Gausson dynamics for logarithmic Schrödinger equations

Abstract

In this paper we study the validity of a Gausson (soliton) dynamics of the logarithmic Schrödinger equation in presence of a smooth external potential.

Keywords

Soliton dynamics logarithmic Schrödinger equation Gausson

1. Introduction

This paper is concerned with the so-called soliton dynamics behaviour for the logarithmic Schrödinger equation with an external potential $\begin{matrix} (1.1) & \{\begin{matrix} i ε \partial_{t} u_{ε} + \frac{ε^{2}}{2} Δ u_{ε} - V (x) u_{ε} + u_{ε} Log | u_{ε} |^{2} = 0, \\ u_{ε} (0, t) = u_{ε, 0} (x), \end{matrix} \end{matrix}$ that is, with the study of the behaviour of the solution $u_{ε}$ , in the semi-classical limit $ε \to 0$ ; namely when the Planck constant $ε = ℏ$ tends to zero, by taking as initial datum for the Cauchy problem (1.1) a function (Gausson type) of the form $\begin{matrix} (1.2) & u_{ε, 0} (x) = e^{\frac{i}{ε} x \cdot v_{0}} R (\frac{x - x_{0}}{ε}), R (x) : = e^{\frac{1 + N}{2}} e^{- | x |^{2}}, x \in R^{N} . \end{matrix}$ Here, $u_{ε} = u_{ε} (x, t)$ is a complex-valued function of $(x, t) \in R^{N} \times R$ , $N ⩾ 1$ , i is the imaginary unit, $V : R^{N} \to R$ is an external potential and $x_{0}$ , $v_{0} \in R^{N}$ , $v_{0} \neq 0$ , are the initial position and velocity for the Newtonian system $\begin{matrix} (1.3) & \{\begin{matrix} \dot{x} (t) = ν (t), x (0) = x_{0}, \\ \dot{ν} (t) = - \nabla V (x (t)), ν (0) = v_{0} . \end{matrix} \end{matrix}$ Notice that the classical Hamiltonian related to (1.3) is $\begin{matrix} (1.4) & H (t) = \frac{1}{2} {| ν (t) |}^{2} + V (x (t)) \end{matrix}$ and is conserved in the time.

Equation (1.1) was proposed by Bialynicki-Birula and Mycielski [4] in 1976 as a model of nonlinear wave mechanics. This NLS equation has wide applications in quantum optics [8], nuclear physics [17], geophysical applications of magma transport [16], effective quantum and gravity, theory of superfluidity, Bose–Einstein condensation and open quantum systems; see [30,31] and the references therein. We refer to [1,2,10,12,13] for a study of existence and stability of standing waves, as well as for a study of the Cauchy problem in a suitable functional framework.

Rigorous results about the soliton dynamics for nonlinear Schrödinguer equation with a power nonlinearity $| u |^{p - 1} u$ were obtained in various papers by J.C. Bronski, R.L. Jerrard [7] and S. Keraani [19]. The main ingredients of the argument are the conservation laws of NLS and of the Hamiltonian (1.4) combined with modulational stability estimates proved by M. Weinstein [27,28]. In recent years, the so-called soliton dynamics has attracted a great deal of attention from both the mathematicians and physicists; see for example [15,22–24,26].

Throughout this paper we assume that the potential V in (1.1) is a $C^{3} (R^{N})$ function bounded with its derivatives. Formally, the NLS (1.1) has the following two conserved quantities. The first conserved quantity is the energy $E_{ε}$ defined by $\begin{matrix} E_{ε} (u) : = \frac{1}{2 ε^{N - 2}} \int_{R^{N}} | \nabla u |^{2} d x + \frac{1}{ε^{N}} \int_{R^{N}} V (x) | u |^{2} d x - \frac{1}{ε^{N}} \int_{R^{N}} | u |^{2} Log | u |^{2} d x . \end{matrix}$ The second conserved quantity is the mass, $\begin{matrix} Q_{ε} (u) : = \frac{1}{ε^{N}} \int_{R^{N}} | u |^{2} d x . \end{matrix}$ Notice that due to the singularity of the logarithm at the origin, the energy fails to be finite as well of class $C^{1}$ on $H^{1} (R^{N})$ . Therefore, we consider the reflexive Banach space $\begin{matrix} (1.5) & W (R^{N}) : = {u \in H^{1} (R^{N}) : | u |^{2} Log | u |^{2} \in L^{1} (R^{N})} . \end{matrix}$ It is well known that the energy $E_{ε}$ is well-defined and of class $C^{1}$ on $W (R^{N})$ (see Section 2). Notice that if $u \in C (R, W (R^{N})) \cap C^{1} (R, W^{'} (R^{N}))$ , then equation (1.1) makes sense in $W^{'} (R^{N})$ , where $W^{'} (R^{N})$ is the dual space of $W (R^{N})$ .

