L^{2}({\mathbb {R}})\), then \(\left\langle i \phi ^{\prime }, w\right\rangle \in L^{1}({\mathbb {R}})\). Let \(\chi, ~\phi _{1}, ~\phi _{2}\) be given by Lemma 2.4. Set \(w_{1}=\chi (w) w\), \(w_{2}=(1-\chi (w)) w\). We have \(\phi =\phi _{1}+\phi _{2}-1\), \(w=w_{1}+w_{2}\), \(\phi _{1}-1 \in {\dot{H}}^{1} \cap L^{\infty }({\mathbb {R}})\) and \(\phi _{2}-1, ~w_{1}, ~w_{2} \in H^{1}({\mathbb {R}})\).

We see that \(\langle i \phi _{2}^{\prime }, w\rangle, ~\langle i(\phi _{2}-1), w^{\prime }\rangle \in L^{1}({\mathbb {R}})\) by Cauchy-Schwarz inequality. We have

$$\begin{aligned} \int _{{\mathbb {R}}}\left\langle i \phi _{2}^{\prime }, w\right\rangle +\left\langle i\left( \phi _{2}-1\right), w^{\prime }\right\rangle d x=0. \end{aligned}$$
(21)

Since \(\phi _{1}-1 \in {\dot{H}}^{1} \cap L^{\infty }({\mathbb {R}})\) and \(w_{1} \in H^{1} \cap L^{\infty }({\mathbb {R}}),\) we have \(\langle i(\phi _{1}-1), w_{1}\rangle \in {\dot{H}}^{1} \cap L^{\infty }({\mathbb {R}})\) and \(\frac{d}{d x}\langle i(\phi _{1}-1), w_{1}\rangle =\langle i \phi _{1}^{\prime }, w_{1}\rangle +\langle i(\phi _{1}-1), w_{1}^{\prime }\rangle\). Since \(w_{1} \in H^{1}({\mathbb {R}})\), then necessarily \(\lim _{|x| \rightarrow \infty } w_{1}(x)=0\) on \({\mathbb {R}}\), and together with \(\phi _{1} - 1 \in L^{\infty }({\mathbb {R}})\) we have

$$\begin{aligned} \int _{{\mathbb {R}}} \frac{d}{d x}\left\langle i\left( \phi _{1}-1\right), w_{1}\right\rangle d x=\left. \left[ \left\langle i\left( \phi _{1}-1\right), w_{1}\right\rangle \right] \right| _{-\infty } ^{\infty }=0. \end{aligned}$$

Then

$$\begin{aligned} \begin{aligned} \int _{{\mathbb {R}}}\left\langle i \phi _{1}^{\prime }, w_{1}\right\rangle +\left\langle i\left( \phi _{1}-1\right), w_{1}^{\prime }\right\rangle d x = 0. \end{aligned} \end{aligned}$$
(22)

Let \(B=\{x \in {\mathbb {R}} ~\big|~||w|-1| \ge \frac{1}{4} \}\). We have \(\frac{1}{16}|B| \le \int _{B}|w|^{2} d x \le \Vert w\Vert _{L^{2}}^{2}\) and B has finite measure. It can be seen that \(w_{2}=0\) and \(w_{2}^{\prime }=0\) a.e. on \({\mathbb {R}} \setminus B\). By Sobolev inequality in 1D ( [9], pp. 212–213), we have \(w_{2} \in L^{\infty }({\mathbb {R}})\). Combined with \(w_{2}^{\prime } \in L^{2}({\mathbb {R}})\), we deduce that \(w_{2} \in L^{1} \cap L^{\infty }({\mathbb {R}})\) and \(w_{2}^{\prime } \in L^{1} \cap L^{2}({\mathbb {R}})\). Using \(\phi _{1} - 1 \in L^{\infty }({\mathbb {R}})\) and \(\phi _{1}^{\prime } \in L^{2}({\mathbb {R}})\), this gives \(\langle i(\phi _{1}-1), w_{2}\rangle \in L^{1} \cap L^{\infty }({\mathbb {R}})\), \(\langle i \phi _{1}^{\prime }, w_{2}\rangle \in L^{1}({\mathbb {R}})\) and \(\langle i(\phi _{1}-1), w_{2}^{\prime }\rangle \in L^{1} \cap L^{2}({\mathbb {R}})\). We have

$$\begin{aligned} \begin{aligned}&\frac{d}{d x}\left\langle i\left( \phi _{1}-1\right), w_{2}\right\rangle =\left\langle i \phi _{1}^{\prime }, w_{2}\right\rangle +\left\langle i\left( \phi _{1}-1\right), w_{2}^{\prime }\right\rangle. \end{aligned} \end{aligned}$$

The above information implies \(\langle i(\phi _{1}-1), w_{2}\rangle\)\(\in W^{1, 1}({\mathbb {R}})\), thus

$$\begin{aligned} \begin{aligned} \int _{{\mathbb {R}}}\left\langle i \phi _{1}^{\prime }, w_{2}\right\rangle +\left\langle i\left( \phi _{1}-1\right), w_{2}^{\prime }\right\rangle d x =\int _{{\mathbb {R}}} \frac{d}{d x}\left\langle i\left( \phi _{1}-1\right), w_{2}\right\rangle d x=0. \end{aligned} \end{aligned}$$
(23)

Now from (21), (22) and (23) we have

$$\begin{aligned} \int _{{\mathbb {R}}}\left\langle i \phi ^{\prime }, w\right\rangle +\left\langle i (\phi - 1), w^{\prime }\right\rangle d x=0. \end{aligned}$$

Since \(\int _{{\mathbb {R}}} \langle -i, w^{\prime } \rangle dx =0\), we have (20). \(\hfill{\square}\)

Corollary 2.10

Let \(\phi _{1}, ~\phi _{2} \in {\mathcal {E}}\) be such that \(\phi _{1} - \phi _{2} \in L^{2}({\mathbb {R}})\). Then

$$\begin{aligned} \begin{aligned} | P(\phi _{1})-P(\phi _{2}) | \le \Vert \phi _{1} - \phi _{2}\Vert _{L^{2}({\mathbb {R}})}\left( \Vert \phi _{1}^{\prime } \Vert _{L^{2}({\mathbb {R}})}+ \Vert \phi _{2}^{\prime } \Vert _{L^{2}({\mathbb {R}})}\right) \end{aligned} \end{aligned}$$
(24)

Proof

The proof uses formula (20) and is the same as ( [27], Corollary 2.6). \(\hfill{\square}\)  

3 Some Preliminary Results

Let \(\Omega \subset {\mathbb {R}}\) be an open set, and it may not be bounded or connected.

Lemma 3.1

Let \(\phi \in {\mathcal {E}}\). For any \(0<\delta _{0}<1\) and \(R>0\), there exists a constant \(M=M(\delta _{0}, R)>0\), such that if \(E_{\Omega }(\phi ) < M\), then

$$\begin{aligned} \begin{aligned} -\delta _{0}<|\phi (x)| - 1 < \delta _{0}, \end{aligned} \end{aligned}$$

for \(x \in \Omega\) satisfies dist\((x, \partial \Omega )> 2R\).        

Proof

Using the 1D Morrey inequality,

$$\begin{aligned} \begin{aligned}|\phi (x)-\phi (y)|&\le \Vert w^{\prime } \Vert _{L^2 (\Omega )}|x-y|^{\frac{1}{2}} \\&\le \left( E_{\Omega }(\phi )\right) ^{\frac{1}{2}}|x-y|^{\frac{1}{2}} \quad \forall x, y \in B\left( x_{0}, R\right). \end{aligned} \end{aligned}$$
(25)

Fix \(\delta _{0}>0\). Suppose dist\((x_{0}, \partial \Omega )> 2R\) and \(||\phi (x_{0})|-1| \ge \delta _{0}\). Let \(r_{\delta _{0}}=\min \{R, \frac{\delta _{0}^{2}}{4 E_{\Omega }(\phi )} \}\). Since \(\big| \small||\phi (x)|-1 \small| - \small||\phi (x_{0})|-1 \small| \big| \le|\phi (x)-\phi (x_{0})|,\) using (25) we get

$$\begin{aligned}||\phi (x)|-1| \ge \frac{\delta _{0}}{2} \quad \forall x \in B\left( x_{0}, r_{\delta _{0}}\right). \end{aligned}$$

We have

$$\begin{aligned} \begin{aligned}&E_{\Omega }(\phi ) \ge \frac{1}{2} \int _{B\left( x_{0}, r_{\delta _{0}}\right) }\left(|\phi|^{2}-1\right) ^{2} d x \\&\ge \frac{1}{2} \int _{B\left( x_{0}, r_{\delta _{0}}\right) }(|\phi|-1)^{2} d x \\&\ge \frac{\delta _{0}^{2}}{4} r_{\delta _{0}}=\frac{\delta _{0}^{2}}{4} \min \left\{ R, \frac{\delta _{0}^{2}}{4 E_{\Omega }(\phi )}\right\}. \end{aligned} \end{aligned}$$
(26)

Solving (26), we have \(E_{\Omega }(\phi ) \ge \frac{\delta _{0}^{2}}{4} \min \left\{ R, 1 \right\}\). Let \(M=M(R, \delta _{0}):= \frac{\delta _{0}^{2}}{4} \min \left\{ R, 1 \right\},\) then the lemma holds. \(\hfill{\square}\)

Lemma 3.2

(i) If \(\phi \in {\mathcal {E}}\) satisfies \(|| \phi| - 1| \le \delta\) with \(\delta \in (0,1)\), then

$$\begin{aligned} E(\phi ) \ge \sqrt{2}(1-\delta )|P(\phi )|. \end{aligned}$$

(ii) Let \(\phi \in {\mathcal {E}}\), \(0 \le v<\sqrt{2}\) and \(\varepsilon \in (0,1-\frac{v}{\sqrt{2}})\). There exists a constant \(M=M(v, \varepsilon )>0\), such that if \(E(\phi )<M\), then

$$\begin{aligned} E(\phi )-v|P(\phi )| \ge \varepsilon E(\phi ). \end{aligned}$$

                      

Proof

(i) Writing \(\phi =\rho e^{i \theta }\), where \(\rho, ~\theta\) are provided by Lemma 2.1. Using (19),

$$\begin{aligned} P(\phi )=-\int _{{\mathbb {R}}}\left( \rho ^{2}-1\right) \theta ^{\prime } d x. \end{aligned}$$

We have the following:

$$\begin{aligned} \begin{aligned} \sqrt{2}(1-\delta )|P(\phi )|&\le \sqrt{2}(1-\delta )\left\| \rho ^{2}-1\right\| _{L^{2}({\mathbb {R}})}\left\| \theta ^{\prime }\right\| _{L^{2}({\mathbb {R}})} \\&\le (1-\delta )^{2}\left\| \theta ^{\prime }\right\| _{L^{2}({\mathbb {R}})}^{2}+\frac{1}{2}\left\| \rho ^{2}-1\right\| _{L^{2}({\mathbb {R}})}^{2} \\&\le E(\phi ). \end{aligned} \end{aligned}$$

(ii) Set \(\varepsilon < 1-\frac{v}{\sqrt{2}}\). Let \(\delta>0\) satisfies \(\varepsilon \le 1 - \frac{v}{\sqrt{2}(1-\delta )}\). Let \(M=M(\delta, 1)\) be given by Lemma 3.1. Let \(\phi \in {\mathcal {E}}\) satisfies \(E(\phi ) < M.\) Using Lemma 3.1, \(-\delta<|\phi|-1 < \delta.\) Using Lemma 3.2 (i), we have

$$\begin{aligned} \begin{aligned} v|P(\phi )| \le \frac{v}{\sqrt{2}(1-\delta )} E(\phi ) \le \left( 1-\varepsilon \right) E(\phi ). \end{aligned} \end{aligned}$$
(27)

Using (27) we obtain

$$\begin{aligned} \begin{aligned} E(\phi )-v|P(\phi )| \ge \varepsilon E(\phi ), \end{aligned} \end{aligned}$$

then (ii) follows. \(\hfill{\square}\)                        

For \(0<q\le \pi\), we define

$$\begin{aligned} \begin{aligned} E_{\text {min}}(q)=\inf _{\phi \in {\mathcal {E}}} \{E(\phi ) ~\big|~ P(\phi )=q\}. \end{aligned} \end{aligned}$$

For any \(\phi \in {\mathcal {E}}\), the function \(\phi _{1}(x)=\phi (-x) \in {\mathcal {E}}\) and \(E(\phi _{1})=E(\phi)\), \(P(\phi _{1})=-P(\phi),\) then \(E_{\min }(-q)=E_{\min }(q)\). That is, \(E_{\min }\) is an even function. This is the reason why we only need to consider \(E_{\min }(q)\) at the interval \(q \in (0,\pi ]\).

Lemma 3.3

Let \(0 < q \le \pi\), we have \(E_{\min }(q)<\sqrt{2} q\).

Proof

From Remark 2.7, we have \(E_{\min }(q) \le E(b_{v(q)}) < v_{s} q\), where v(q) is the unique velocity v such that \(P(b_{v})=q\). \(\hfill{\square}\)

Lemma 3.4

For any \(\varepsilon>0\), there exists \(q_1(\varepsilon )>0\) with

$$\begin{aligned} E_{\min }(q) \ge \left( \sqrt{2} -\varepsilon \right) q \quad \forall q \in (0, q_1(\varepsilon )). \end{aligned}$$

      

Proof

Lemma 3.2 (ii) implies

$$\begin{aligned} E(\phi ) \ge (\sqrt{2} - \varepsilon )|P(\phi )| \end{aligned}$$

for all \(\phi \in {\mathcal {E}}\) verifying \(E(\phi )<M(\varepsilon )\). Set \(q_1(\varepsilon )=\frac{M(\varepsilon )}{\sqrt{2} + c_1}<\pi\), where \(c_1\) is a positive constant. Fix \(q \in (0, q_1(\varepsilon ) )\). There exists \(\phi \in {\mathcal {E}}\) satisfying \(P(\phi )=q\), \(E(\phi )<E_{\min }(q) + c_1 q\). Using Lemma 3.3, we have

$$\begin{aligned} E(\phi )<\left( \sqrt{2} + c_1\right) q < \left( \sqrt{2} + c_1\right) q_1(\varepsilon )=M(\varepsilon ), \end{aligned}$$

thus \(E(\phi ) \ge (\sqrt{2} - \varepsilon )|P(\phi )|=(\sqrt{2} - \varepsilon ) q\). This yields \(E_{\min }(q) \ge (\sqrt{2} - \varepsilon ) q\). \(\hfill{\square}\)

Lemma 3.5

(i) For any \(0 \le q_1 \le q \le \pi\), we have \(E_{\min }(q) \le E_{\min }(q_1)+E_{\min }(q-q_1)\).

(ii) \(E_{\min }\) is nondecreasing. It is continuous with best Lipchitz constant \(\sqrt{2}\). It is concave.

(iii) The conclusion of (i) can be upgraded to strictly subadditive, i.e., for any \(0< q_1< q < \pi\), \(E_{\min }(q)<E_{\min }(q_1)+E_{\min }(q-q_1)\).

Proof

(i) Corollary A.2 in Appendix A provides \(\phi _1, ~\phi _2 \in {\mathcal {E}}\) with \(\begin{aligned}P(\phi _1)&=q_1, ~P(\phi _2)=q - q_1, ~E(\phi _1)<E_{\text{ min }}(q_1)+\frac{\varepsilon }{2}, ~E(\phi _2)<E_{\text{ min }}(q - q_1)\\&+\frac{\varepsilon }{2},\end{aligned}\) where \(\varepsilon>0\)\(\phi _1=1\) on \([R_1, \infty ), ~\phi _2=1\) on \((-\infty, R_2].\) Define \(\phi (x)=\left\{ \begin{array}{l}\phi _1(x), \quad \text{ if } x \le R_1 \\ \phi _2(x-2(R_1+R_2)) \text{ otherwise. } \end{array}\right.\) Then \(\phi \in {\mathcal {E}}, ~P(\phi )=P(\phi _1)+P(\phi _2)=q\) and \(E(\phi )=E(\phi _1)+E(\phi _2)\). Thus \(E_{\min }(q) \le E(\phi )<E_{\text{ min }}(q_1)+E_{\text{ min }}(q-q_1)+\varepsilon\). This gives (i).

(ii) Let \(0<q_1<q_2 < \pi\) and \(\sigma =\frac{q_1}{q_2}<1\). Assume that \(\phi \in {\mathcal {E}}\) satisfies \(\inf _{x \in {\mathbb {R}}}|\phi (x)|> 0\) and \(P(\phi )=q_2\) (such \(\phi\) exists according to Remark 2.7). We write \(\phi =\rho e^{i \theta }\), by Theorem 1 in ( [10], p. 37). Then for \(\phi _{\sigma }=\rho e^{i \sigma \theta }\) we have \(P(\phi _{\sigma })= P(\rho e^{i \sigma \theta })= \sigma P(\phi )=q_1\). Using (9) we have \(E_{\min }(q_1) \le E(\phi _{\sigma } ) \le E(\phi ).\) Taking the infimum over all \(\phi\) satisfying \(P(\phi )=q_2\), we see that \(E_{\min }(q_1) \le E_{\min }(q_2).\) We thus have that \(E_{\text{ min }}\) is nondecreasing.      