We see that the well-posedness of the Cauchy Problem for (1.1) in $W^{'} (R^{N})$ and the conservation laws follow by a standard compactness method developed in [11, Chapter 9].

Proposition 1.1.
Let $ε > 0$ . For every $u_{ε, 0} \in W (R^{N})$ , there is a unique global solution $u_{ε} \in C (R, W (R^{N})) \cap C^{1} (R, W^{'} (R^{N}))$ of Eq. ( 1.1 ) such that $u_{ε} (x, 0) = u_{ε, 0}$ and ${sup}_{t \in R} ‖ u_{ε} (t) ‖_{W (R^{N})} < \infty$ . Furthermore, the solution $u_{ε} (t)$ satisfies the conservation laws: $\begin{matrix} E_{ε} (u_{ε} (t)) = E (u_{ε, 0}) and Q_{ε} (u_{ε} (t)) = Q_{ε} (u_{ε, 0}) for all t \in R . \end{matrix}$

We denote $\begin{matrix} Σ (R^{N}) : = {u \in H^{1} (R^{N}) : | x |^{} u \in L^{2} (R^{N})} . \end{matrix}$ It is well known that $Σ (R^{N})$ is a Hilbert space when is equipped with the norm $\begin{matrix} ‖ u ‖_{Σ (R^{N})} : = \sqrt{\int_{R^{N}} (| \nabla u |^{2} + | x |^{2} | u |^{2} + | u |^{2}) d x}, \end{matrix}$ and it is continuously embedded in $H^{1} (R^{N})$ . From [9, Theorem 1.5] we have that if initial data $u_{ε, 0}$ belong to $Σ (R^{N})$ , then the solution $u_{ε} (x, t)$ of Eq. (1.1) belong to $L_{loc}^{\infty} (R, Σ (R^{N}))$ . Moreover, if $u_{ε, 0} \in H^{2} (R^{N})$ , then $u_{ε} (t) \in H^{2} (R^{N})$ and $\partial_{t} u_{ε} (t) \in L^{2} (R^{N})$ , for all $t \in R$ .

Notice that the initial data $u_{ε, 0}$ in (1.2) belong to $Σ (R^{N}) \cap H^{2} (R^{N})$ . On the other hand, it is not hard to show that one has the following chain of continuous embeddings (see Lemma 2.1 below) $\begin{matrix} Σ (R^{N}) ↪ W (R^{N}) ↪ H^{1} (R^{N}) . \end{matrix}$ In particular, since $E_{ε}$ is of class $C^{1}$ on $W (R^{N})$ , it follows that $E_{ε}$ is of class $C^{1}$ on $Σ (R^{N})$ .

Let $ω \in R$ and $φ \in W^{} (R^{N})$ be solution of the semilinear elliptic equation $\begin{matrix} (1.6) & - \frac{1}{2} Δ φ + ω φ - φ Log | φ |^{2} = 0, x \in R^{N} . \end{matrix}$ It is well known that the Gausson $\begin{matrix} ϕ_{ω} (x) : = e^{\frac{ω - 1}{2}} R (x), \end{matrix}$ where the function $R (x)$ is defined in (1.2), solves the problem (1.6) for any dimension N. Furthermore, $ϕ_{ω} (x)$ is the unique, up to translations, strictly positive $C^{2}$ -solution for (1.6) such that $ϕ_{ω} (x) \to 0$ as $| x | \to \infty$ ; see [15, Theorem 1.2].

Orbital stability of Gaussons solutions $ϕ_{ω} (x)$ have been studied in a suitable functional framework. More specifically, Cazenave [10]; Cazenave and Lions [13]; Ardila [1]; Blanchard and co. [5,6]; research the orbital stability of stationary solutions of (1.1) in the energy space $W (R^{N})$ .

As mentioned above, the modulational stability property of ground states plays an important role in soliton dynamics; however, due to the singularity of the logarithm at the origin, it is not clear whether the energy functional is of class $C^{2}$ in a tubular neighbourhood of the Gausson R. In particular, it is an open problem to determine whether the Gausson R satisfies the modulational stability estimates.

Consider $H^{1} (R^{N})$ equipped with the scaled norm $\begin{matrix} ‖ ϕ ‖_{H_{ε}^{1}}^{} = \sqrt{ε^{2 - N} ‖ \nabla ϕ ‖_{L^{2}}^{2} + ε^{- N} ‖ ϕ ‖_{L^{2}}^{2}} . \end{matrix}$ The following result is obtained by using the only information that the minimizing sequences for the constrained variational problem associated to (1.6) are precompact in $W (R^{N})$ .
Theorem 1.2.
Let $u_{ε} \in Σ (R^{N})$ be the family of solutions to the Cauchy problem ( 1.1 ) with initial data ( 1.2 ), for some $x_{0}$ , $v_{0} \in R^{N}$ . Then there exist a positive constant C, independent of $ε > 0$ , such that $\begin{matrix} sup_{t \in R} {‖ \nabla u_{ε} (t) ‖}_{L^{2}}^{2} ⩽ C ε^{N - 2} . \end{matrix}$ Moreover, for any $η > 0$ there exist $ε > 0$ , a time $T_{ε}^{} > 0$ and continuous functions* $\begin{matrix} θ_{ε} : [0, T_{ε}^{}] \to [0, 2 π], y_{ε} : [0, T_{ε}^{}] \to R^{N}, \end{matrix}$ such that $\begin{matrix} (1.7) & u_{ε} (x, t) = e^{\frac{i}{ε} (ν (t) \cdot x + θ_{ε} (t))} e^{\frac{1 + N}{2}} e^{- \frac{1}{ε^{2}} | x - y_{ε} (t) |^{2}} + ω_{ε} (x, t), \end{matrix}$ where $‖ ω_{ε} (t) ‖_{H_{ε}^{1}}^{} < η^{2}$ for all $t \in [0, T_{ε}^{})$ .

Here* $y_{ε} = x (t) + ε z_{ε} (t)$ for some continuous function $z_{ε} : [0, T_{ε}^{}] \to R^{N}$ , where* $(x (t), ν (t))$ is the solution of the classical Hamiltonian system ( 1.3 ).

As it is well known, to prove the modulational stability property of ground states, it is necessary to study the spectral structure of the complex self-adjoint operator $E^{″} (R)$ , where $\begin{matrix} (1.8) & E (u) = \frac{1}{2} \int_{R^{N}} | \nabla u |^{2} d x - \int_{R^{N}} | u |^{2} Log | u |^{2} d x . \end{matrix}$ Notice that $E$ is of class $C^{1}$ on $Σ (R^{N})$ . Since $E^{″} (R)$ is a bounded operator defined on $Σ (R^{N})$ with values in $Σ^{'} (R^{N})$ (see Section 4 for more details), it is natural to assume that the energy functional is of class $C^{2}$ in a neighbourhood $V_{ε} (R)$ of R, of size $ε > 0$ , where $\begin{matrix} V_{ε} (R) : = {u \in Σ (R^{N}) : ‖ u - R ‖_{H^{1}}^{2} < ε} . \end{matrix}$
Remark 1.3.
The proof that the functional $E$ is smooth on $V_{ε} (R)$ seems very difficult because of the technical complications related to the singularity of the logarithm at the origin.
Open Problem 1.4.
Prove or disprove that $E$ is of class $C^{2}$ on $V_{ε} (R)$ .
Proposition 1.5.
Suppose that $E$ is of class $C^{2}$ on $V_{ε} (R)$ , for any ε small enough. Then the modulational stability property holds. That is, there exist two constants $C > 0$ and $h > 0$ , such that $\begin{matrix} inf_{y \in R^{N}, θ \in R} {‖ ϕ - e^{i θ} R (\cdot - y) ‖}_{H^{1}}^{2} ⩽ C (E (ϕ) - E (R)) \end{matrix}$ for all $ϕ \in Σ (R^{N})$ , such that $‖ ϕ ‖_{L^{2}}^{2} = ‖ R ‖_{L^{2}}^{2}$ and $E (ϕ) - E (R) < h$ .