The conclusion of (i) and Lemma 3.3 implies

$$\begin{aligned} \begin{aligned} E_{\min }(q_2) - E_{\min }(q_1) \le \sqrt{2} (q_2-q_1). \end{aligned} \end{aligned}$$

Combining with Lemma 3.4, we see that \(E_{\min }\) is Lipchitz continuous with best Lipchitz constant \(\sqrt{2}\).

For \(f: {\mathbb {R}} \rightarrow {\mathbb {C}}\) and \(c \in {\mathbb {R}}\), denote

$$\begin{aligned} \begin{aligned}&Q_c^{+} f(x)= {\left\{ \begin{array}{ll}f(x) & \text{ if } x \ge c \\ e^{i \theta } \overline{f(2 c-x)} & \text{ if } x<c,\end{array}\right. } \\&Q_c^{-} f(x)= {\left\{ \begin{array}{ll}e^{i \theta } \overline{f(2 c-x)} & \text{ if } x \ge c \\ f(x) & \text{ if } x<c,\end{array}\right. } \end{aligned} \end{aligned}$$

where \(\theta \in {\mathbb {R}}\) is a constant satisfying \(f(c)=e^{i \theta } \overline{f(c)}\), which ensures that \(Q_c^{+} f(x), Q_c^{-} f(x)\) is continuous at \(x=c\). For any \(\phi \in {\mathcal {E}}\) we have \(Q_c^{+} \phi, Q_c^{-} \phi \in {\mathcal {E}}, ~E(Q_c^{+} \phi )+E(Q_c^{-} \phi )=2 E(\phi )\) and \(P(Q_c^{+} \phi )+P(Q_c^{-} \phi )=2 P(\phi ).\) The map \(c \mapsto P(Q_c^{+} \phi )\) is continuous on \({\mathbb {R}}\), goes to 0 as \(c \rightarrow \infty\) and to \(2 P(\phi )\) as \(c \rightarrow -\infty\). Then proceeding similarly as in ( [13], p. 176), we can show the concavity of \(E_{\min }\).

(iii) Let \(0< q_1 < q \le \pi\). The result of (ii) implies that \(E_{\min }(q_1) \ge \frac{q_1}{q} E_{\min }(q),\) with equality holds if and only if \(E_{\text{ min }}(q_1)=a_1 q_1\) for a constant \(a_1 \in {\mathbb {R}}\). Using Lemma 3.3, we see that \(a_1<\sqrt{2}\). However, using Lemma 3.4 we see that \(a_1 \ge \sqrt{2} - \varepsilon\). Hence \(a_1\) doesn’t exist. This means that we have the strict inequality \(E_{\min}(q_1) > \frac{q_1}{q} E_{\min}(q).\) \(\hfill{\square}\)  

4 Minimizing E at Fixed P

We will implement \(b_v\) as solution of the constrained minimization problem using concentration-compactness principle. We will show the precompactness of minimizing sequences.

Theorem 4.1

Set \(0 < q \le \pi\). Let \(\left( \phi _{n}\right) _{n \ge 1} \subset {\mathcal {E}}\) be a minimizing sequence, that is, suppose that

$$\begin{aligned} P\left( \phi _{n}\right) \rightarrow q \quad \text{ and } \quad E\left( \phi _{n}\right) \rightarrow E_{\min }(q). \end{aligned}$$

Then, up to a subsequence and translations, we have the following:

(i) there exist \(\phi \in {\mathcal {E}}\) such that \(\phi _{n} \rightarrow \phi\) a.e. on \({\mathbb {R}}\) and \(d_{0}(\phi _{n}, \phi ) \rightarrow 0\), i.e.,

$$\begin{aligned} \begin{aligned}&\left\| \phi _{n}^{\prime } - \phi ^{\prime } \right\| _{L^{2}({\mathbb {R}})} \rightarrow 0, \\&\left\| \left| \phi _{n} \right| -|\phi|\right\| _{L^{2}({\mathbb {R}})} \rightarrow 0 \quad \text{ as } n \rightarrow \infty. \end{aligned} \end{aligned}$$

(ii) \(P(\phi )=q, ~E(\phi )=E_{\min }(q)\).

Proof

Let \(\beta _{0} = E_{\min }(q)\). We have that \(E\left( \phi _{n}\right) \rightarrow \beta _{0}>0\) as \(n \rightarrow \infty\).  

The concentration-compactness principle [26] will be used. Let \(\xi _{n}(t)\) be the concentration function of \(E(\phi _{n})\):

$$\begin{aligned} \begin{aligned} \xi _{n}(t)=\sup _{y \in {\mathbb {R}}} E_{B(y, t)} ( \phi _{n} ). \end{aligned} \end{aligned}$$

Following [26], up to a subsequence, there exists \(\xi:[0, \infty ) \rightarrow {\mathbb {R}}\) and \(\beta \in \left[ 0, \beta _{0}\right]\) satisfying

$$\begin{aligned} \begin{aligned} \xi _{n}(t) \rightarrow \xi (t) \text { when } n \rightarrow \infty \quad \text { and } \quad \xi (t) \rightarrow \beta \text { when } t \rightarrow \infty. \end{aligned} \end{aligned}$$

Using similar arguments as Theorem 5.3 in [27], there exists a nondecreasing sequence \(r_{n} \rightarrow \infty\) satisfying

$$\begin{aligned} \begin{aligned} \lim _{n \rightarrow \infty } \xi _{n}\left( r_{n}\right) =\lim _{n \rightarrow \infty } \xi _{n}\left( \frac{r_{n}}{2}\right) = \beta. \end{aligned} \end{aligned}$$
(28)

Step 1 (Ruling out vanishing) We will prove that vanishing will not hold, i.e., there exists a constant \(c_{1}>0\) such that \(\sup _{y \in {\mathbb {R}}} E_{B(y, 1)}\left( \phi _{n}\right) \ge c_{1}\) as \(n \rightarrow \infty\). Suppose in contradiction that up to a subsequence

$$\begin{aligned} \begin{aligned} \displaystyle \lim _{n \rightarrow \infty } \sup _{y \in {\mathbb {R}}} E_{B(y, 1)} (\phi _{n} )=0, \end{aligned} \end{aligned}$$
(29)

then we show that \(\left\| \left| \phi _{n}\right| - 1 \right\| _{L^{\infty }({\mathbb {R}})} \rightarrow 0\) as \(n \rightarrow \infty\).

Since \(\left( E(\phi _{n})\right) _{n \ge 1}\) is bounded, then \(\Vert \phi _{n}^{\prime } \Vert _{L^{2}({\mathbb {R}})}\) is bounded for any n. Using Morrey inequality, there exists \(C_{1}>0\) such that

$$\begin{aligned} \begin{aligned} \left| \phi _{n}(x)-\phi _{n}(y)\right| \le C_{1}|x-y|^{\frac{1}{2}} \quad \forall \text { x, y } \in {\mathbb {R}}. \end{aligned} \end{aligned}$$
(30)

Since \(\phi _{n} \in {\mathcal {E}}\), using Lemma 2.2, \(\phi _{n} \in L^{\infty }({\mathbb {R}})\). Let \(\delta _{n}=\Vert|\phi _{n}|-1\Vert _{L^{\infty }({\mathbb {R}})}\). Choose \(x_{n} \in {\mathbb {R}}\) such that \(||\phi _{n}(x_{n})|-1| \ge \frac{\delta _{n}}{2}\). From (30) we have \(||\phi _{n}(x)|-1| \ge \frac{\delta _{n}}{4}\) for any \(x \in B(x_{n}, r_{n})\), with \(r_{n}= (\frac{\delta _{n}}{4 C_{1}})^{2}\). We have

$$\begin{aligned} \begin{aligned} \int _{B\left( x_{n}, r_{n}\right) }\left( \left| \phi _{n}\right| ^{2}-1\right) ^{2} d x \ge \frac{\delta _{n}^{2}}{8} r_{n}. \end{aligned} \end{aligned}$$
(31)

Combining (29) with (31), \(\lim _{n \rightarrow \infty } \delta _{n}^{2} r_{n}=0\). Clearly this implies \(\lim _{n \rightarrow \infty } \delta _{n}=0\). Then Lemma 3.2 (i) implies

$$\begin{aligned} \begin{aligned} E\left( \phi _{n}\right) \ge \sqrt{2}\left( 1-\delta _{n}\right) \left| P\left( \phi _{n}\right) \right|.\end{aligned} \end{aligned}$$

Letting \(n \rightarrow \infty\), we have

$$\begin{aligned} \begin{aligned} \displaystyle \liminf _{n \rightarrow \infty } E\left( \phi _{n}\right) \ge \sqrt{2} q. \end{aligned} \end{aligned}$$
(32)

However, using Lemma 3.3,

$$\begin{aligned} \begin{aligned} \displaystyle \limsup _{n \rightarrow \infty } E\left( \phi _{n}\right) < \sqrt{2} q. \end{aligned} \end{aligned}$$
(33)

We see that (32) contradicts with (33).              