In light of Proposition 1.5, the soliton dynamics in Theorem 1.2 can be improved. Indeed, we have the following result.
Theorem 1.6.
Let $u_{ε} \in Σ (R^{N})$ be the family of solutions to the Cauchy problem ( 1.1 ) with initial data ( 1.2 ). Furthermore, let $(x (t), ν (t))$ be the solution of the Hamiltonian system ( 1.3 ). Under the hypothesis of Proposition 1.5 , there exist $θ_{ε} : R^{+} \to [0, 2 π]$ such that, $\begin{matrix} u_{ε} (x, t) = e^{\frac{i}{ε} (ν (t) \cdot x + θ_{ε} (t))} e^{\frac{1 + N}{2}} e^{- \frac{1}{ε^{2}} | x - x (t) |^{2}} + ω_{ε} (x, t), \end{matrix}$ locally uniformly in time $t \in R$ , where $ω_{ε} \in H_{ε}^{1}$ and $‖ ω_{ε} (t) ‖_{H_{ε}^{1}}^{} = O (ε)$ , as $ε \to 0$ .

The paper is organized as follows.

In Section 2 we provide, by variational techniques, a characterization of the Gausson R.

In Section 3 we prove Theorem 1.2.

In Section 4, we show some delicate estimates for $E^{″} (R)$ (Proposition 1.5).

Finally, in Section 5 we give a sketch of proof of Theorem 1.6. Notation.
$⟨ \cdot, \cdot ⟩$ is the duality pairing between $B^{'}$ and B, where B is a Banach space and $B^{'}$ is its dual. The space $L^{2} (R^{N}, C)$ will be denoted by $L^{2} (R^{N})$ and its norm by $‖ \cdot ‖_{L^{2}}$ . This space will be endowed with the real scalar product $\begin{matrix} {(u, v)}_{L^{2}} = Re \int_{R^{N}} u \overline{v} d x, for u, v \in L^{2} (R^{N}) . \end{matrix}$ We denote by $‖ \cdot ‖_{H^{1}}$ the $H_{C}^{1} (R^{N})$ -norm. If L is a linear operator acting on some space $⟨ L v, v ⟩$ denotes the value of the quadratic form associated with L evaluated at v. Finally, throughout this paper, the letter C will denote positive constants whose value may change form line to line.

2. Variational analysis

In this section we establish some results that will be used later in the paper. In particular, we provide a characterization of the Gausson R as minimizer of the energy functional $E$ among functions with the same mass.

We first need to introduce some notation which facilitates the subsequent discussion. Following [10], we define the functions Φ, Ψ on $[0, \infty)$ by $\begin{matrix} (2.1) & Φ (s) : = \{\begin{matrix} - s^{2} Log (s^{2}), & if 0 ⩽ s ⩽ e^{- 3}; \\ 3 s^{2} + 4 e^{- 3} s^{} - e^{- 6}, & if s ⩾ e^{- 3}; \end{matrix} Ψ (s) : = F (s) + Φ (s), \end{matrix}$ where $\begin{matrix} F (s) : = s^{2} Log s^{2} for all s \in R . \end{matrix}$ Notice that Φ is a Young function (see Lemma 1.3 in [10]). Then we define the associate Orlicz space by considering $\begin{matrix} L^{Φ} (R^{N}) : = {f \in L_{loc}^{1} (R^{N}) : Φ (| f |) \in L_{}^{1} (R^{N})}, \end{matrix}$ and we endowed this space with Luxemburg norm $\begin{matrix} ‖ f ‖_{L^{Φ}} = inf {k > 0 : \int_{R^{N}} Φ (k^{- 1} | f (x) |) d x ⩽ 1} . \end{matrix}$ In light of [10, Lemma 2.1], we have that $(L^{Φ} (R^{N}), ‖ \cdot ‖_{L^{Φ}})$ is a separable reflexive Banach space. Finally, we define the Banach space $(W (R^{N}), ‖ \cdot ‖_{W (R^{N})}^{}) = (H_{C}^{1} (R^{N}) \cap L^{Φ} (R^{N}), ‖ \cdot ‖_{W (R^{N})}^{})$ where $\begin{matrix} ‖ f ‖_{W (R^{N})}^{} : = ‖ f ‖_{H^{1} (R^{N})}^{} + ‖ f ‖_{L^{Φ}}^{} for all f \in W (R^{N}) . \end{matrix}$ Since $E \in C^{1} (W (R^{N}), R)$ (see Proposition 2.7 in [10]), it follows from Lemma 2.1(ii) below that $E \in C^{1} (Σ (R^{N}), R)$ .

Lemma 2.1.
The following assertions hold.
The embedding $Σ (R^{N}) ↪ L^{q} (R^{N})$ is compact, where $2 ⩽ q < 2^{}$ .