Step 2 (Ruling out dichotomy) We will prove that \(\beta \notin \left( 0, \beta _{0}\right)\). Suppose that \(0<\beta <\beta _{0}\). Let \(r_{n}\) be as in (28) and set \(R_{n}=\frac{r_{n}}{2}\). After translation, we have \(E_{B(0, R_{n})}(\phi _{n}) \ge \xi _{n}(R_{n})-\frac{1}{n}\). Using (28) we obtain

$$\begin{aligned} \begin{aligned} \varepsilon _{n}:= E_{B(0, r_{n}) \setminus B(0, R_{n})}\left( \phi _{n}\right) \le \xi _{n}\left( r_{n}\right) -\left( \xi _{n}\left( R_{n}\right) -\frac{1}{n}\right) \rightarrow 0. \end{aligned} \end{aligned}$$

Applying Lemma A.1 (in Appendix A), set \(R=R_{n}, ~A=2, ~\varepsilon =\varepsilon _{n}\) in that Lemma, then there exist two functions \(\phi _{n, 1}, ~\phi _{n, 2}\) such that \(E\left( \phi _{n, 1}\right) \ge E_{B(0, R_{n})}(\phi _{n})\)\(\ge \xi _{n}(R_{n})-\frac{1}{n}, ~E(\phi _{n, 2}) \ge E_{{\mathbb {R}} \setminus B ( 0,2 R_{n}) } (\phi _{n}) \ge\)\(E(\phi _{n})-\xi (2 R_{n})\) and

$$\begin{aligned} \begin{aligned} \left| E\left( \phi _{n}\right) -E\left( \phi _{n, 1}\right) -E\left( \phi _{n, 2}\right) \right| \rightarrow 0 \quad \text{ as } n \rightarrow \infty. \end{aligned} \end{aligned}$$

From (28), we deduce that necessarily

$$\begin{aligned} \begin{aligned} E\left( \phi _{n, 1}\right) \rightarrow \beta \quad \text{ and } \quad E\left( \phi _{n, 2}\right) \rightarrow \beta _{0} - \beta \quad \text{ as } n \rightarrow \infty. \end{aligned} \end{aligned}$$

From Lemma A.1 (v) (in Appendix A) we have

$$\begin{aligned} \begin{aligned} \left| P\left( \phi _{n}\right) -P\left( \phi _{n, 1}\right) -P\left( \phi _{n, 2}\right) \right| \rightarrow 0 \quad \text{ as } n \rightarrow \infty. \end{aligned} \end{aligned}$$
(34)

Proceeding as ( [13], p. 181), we infer that up to a subsequence, there exists \(q_{1}, ~q_{2} \in (0, q)\), such that \(P(\phi _{n, 1}) \rightarrow q_{1}\) and \(P(\phi _{n, 2}) \rightarrow q_{2}\) and \(q_{1}+q_{2}=q\).

Since \(E(\phi _{n, 1}) \ge E_{\min }(P(\phi _{n, 1}))\) and \(E(\phi _{n, 2}) \ge E_{\min }(P(\phi _{n, 2}))\), taking limit we obtain \(\beta \ge E_{\text{ min }}(q_{1}), ~\beta _{0} - \beta \ge E_{\min }(q_{2})\). We then have

$$\begin{aligned} E_{\min }(q)=\beta +\left( \beta _{0}-\beta \right) \ge E_{\min }\left( q_{1}\right) +E_{\min }\left( q_{2}\right), \end{aligned}$$

which is in contradiction with strictly subadditivity of \(E_{\min }\) (Lemma 3.5 (iii)). Thus we have \(\beta \notin \left( 0, \beta _{0}\right)\).

Step 3 (Concentration-compactness) After finishing step 1 and step 2, we thus have concentration, i.e., \(\beta =\beta _{0}\). Then after translation, for any \(\varepsilon>0\), there exists positive \(A_{\varepsilon }\) and \(n_{\varepsilon } \in {\mathbb {N}}\) satisfying

$$\begin{aligned} \begin{aligned} E_{ {\mathbb {R}} \setminus B\left( 0, A_{\varepsilon }\right) }({\phi }_{n}) < \varepsilon \quad \forall n \ge n_{\varepsilon }. \end{aligned} \end{aligned}$$
(35)

Let \(\chi\) be provided by Lemma 2.4 and set \(\phi _{n, 1}= \chi (\phi _{n})(\phi _{n}-1)+1\), \(\phi _{n, 2}=(1-\chi (\phi _{n}))(\phi _{n}-1)+1\). From Lemma 2.4 we see that \((\phi _{n, 1})_{n \ge 1} \subset {\mathcal {E}},\)\((\phi _{n, 2}-1)_{n \ge 1} \subset H^{1}({\mathbb {R}})\) and \((E(\phi _{n, 1}))_{n \ge 1}\), \((E(\phi _{n, 2}))_{n \ge 1}\) are bounded. Using Lemma 2.1, write \(\phi _{n, 1}=\rho _{n} e^{i \theta _{n}}\) with \(\frac{1}{2} \le \rho _{n} \le \frac{3}{2}\) and \(\theta _{n} \in {\dot{H}}^{1}({\mathbb {R}}),\)\(\left( \rho _{n}-1\right) _{n \ge 1} \subset H^{1}({\mathbb {R}})\). \(( \phi _{n} )^{\prime }_{n \ge 1} \subset L^{2}({\mathbb {R}})\) and \((\phi _{n})_{n \ge 1} \subset L^{2}(B(0, A))\) for any \(A>0\) (using Lemma 2.2). We see that up to a subsequence \((n_{k})_{k \ge 1},\) there exist \(\phi \in H_{\text {loc}}^{1}({\mathbb {R}})\) with \(\phi ^{\prime } \in L^{2}({\mathbb {R}}), ~\phi _{1} \in H^{1}_{\text {loc}}({\mathbb {R}})\) with \(\phi _{1}^{\prime } \in L^{2}({\mathbb {R}})\), \(\phi _{2}-1 \in H^{1}({\mathbb {R}}), ~\theta \in {\dot{H}}^{1}({\mathbb {R}}), ~\rho -1 \in H^{1}({\mathbb {R}})\) such that

$$\begin{aligned}&(\phi _{n_{k}})^{\prime } \rightharpoonup \phi ^{\prime }, ~ ( \phi _{n_{k}, 1})^{\prime } \rightharpoonup \phi _{1}^{\prime }, ~ \text { and } ~ ( \theta _{n_{k}} )^{\prime } \rightharpoonup \theta ^{\prime } \text { weakly in } L^{2}({\mathbb {R}}), \\&\phi _{n_{k}, 2}-1 \rightharpoonup \phi _{2}-1 ~ \text { and } ~ \rho _{n_{k}}-1 \rightharpoonup \rho -1 \text { weakly in } H^{1}({\mathbb {R}}), \\&\phi _{n_{k}} \rightharpoonup \phi \text { weakly in } H^{1}(B(0, A)) \quad \forall A>0, \\&\phi _{n_{k}, 1} \rightarrow \phi _{1}, ~ \phi _{n_{k},2} \rightarrow \phi _{2}, ~ \theta _{n_{k}} \rightarrow \theta, ~ \rho _{n_{k}}-1 \rightarrow \rho -1, ~ \phi _{n_{k}} \rightarrow \phi \\&\text { strongly in } L^{p}(B(0, A)) \text { and a.e. on } {\mathbb {R}}, \quad \forall A>0, ~p \in [1, \infty ]. \end{aligned}$$
(36)

Weak convergence implies

$$\begin{aligned} \begin{aligned} \int _{{\mathbb {R}}}| \phi ^{\prime }|^{2} d x \le \liminf _{k \rightarrow \infty } \int _{{\mathbb {R}}}| ( \phi _{n_{k}} )^{\prime }|^{2} d x. \end{aligned} \end{aligned}$$
(37)

Fatou’s Lemma implies

$$\begin{aligned} \begin{aligned} V(|\phi|^{2}) \le \liminf _{k \rightarrow \infty } V(|\phi _{n_{k}}|^{2}). \end{aligned} \end{aligned}$$
(38)

From (37) and (38),

$$\begin{aligned} \begin{aligned} E(\phi ) \le \liminf _{k \rightarrow \infty } E({\phi }_{n_{k}})=E_{\min }(q). \end{aligned} \end{aligned}$$
(39)

Step 4: Lemmas 4.2 and 4.3 will be used.  

Lemma 4.2

Suppose the following hold for \(\left( \omega _{n}\right) _{n \ge 1} \subset {\mathcal {E}}\):

(i) \(\left( E\left( \omega _{n}\right) \right) _{n \ge 1}\) is bounded, and (35) holds for \(\omega _{n}\);

(ii) There exists \(\omega \in {\mathcal {E}}\) such that \(\Vert \omega _{n} - \omega \Vert _{L^{2}(B(0, A))} \rightarrow 0\) for \(A>0\) and \(\omega _{n} \rightarrow \omega\) a.e. on \({\mathbb {R}}\).

Then we have \(|\omega _{n}| \rightarrow|\omega|\) in \(L^{2}({\mathbb {R}})\), \((1 -|\omega _{n}|^{2})^2 \rightarrow (1 -| \omega|^{2})^2\) in \(L^{1}({\mathbb {R}})\) as \(n \rightarrow \infty\).

Proof

Fix \(\varepsilon>0\), assumption (i) implies that

$$\begin{aligned} \begin{aligned} \Vert|\omega _{n}|^{2}-1\Vert _{L^{2}({\mathbb {R}} \setminus B(0, A_{\varepsilon }))}^{2} \le 2 \varepsilon \quad \text{ for } n \ge n_{\varepsilon }. \end{aligned} \end{aligned}$$
(40)

\(\omega\) has a similar estimate. From 40 we have

$$\begin{aligned} \begin{aligned}&\left\| \left| \omega _{n}\right| -| \omega|\right\| _{L^{2}\left( {\mathbb {R}} \setminus B(0, A_{\varepsilon }) \right) } \\&\le \Vert| \omega _{n}|^{2}-1\Vert _{L^{2}({\mathbb {R}} \setminus B(0, A_{\varepsilon }))} +\left\|| \omega|^{2}-1\right\| _{L^{2}\left( {\mathbb {R}} \setminus B\left( 0, A_{\varepsilon }\right) \right) } \le 2 \sqrt{2} \sqrt{\varepsilon }. \end{aligned} \end{aligned}$$
(41)

Using assumption (ii) and the fact that \(|\omega _n| \in L^p(B(0, A))\) for \(1 \le p \le \infty\) (using Lemma 2.2), we obtain \(\omega _{n} \rightarrow \omega\) in \(L^{p}(B(0, A))\) for \(1 \le p \le \infty\). Therefore for large n, we have \(\Vert| \omega _{n}|-| \omega|\Vert _{L^{2}(B(0, A_{\varepsilon }))} \le \varepsilon\), \(\Vert V(|\omega _{n}|^{2})-V(| \omega|^{2})\Vert _{L^{1}(B(0, A_{\varepsilon }))} \le \varepsilon,\) Combining with (40) and (41), we have \(\Vert| \omega _{n}|-| \omega|\Vert _{L^{2}({\mathbb {R}})} \le 2 \sqrt{2} \sqrt{\varepsilon }+\varepsilon\), \(\Vert V(| \omega _n|^{2})- V(| \omega|^{2})\Vert _{L^{1}({\mathbb {R}})} \le 3 \varepsilon\) for large n. Lemma 4.2 follows when letting \(\varepsilon\) goes to 0. \(\hfill{\square}\)            

The following lemma is a 1D counterpart of Lemma 4.12 in [13], where the space dimension is assumed to be \(N \ge 2\). The conformal transform method is used in the prove of Lemma 4.12 in [13], however, this method is not valid for the 1D case. We use a method which is inspired by ( [27], pp. 163–164).

Lemma 4.3

Suppose the following hold for \(\left( \omega _{n}\right) _{n \ge 1} \subset {\mathcal {E}}\):

(i) \(\left( E\left( \omega _{n}\right) \right) _{n \ge 1}\) is bounded, and (35) holds for \(\omega _{n}\);

(ii) There is \(\omega \in {\mathcal {E}}\) with \(\omega _n^{\prime } \rightharpoonup \omega ^{\prime }\) weakly in \(L^2({\mathbb {R}})\), and \(\Vert \omega _n - \omega \Vert _{L^2( B(0, A))} \rightarrow 0\) for any \(A> 0\)

Then \(P( \omega _n ) \rightarrow P( \omega )\) as \(n \rightarrow \infty\).

Proof

Consider a subsequence of \((\omega _{n})_{n \ge 1}\). For simplicity, we still denote it by \((\omega _{n})_{n \ge 1}\). Let \(\varepsilon, ~A_{\varepsilon }, ~n_{\varepsilon }\) be as in (35). From 12 we get

$$\begin{aligned} \begin{aligned} \left\| \left( 1-\chi ^{2}\left( \omega _{n}\right) \right) \left( \omega _{n}-1\right) \right\| _{L^{2}({\mathbb {R}})} \le C\left\| \left| \omega _{n}\right| ^{2}-1\right\| _{L^{2}({\mathbb {R}})} \le C (E\left( \omega _{n}\right) )^{\frac{1}{2}}. \end{aligned} \end{aligned}$$

The Cauchy-Schwartz inequality implies

$$\begin{aligned} \begin{aligned}&\int _{{\mathbb {R}} \setminus B\left( 0, A_{\varepsilon }\right) }\left| \left( 1-\chi ^{2}\left( \omega _{n}\right) \right) \left\langle i \omega _{n}^{\prime }, \omega _{n}-1\right\rangle \right| d x \\ \le&\left\| \left( 1-\chi ^{2}\left( \omega _{n}\right) \right) \left( \omega _{n}-1\right) \right\| _{L^{2}({\mathbb {R}})}\left\| \omega _{n}^{\prime }\right\| _{L^{2}\left( {\mathbb {R}} \setminus B\left( 0, A_{\varepsilon }\right) \right) } \\ \le&C \sqrt{M} \sqrt{\varepsilon } \end{aligned} \end{aligned}$$
(42)

for any \(n \ge n_{\varepsilon }\), and \(M>0\) is such that \(E\left( \omega _{n}\right) \le M\) for any n.  

Let \(\chi\) be provided by Lemma 2.4 and set \(\omega _{n, 1}=\chi (\omega _{n})(\omega _{n}-1)+1\), \(\omega _{n, 2}=(1-\chi (\omega _{n}))(\omega _{n}-1)+1\). From Lemma 2.4, we see that \((\omega _{n, 1})_{n \ge 1} \subset {\mathcal {E}},\) \(~(\omega _{n, 2}-1)_{n \ge 1} \subset H^{1}({\mathbb {R}})\). Using Lemma 2.1, we write \(\omega _{n, 1}=\rho _{n} e^{i \theta _{n}}\) with \(\frac{1}{2} \le \rho _{n} \le \frac{3}{2}\), \(\theta _{n} \in {\dot{H}}^{1}({\mathbb {R}})\). Using assumption (i) and (ii), we deduce that up to a subsequence, there exist \(\{\rho _{n}\}\), \(\{\theta _{n}\}\), \(\{\omega _{n}\}\), \(\{\omega _{n, 1}\}\), \(\{\omega _{n, 2}\}\), \(\rho\), \(\theta\), \(\omega\) that satisfy (36).

From (13) we have

$$\begin{aligned} \Vert \rho _{n}^{2}-1\Vert _{L^{2}({\mathbb {R}})} \le C (E(\omega _{n}))^{\frac{1}{2}} \le C M^{\frac{1}{2}}. \end{aligned}$$

Using (9) and (11) we get

$$\begin{aligned} \left| \theta _{n}^{\prime }\right| \le 2\left| \frac{d\left( \chi \left( \omega _{n}\right) \left( \omega _{n}-1\right) \right) }{d x}\right| \le C\left| \omega _{n}^{\prime }\right|. \end{aligned}$$

Then assumption (i) implies that

$$\begin{aligned} \Vert \theta _{n}^{\prime } \Vert _{L^{2}({\mathbb {R}} \setminus B\left( 0, A_{\varepsilon }\right) )} \le C \sqrt{\varepsilon } \quad \forall n \ge n_{\varepsilon }. \end{aligned}$$

We have

$$\begin{aligned} \begin{aligned} \int _{{\mathbb {R}} \setminus B\left( 0, A_{\varepsilon }\right) }\left| \left( \rho _{n}^{2}-1\right) \theta _{n}^{\prime }\right| d x \le \left\| \rho _{n}^{2}- 1 \right\| _{L^{2}({\mathbb {R}})}\left\| \theta _{n}^{\prime }\right\| _{L^{2}\left( {\mathbb {R}} \setminus B\left( 0, A_{\varepsilon }\right) \right) } \le C \sqrt{M} \sqrt{\varepsilon } \end{aligned} \end{aligned}$$
(43)

\(\forall n \ge n_{\varepsilon }\). We see that (42) and (43) also hold with \(\omega,~ \rho\), and \(\theta\) replacing \(\omega _{n},~ \rho _{n}\) and \(\theta _{n}\).