The inclusion map* $Σ (R^{N}) ↪ L^{2 - δ} (R^{N})$ is continuous, where $δ = 1 / N$ . In particular, the embedding $Σ (R^{N}) ↪ W (R^{N})$ is continuous.

Proof.
Statement (i) is proved in [29, Lemma 3.1]. Next let $u \in Σ (R^{N})$ . By Hölder’s inequality with conjugate exponents $2 N / (2 N - 1)$ , $2 N$ we obtain $\begin{matrix} \int_{R^{N}} {| u (x) |}^{2 - \frac{1}{N}} d x ⩽ {(\int_{R^{N}} \frac{1}{{(1 + | x |^{2})}^{α}} d x)}^{\frac{1}{2 N}} {(\int_{R^{N}} (1 + | x |^{2}) {| u (x) |}^{2} d x)}^{\frac{2 N - 1}{2 N}}, \end{matrix}$ where $α = 2 N - 1$ . Since $α > N / 2$ , it follows that there exists a constant $C > 0$ depending only on N such that $‖ u ‖_{L^{2 - 1 / N} (R^{N})} ⩽ C ‖ u ‖_{Σ (R^{N})}$ ; that is, the embedding $\begin{matrix} (2.2) & Σ (R^{N}) ↪ L^{2 - δ} (R^{N}) \end{matrix}$ is continuous. On the other hand, by [10, Proposition 2.2], we see that $\begin{matrix} (2.3) & \int_{R^{N}} | Ψ (| u |) - Ψ (| v |) | d x ⩽ C (1 + ‖ u ‖_{H^{1} (R^{N})}^{2} + ‖ v ‖_{H^{1} (R^{N})}^{2}) ‖ u - v ‖_{L^{2}} \end{matrix}$ for all u, $v \in H^{1} (R^{N})$ . Moreover, it follows from (2.1) that for every $N \in N$ , there exists $C > 0$ depending only on N such that $\begin{matrix} Φ (| z |) ⩽ C (| z |^{2 + \frac{1}{N}} + | z |^{2 - \frac{1}{N}}) + Ψ (| z |) for any z \in C . \end{matrix}$ Since $2 < 2 + 1 / N < 2^{}$ , it follows from statement (i), (2.2), and (2.3) that if $u_{n} \to 0$ strongly in $Σ (R^{N})$ , then $\int_{R^{N}} Φ (| u_{n} |) d x \to 0$ as n goes to $+ \infty$ . By [10, inequality (2.2) in Lemma 2.1], we see that $u_{n} \to 0$ strongly in $L^{Φ} (R^{N})$ . Therefore, the embedding $Σ (R^{N}) ↪ L^{Φ} (R^{N})$ is continuous. This concludes the proof. □

We have the following.
Proposition 2.2.
For every* $η > 0$ , there exist $h > 0$ such that, if $ϕ \in Σ (R^{N})$ , $‖ ϕ ‖_{L^{2}}^{2} = ‖ R ‖_{L^{2}}^{2}$ and $E (ϕ) - E (R) < h$ , then $\begin{matrix} inf_{y \in R^{N}, θ \in R} {‖ ϕ - e^{i θ} R (\cdot - y) ‖}_{H^{1}}^{2} < η^{2} . \end{matrix}$

Our next goal is to prove Proposition 2.2. In this aim, we study the constrained problem $\begin{matrix} (2.4) & ℓ_{R} : = inf {E (u) : u \in W (R^{N}), ‖ u ‖_{L^{2}}^{2} = ‖ R ‖_{L^{2}}^{2}}, \end{matrix}$ where R is the Gausson defined in (1.2).
Lemma 2.3.
Every minimizing sequence of $ℓ_{R}$ is relativity compact in $W (R^{N})$ . More precisely, if a sequence ${u_{n}} \subset W (R^{N})$ is such that $‖ u_{n} ‖_{L^{2}}^{2} = ‖ R ‖_{L^{2}}^{2}$ and $E (u_{n}) \to ℓ_{R}$ as $n \to \infty$ , then there exist a family ${y_{n}}_{n \in N} \subset R^{N}$ and ${θ_{n}}_{n \in N} \subset R$ such that, up to a subsequence, $\begin{matrix} e^{- i θ_{n}} u_{n} (\cdot + y_{n}) \to R strongly in W (R^{N}) . \end{matrix}$ In particular, $E (R) = ℓ_{R}$ and $‖ u_{n} - e^{i θ_{n}} R (\cdot - y_{n}) ‖_{H^{1}}^{2} \to 0$ , as n goes to $+ \infty$ .