Since \(\omega _{n} \rightarrow \omega\) and \(\rho _{n}-1 \rightarrow \rho -1\) in \(L^{2} ( B(0, A_{\varepsilon }) )\) and a.e., then

$$\begin{aligned} \left( 1-\chi ^{2}\left( \omega _{n}\right) \right) \left( \omega _{n}-1\right) \rightarrow \left( 1-\chi ^{2}(\omega )\right) (\omega - 1) \quad \text { and } \quad \rho _{n}^{2}-1 \rightarrow \rho ^{2}-1 \end{aligned}$$

in \(L^{2}(B(0, A_{\varepsilon }))\). Combining with the fact that \(\omega _{n}^{\prime } \rightharpoonup \omega ^{\prime }\) and \(\theta _{n}^{\prime } \rightharpoonup \theta ^{\prime }\) weakly, we have

$$\begin{aligned} \begin{aligned} \int _{B\left( 0, A_{\varepsilon }\right) }\left\langle i \omega _{n}^{\prime },\left( 1-\chi ^{2}\left( \omega _{n}\right) \right) \left( \omega _{n}-1\right) \right\rangle d x \rightarrow \int _{B\left( 0, A_{\varepsilon }\right) }\left\langle i \omega ^{\prime },\left( 1-\chi ^{2}(\omega )\right) (\omega -1\rangle d x\right. \end{aligned} \end{aligned}$$
(44)

and

$$\begin{aligned} \begin{aligned} \int _{B\left( 0, A_{\varepsilon }\right) }\left( \rho _{n}^{2}-1\right) \theta _{n}^{\prime } d x \rightarrow \int _{B\left( 0, A_{\varepsilon }\right) }\left( \rho ^{2}-1\right) \theta ^{\prime } d x. \end{aligned} \end{aligned}$$
(45)

Using (42)-(45) and (18), we deduce that there exist \(n_{1}(\varepsilon ) \ge n_{\varepsilon }\) such that for any \(n \ge n_{1}(\varepsilon )\),

$$\begin{aligned} \left| P\left( \omega _{n}\right) -P(\omega )\right| \le C \sqrt{\varepsilon }. \end{aligned}$$

Since every subsequence of \((\omega _{n})_{n \ge 1}\) includes a further subsequence satisfying \(P\left( \omega _{n}\right) \rightarrow P(\omega )\) as \(n \rightarrow \infty\), thus Lemma 4.3 follows. \(\hfill{\square}\)

We will finish the proof of Theorem 4.1. From (35), (36) and Lemma 4.3 we see that \(q=\lim _{k \rightarrow \infty } P(\phi _{n_{k}})=P(\phi )\). Necessarily we have \(\lim _{k \rightarrow \infty } E(\phi _{n_{k}} ) = E_{\min }(q) \le E(\phi )\). Together with (39), we see that \(E(\phi )=E_{\min }(q)\). From (35), (36) and Lemma 4.2, we see that \(|\phi _{n_{k}}| \rightarrow|\phi|\) in \(L^{2}({\mathbb {R}}),\) \(V(| \phi _{n_{k}}|^{2}) \rightarrow V(|\phi|^{2})\) in \(L^{1}({\mathbb {R}})\). Combining (37), (38) and \(E(\phi _{n_{k}}) \rightarrow E(\phi )\) leads to \(\int _{{\mathbb {R}}}| ( \phi _{n_{k}} )^{\prime }|^{2} d x \rightarrow \int _{{\mathbb {R}}}| \phi ^{\prime }|^{2} d x\). Combining with the weak convergence \(( \phi _{n_{k}})^{\prime } \rightharpoonup \phi ^{\prime }\) in \(L^{2}({\mathbb {R}})\), we have the strong convergence \(\Vert ( \phi _{n_{k}} )^{\prime } - \phi ^{\prime } \Vert _{L^{2}({\mathbb {R}})} \rightarrow 0\) as \(k \rightarrow \infty\). \(\hfill{\square}\)

Corollary 4.4

The momentum P and energy E defined on \({\mathcal {E}}\) are continuous functionals, under the semi-distance \(d_0\).

Proof

For momentum P, the proof uses Lemma 4.3 and Corollary 2.8. For energy E, the proof uses Lemma 4.2. The details are similar to ( [13], Corollary 4.13) and we omit it. \(\hfill{\square}\)

Remark 4.5

(1) It is proved in Lemma 2.7 of [7] that the momentum P is locally Lipschitz continuous on \({\mathcal {E}}\) for the distance \(d_A\) defined as Eq. (7). It is proved in ([6], pp. 75–76) that P is continuous on \({\mathcal {E}}\) for the distance \(d_A\). Hence, Corollary 4.4 is an improvement of these results.

(2) For \(0< q < \pi\), assume \(\phi \in {\mathcal {E}}\) satisfies \(P(\phi )=q\), \(E(\phi )=E_{\min }(q)\). For \((\phi _{n})_{n \ge 1} \subset {\mathcal {E}}\) such that \(d_{0}(\phi _{n}, \phi ) \rightarrow 0\), by Corollary 4.4, we have \(P(\phi _{n}) \rightarrow q\) and \(E(\phi _{n}) \rightarrow E_{\min }(q)\), modulo translation. Therefore, Theorem 4.1 offers an optimal convergence result. The corresponding optimality in dimension \(N \ge 2\) is pointed out by ( [13], p. 187).

Now we will show that the minimizers are traveling waves \(b_v\).

Proposition 4.6

Let \(0 < q \le \pi\). Suppose \(\phi \in {\mathcal {E}}\) minimizes E subject to \(P(\phi )=q\). Then

(i) There exists v such that

$$\begin{aligned} \begin{aligned} i v \phi ^{\prime }+ \phi ^{\prime \prime } +\phi \left( 1 -| \phi|^{2}\right) =0 \quad \text{ in } D^{\prime }({\mathbb {R}}). \end{aligned} \end{aligned}$$
(46)

(ii) There exist constants \(\theta _{0}\in {\mathbb {R}}\), \(x_0 \in {\mathbb {R}}\) and \(\phi = e^{i \theta _{0}} b_{v}(\cdot + x_{0}) \in {\mathcal {E}}\) such that \(P(\phi )=q, ~E(\phi )=E_{\min }(q)\) and \(\phi\) satisfies (46) with speeds \(v=E_{\min }^{\prime }(q)\) for \(0< q < \pi\) and \(v=d^{-} E_{\min }(\pi )=0\) for \(q = \pi\) (\(d^{-} E_{\min }(\pi )\) is the left derivative of \(E_{\min }\) at \(\pi\)), where \(b_v\) is given by (3).

More precisely, for \(0 < q \le \pi\),

$$\begin{aligned} U_{q}=\left\{ \phi \in {\mathcal {E}} ~\big|~ P(\phi )=q \text{, } \text{ and } E(\phi )=E_{\min }(q)\right\} \end{aligned}$$

has a unique element \(b_{v(q)}\) (up to translations and rotations), where v(q) denote the unique speed v such that \(P(b_{v})=q\).

Proof

(i) Proceeding exactly as Proposition 4.14 in ( [13], pp. 187–188), for any \(\psi \in C_{c}^{\infty }({\mathbb {R}})\), there exists v such that

$$\begin{aligned} \int _{{\mathbb {R}}}\left\langle i v \phi ^{\prime }+ \phi ^{\prime \prime } +\phi \left( 1-|\phi|^{2}\right), \psi \right\rangle d x=0, \end{aligned}$$

and this implies (46).

(ii) Consider a sequence \(q_{n} \rightarrow q\) (when \(q=\pi\), this sequence should be \(q_{n} \uparrow q\)). Assume \(q_{n}>0\). Let \(\phi _{n} \in {\mathcal {E}}\) be such that \(P(\phi _{n})=q_{n} \rightarrow q\) and \(E(\phi _{n})=E_{\text{ min }}(q_{n}) \rightarrow E_{\min }(q)\) (using continuity of \(E_{\text{min}}\)). Using Theorem 4.1, we see that up to translation and subsequence, there exist \(\phi _1 \in {\mathcal {E}}\) verifying \(P(\phi _1)=q, ~E(\phi _1)=E_{\min }(q)\) and \(\phi _{n} \rightarrow \phi _1\) a.e. on \({\mathbb {R}}\) and

$$\begin{aligned} d_{0}\left( \phi _{n}, \phi _1 \right) \rightarrow 0 \quad \text{ when } n \rightarrow \infty. \end{aligned}$$

Using (i), \(\phi _{n}\) satisfies (46). Taking limit \(n \rightarrow \infty\), we see that \(\phi _1\) satisfies (46). Using the fact that (46) is integrable, we infer that \(\phi =\phi _1\), and there exist constants \(\theta _{0} \in {\mathbb {R}}\) and \(x_{0} \in {\mathbb {R}}\), such that

$$\begin{aligned} \phi =e^{i \theta _{0}} b_{v}(\cdot + x_0) \end{aligned}$$

and the statement in Proposition 4.6 (ii) holds. \(\hfill{\square}\)    

5 Orbital Stability

The Cauchy problem of (1) was solved in [28], see Theorem 2.3 in ( [15], p. 142) for a summary of the case in space dimension \(N=1\).

Theorem 5.1

( [15]). For any \(\phi _{0} \in {\mathcal {E}}\), there exists a unique solution \(\phi (t) \in C([0, \infty ), {\mathcal {E}})\) of (1) with \(\phi (0)=\phi _{0}\). The following properties of solution hold:

(1) For any \(T>0\), if \(d_{{\mathcal {E}}} (\phi _{0}^{n}, \phi _{0}) \rightarrow 0\), then \(d_{{\mathcal {E}}} (\phi _{n}(t), \phi (t)) \rightarrow 0\) uniformly on [0, T] as \(n \rightarrow \infty\), where \(\phi _{n}(t)\) is solution with initial data \(\phi _{0}^{n}\).

(2) For any \(t \in [0, \infty )\), \(E(\phi (t))=E(\phi _{0})\).

(3) \(\phi - \phi _{0} \in C([0, \infty ), H^{1}({\mathbb {R}}))\).

(4) If \(\Delta \phi _{0} \in L^{2}({\mathbb {R}})\), then \(\Delta \phi \in C([0, \infty ), L^{2}({\mathbb {R}}))\).

The following two Lemmas 5.2 and 5.3 are a regularization of functions in \({\mathcal {E}}\). The regularization technique was exploited in [13, 27, 2].

For \(\phi \in {\mathcal {E}}\) and \(s>0\), consider

$$\begin{aligned} G_{s, \Omega }^{\phi }(\gamma )=E_{\Omega }(\gamma )+\frac{1}{s^{2}} \int _{\Omega }|\gamma - \phi|^{2} d x. \end{aligned}$$

We see that \(G_{s, \Omega }^{\phi }(\gamma ) < \infty\) when \(\gamma \in {\mathcal {E}}\) and \(\gamma -\phi \in L^{2}(\Omega ),\) We define \(H_{0}^{1}(\Omega ):=\left\{ w \in H^{1}({\mathbb {R}}) ~|~ w=0 ~in~ {\mathbb {R}} \setminus \Omega \right\}\), and \(H_{\phi }^{1}(\Omega ):=\left\{ \gamma \in {\mathcal {E}} ~|~ \gamma -\phi \in H_{0}^{1}(\Omega )\right\}.\)

Lemma 5.2

(i) There exists a minimizer of \(G_{s, \Omega }^{\phi }\) in \(H_{\phi }^{1}(\Omega )\).

(ii) Denote the minimizer provided by (i) by \(\gamma _{s}\). Then

$$\begin{aligned} \begin{aligned} E_{\Omega }\left( \gamma _{s}\right) \le E_{\Omega }(\phi ); \end{aligned} \end{aligned}$$
(47)
$$\begin{aligned} \begin{aligned} \left\| \gamma _{s}-\phi \right\| _{L^{2}(\Omega )}^{2} \le s^{2} E_{\Omega }(\phi ). \end{aligned} \end{aligned}$$
(48)

(iii) Denote \(F(z)=z(|z|^{2}-1)\) for \(z \in {\mathbb {C}}\). Then

$$\begin{aligned} \begin{aligned} -\gamma _{s}^{\prime \prime }+F\left( \gamma _{s}\right) +\frac{1}{s^{2}}\left( \gamma _{s} - \phi \right) =0 \qquad \text { in } D^{\prime }(\Omega ). \end{aligned} \end{aligned}$$
(49)

For set \(\Omega _{1} \subset \subset \Omega\), \(\gamma _{s} \in W^{2, p}(\Omega _{1})\), \(\forall p \in (1, \infty )\). Hence, \(\gamma _{s} \in C^{1, \alpha }(\Omega _{1})\) for \(\alpha \in (0,1)\).

Proof

(i) We see that \(\phi \in H_{\phi }^{1}(\Omega )\). Let \((\gamma _{n})_{n \ge 1}\) be a minimizing sequence for \(G_{s, \Omega }^{\phi }\) in \(H_{\phi }^{1}(\Omega )\). Suppose \(G_{s, \Omega }^{\phi }(\gamma _{n}) \le G_{s, \Omega }^{\phi }(\phi )=E_{\Omega }(\phi )\). This implies \(\int _{\Omega }| \gamma _{n}^{\prime }|^{2} d x \le E_{\Omega }(\phi ).\) We have

$$\begin{aligned} \begin{aligned} \int _{\Omega }\left| \gamma _{n}-\phi \right| ^{2} d x \le s^{2} E_{\Omega }(\phi ). \end{aligned} \end{aligned}$$

It follows that \(\gamma _{n}-\phi \in H_{0}^{1}(\Omega )\). Then, up to a subsequence, there exists \(w \in H_{0}^{1}(\Omega )\) such that \(\gamma _{n}-\phi \rightharpoonup w\) weakly in \(H_{0}^{1}(\Omega )\), \(\gamma _{n}-\phi \rightarrow w\) a.e. and \(\gamma _{n}-\phi \rightarrow w\) in \(L_{\text{ loc }}^{p}(\Omega )\) with \(p \in [1, \infty ]\). Let \(\gamma =\phi +w\), we have \(\gamma _{n}^{\prime } \rightharpoonup \gamma ^{\prime }\) weakly in \(L^{2}({\mathbb {R}})\), together with an application of Fatou’s Lemma, we have \(G_{s, \Omega }^{\phi }(\gamma ) \le \liminf _{n \rightarrow \infty } G_{s, \Omega }^{\phi }(\gamma _{n})\). Hence, \(\gamma\) is a minimizer.

(ii) We see that \(G_{s, \Omega }^{\phi }\left( \gamma _{s}\right) \le G_{s, \Omega }^{\phi }(\phi )=E_{\Omega }(\phi )\), then (47) and (48) hold.

(iii) Since \(\left. \frac{d}{d h}\right| _{h=0}(G_{s, \Omega }^{\phi }(\gamma _{s}+h \zeta ))=0\), \(\forall \zeta \in C_{c}^{\infty }(\Omega )\), we then have (49).

Since \(\gamma _{s} \in {\mathcal {E}}\), we have \(|\gamma _{s}|^{2}- 1 \in L^{2}({\mathbb {R}})\). We also have \(\gamma _{s} \in L^{\infty }\) by Lemma 2.2. Using \(\Vert F(\gamma _{s})\Vert _{L^{\infty }} \le \Vert \gamma _{s}\Vert _{L^{\infty }}(\Vert \gamma _{s}\Vert _{L^{\infty }}^{2}+1)\), we have \(F(\gamma _{s}) \in L^{\infty }({\mathbb {R}})\). We then have

$$\begin{aligned} \left\| F\left( \gamma _{s}\right) \right\| _{L^{2}({\mathbb {R}})} \le \left\| \gamma _{s}\right\| _{L^{\infty }({\mathbb {R}})} \Vert| \gamma _{s}|^{2} -1 \Vert _{L^{2}({\mathbb {R}})}, \end{aligned}$$

this gives \(F(\gamma _{s}) \in L^{2}({\mathbb {R}})\). Then \(F(\gamma _{s}) \in L^{2} \cap L^{\infty }({\mathbb {R}})\). We have \(\gamma _{s}, ~\phi \in H_{\text {loc}}^{1}({\mathbb {R}}).\) We deduce that \(\gamma _{s}, ~\phi \in L_{\text {loc}}^{p}({\mathbb {R}})\) for \(p \in [1, \infty ]\) by 1D Sobolev embedding. Using (49) we deduce that \(\gamma _{s}^{\prime \prime } \in L_{\text {loc}}^{p}(\Omega )\) for \(p \in [1, \infty ]\). Then using the elliptic estimates ([17], Theorem 9.11), we get (iii). \(\hfill{\square}\)

The following lemma provides a way of using higher regularity functions to approximate the functions in \({\mathcal {E}}\).

Lemma 5.3

Fix \(\phi \in {\mathcal {E}}\) and \(k \in {\mathbb {N}}\). For any \(\varepsilon>0\), there exists \(\gamma \in {\mathcal {E}}\) satisfying \(\gamma ^{\prime } \in H^{k}({\mathbb {R}}), ~E(\gamma ) \le E(\phi )\) and \(\Vert \gamma - \phi \Vert _{H^{1}({\mathbb {R}})}<\varepsilon\).

Proof

The proof uses Lemma 5.2 and is similar to Lemma 3.5 in ( [13], pp. 170–171). \(\hfill{\square}\)

Lemma 5.4

(Conservation of the momentum) Let \(\phi\) solves (1) (provided by Theorem 5.1) with initial condition \(\phi _{0} \in {\mathcal {E}}\). Then

$$\begin{aligned} P(\phi (t))=P\left( \phi _{0}\right) \quad \forall t \in [0, \infty ). \end{aligned}$$

Proof

We first assume that \(\Delta \phi _{0} \in L^{2}({\mathbb {R}})\). By Theorem 5.1 (4) we have \(\phi _{x} \in C\left( [0, \infty ), H^{1}({\mathbb {R}})\right)\). For \(t, ~ t+ t_{1}>0\), Theorem 5.1 (3) says \(\phi (t+ t_{1})-\phi (t) \in\)\(H^{1}({\mathbb {R}})\), we thus have \(\langle i \phi _{x}(t+ t_{1})+i \phi _{x}(t)\), \(\phi (t + t_{1}) - \phi (t)\rangle \in L^{1}({\mathbb {R}})\). Using (20) we get

$$\begin{aligned} \frac{1}{t_{1}}(P(\phi (t+ t_{1}))-P(\phi (t)))= \int _{{\mathbb {R}}}\langle i \phi _{x}(t+ t_{1})+i \phi _{x}(t), \frac{1}{t_{1}}(\phi (t+ t_{1})-\phi (t))\rangle d x. \end{aligned}$$

Taking limit \(t_{1} \rightarrow 0\) and using (1),

$$\begin{aligned} \begin{aligned} \frac{d}{d t} P(\phi (t))=2 \int _{{\mathbb {R}}}\left\langle \frac{\partial \phi (t)}{\partial x}, \phi _{x x}(t)+\phi (t)\left( 1-|\phi|^{2}\right) \right\rangle d x. \end{aligned} \end{aligned}$$
(50)

Since \(\phi _{x}(t) \in H^{1}({\mathbb {R}})\), we have

$$\begin{aligned} \begin{aligned} \int _{{\mathbb {R}}}\left\langle \phi _{x}(t), \phi _{x x}(t)\right\rangle d x=\frac{1}{2} \int _{{\mathbb {R}}} \frac{\partial }{\partial x}\left( \left| \phi _{x}(t)\right| ^{2}\right) d x. \end{aligned} \end{aligned}$$
(51)

Since \(|\phi _{x}|^{2} \in L^{1}({\mathbb {R}})\) and \(\frac{\partial }{\partial x}(|\phi _{x}|^{2})\)\(=2\langle \phi _{x}, \phi _{x x}\rangle \in L^{1}({\mathbb {R}})\), hence \(|\phi _{x}|^{2} \in W^{1, 1}({\mathbb {R}})\). Using (51) we get \(\int _{{\mathbb {R}}}\langle \phi _{x}, \phi _{x x}\rangle d x=0\).

We have \(2\langle \phi _{x}, \phi (1-|\phi|^{2})\rangle = -\frac{1}{2} \frac{\partial }{\partial x}(1-|\phi|^{2})^{2}\). Since \(\phi _{x} \in L^{2}({\mathbb {R}})\), and \(\phi (1-|\phi|^{2}) \in L^{2}({\mathbb {R}})\) by Lemma 2.2, we have \(\frac{\partial }{\partial x}(1-| \phi|^{2})^{2}=-4\langle \phi _{x}, \phi (1-|\phi|^{2})\rangle \in\)\(L^{1}({\mathbb {R}})\), hence \((1-|\phi|^{2})^{2} \in W^{1, 1}({\mathbb {R}})\). Thus, \(\int _{{\mathbb {R}}} \frac{\partial }{\partial x}(1-|\phi|^{2})^{2} d x=0\). Then we obtain \(\frac{d}{d t} P(\phi (t))=0\) using (50), i.e., \(P(\phi (\cdot ))\) is constant on \([0, \infty )\).

Then we deal with arbitrary function \(\phi _{0} \in {\mathcal {E}}\). By Lemma 5.3, there exists \((\phi _{0}^{n})_{n \ge 1} \subset {\mathcal {E}}\) with \(\left( \phi _{0}^{n}\right) _{x} \in H^{2}({\mathbb {R}})\), \(\Vert \phi _{0}^{n}-\phi _{0} \Vert _{H^{1}({\mathbb {R}})} \rightarrow 0\) as \(n \rightarrow \infty\) (thus, \(d_{{\mathcal {E}}} \left( \phi _{0}^{n}, \phi _{0}\right) \rightarrow 0\)). From Theorem 5.1 (1), for any \(T>0\), \(d_{\mathcal {E}} \left( \phi _{n}(t), \phi (t)\right) \rightarrow 0\) uniformly on [0, T] for large n, where \(\phi _{n}\) solves (1) with initial condition \(\phi _{0}^{n}\). Then we have \(d_{0} \left( \phi _{n}(t), \phi (t)\right) \rightarrow 0\) uniformly on [0, T]. We deduce that \(P(\phi _{n}(t)) \rightarrow P(\phi (t))\) by Corollary 4.4. We get \(P(\phi _{n}(t))=P(\phi _{0}^{n})\) using the conclusion of the first part of the proof. Since \(\Vert \phi _{0}^{n}-\phi _{0} \Vert _{H^{1}({\mathbb {R}})} \rightarrow 0\), using Corollary 2.10 we get \(P\left( \phi _{0}^{n}\right) \rightarrow P\left( \phi _{0}\right).\) Thus, we have \(P(\phi (t))=P\left( \phi _{0}\right)\). \(\hfill{\square}\)

Using the arguments in [11], we have the following orbital stability result, with respect to the semi-distance \(d_{0}\).

Theorem 5.5

Let \(0<q \le \pi\), and let

$$\begin{aligned} U_{q}=\left\{ \phi \in {\mathcal {E}} ~\big|~ E(\phi )=E_{\min }(q), ~P(\phi )=q\right\} \end{aligned}$$

be defined as in Proposition 4.6. Then \(U_{q}\) is orbitally stable, under the semi-distance \(d_{0}\). That is, for any \(\varepsilon>0\) there exists \(\delta>0\), if \(d_{0}\left( \phi _{0}, U_{q}\right) <\delta\), then \(d_{0}\left( \phi (t), U_{q}\right) <\varepsilon\) for any \(t>0\), where \(\phi (t)\) is a solution with initial condition \(\phi _{0}\).

Proof

If the converse is true, then there exists \(\varepsilon _{0}>0\) and \(\phi _{0}^{n} \in {\mathcal {E}}\) satisfying \(d_{0}(\phi _{0}^{n}, U_{q})<\frac{1}{n}\) for any \(n \ge 1\), \(d_{0}(\phi _{n}(t_{n}), U_{q}) \ge \varepsilon _{0}\) for some \(t_{n}>0\), where \(\phi _{n}\) is the solution of (1) with \(\phi _{n}(0)=\phi _{0}^{n}\).            

We claim that \(E(\phi _0^n) \rightarrow E_{\text{ min }}(q)\), \(P(\phi _0^n) \rightarrow q\). Consider an arbitrary subsequence of \((\phi _0^n)_{n \ge 1}\) (still use \((\phi _0^n)_{n \ge 1})\). Using Theorem 4.1, we see that up to subsequence and translation, there exist \(\phi \in U_{q}\) such that \(d_0(\phi _0^n, \phi ) \rightarrow 0\). Using Corollary 4.4 we get \(P(\phi _0^n) \rightarrow P(\phi )=q\) and \(E(\phi _0^n) \rightarrow E(\phi )=E_{\text{ min }}(q)\). Because any subsequence of \((\phi _0^n)_{n \ge 1}\) includes a further subsequence satisfying the property, we conclude that the claim holds.

By Theorem 5.1 (2): \(E(\phi _{n}(t_{n}))=E(\phi _{0}^{n}) \rightarrow E_{\min }(q)\). Lemma 5.4 implies \(P(\phi _{n}(t_{n}))=P(\phi _{0}^{n}) \rightarrow q\). Using again Theorem 4.1, we see that up to translation, there exist a subsequence \((\phi _{n_{k}})_{k \ge 1}\) and \(\phi _{1} \in U_{q}\) satisfying \(d_{0}(\phi _{n_{k}}(t_{n_{k}}), \phi _{1}) \rightarrow 0\), which contradicts \(d_{0}(\phi _{n}(t_{n}), U_{q}) \ge \varepsilon _{0}\) for all n. \(\hfill{\square}\)