Before giving the proof of Lemma 2.3, we need to establish some preliminaries. We define the following functionals of class $C^{1}$ on $W (R^{N})$ : $\begin{array}{l} (2.5) & S (u) : = \frac{1}{4} \int_{R^{N}} | \nabla u |^{2} d x + \int_{R^{N}} | u |^{2} d x - \frac{1}{2} \int_{R^{N}} | u |^{2} Log | u |^{2} d x, \\ I (u) : = \frac{1}{2} \int_{R^{N}} | \nabla u |^{2} d x + \int_{R^{N}} | u |^{2} d x - \int_{R^{N}} | u |^{2} Log | u |^{2} d x . \end{array}$ Notice that (1.6) with $ω = 1$ is equivalent to $S^{'} (u) = 0$ , and $I (u) = ⟨ S^{'} (u), u ⟩$ is the so-called Nehari functional. Finally, let us consider the minimization problem $\begin{array}{l} (2.6) & \begin{aligned} d & : = inf {S (u) : u \in W (R^{N}) ∖ {0}, I (u) = 0} \\ = \frac{1}{2} inf {‖ u ‖_{L^{2}}^{2} : u \in W (R^{N}) ∖ {0}, I (u) = 0} . \end{aligned} \end{array}$

Now we recall the following fact from [1, Proposition 1.4 and Lemma 3.1].
Theorem 2.4.
Let ${u_{n}}_{n \in N} \subset W^{} (R^{N})$ be a minimizing sequence for d. Then there exist a family $(y_{n}) \subset R^{N}$ and ${θ_{n}}_{n \in N} \subset R$ such that, up to a subsequence, $e^{- i θ_{n}} u_{n} (\cdot + y_{n}) \to R$ strongly in $W (R^{N})$ . In particular, $I (R) = 0$ and $S (R) = d$ .
Lemma 2.5.
The following fact hold, $\begin{matrix} (2.7) & S (R) = inf {S (u) : u \in W (R^{N}), ‖ u ‖_{L^{2}}^{2} = ‖ R ‖_{L^{2}}^{2}} . \end{matrix}$ In particular, $E (R) = ℓ_{R}$ where $ℓ_{R}$ is defined in ( 2.4 ).
Proof.
First, it is clear that $inf {S (u) : u \in W (R^{N}), ‖ u ‖_{L^{2}}^{2} = ‖ R ‖_{L^{2}}^{2}} ⩽ S (R)$ . Next we claim that if $‖ u ‖_{L^{2}}^{2} = ‖ R ‖_{L^{2}}^{2}$ , then $I (u) ⩾ 0$ , where I is the Nehari functional. Indeed, assume by contradiction that $I (u) < 0$ . It is not hard to show that there exists λ, $0 < λ < 1$ , such that $I (λ u) = 0$ . But then, $‖ λ u ‖_{L^{2}}^{2} < ‖ u ‖_{L^{2}}^{2} = ‖ R ‖_{L^{2}}^{2}$ and $I (λ u) = 0$ , which is a contradiction with Theorem 2.4. Therefore, if $‖ u ‖_{L^{2}}^{2} = ‖ R ‖_{L^{2}}^{2}$ , then $I (u) ⩾ 0$ and $\begin{matrix} S (u) = \frac{1}{2} I (u) + \frac{1}{2} ‖ u ‖_{L^{2}}^{2} ⩾ \frac{1}{2} ‖ u ‖_{L^{2}}^{2} = \frac{1}{2} ‖ R ‖_{L^{2}}^{2} = S (R); \end{matrix}$ this implies (2.7). Finally, the proof of the last assertion of lemma immediately follows from (2.7). This concludes the proof of Lemma 2.5. □

Now we give the proof of Lemma 2.3.
Proof of Lemma 2.3.
Let ${u_{n}}_{n \in N} \subset W (R^{N})$ be a minimizing sequence for the value $ℓ_{R}$ , that is $‖ u_{n} ‖_{L^{2}}^{2} = ‖ R ‖_{L^{2}}^{2}$ and $E (u_{n}) \to ℓ_{R}$ , as $n \to \infty$ . From Lemma 2.5, we have as $n \to \infty$ that $\begin{matrix} I (u_{n}) = E (u_{n}) + ‖ u_{n} ‖_{L^{2}}^{2} \to E (R) + ‖ R ‖_{L^{2}}^{2} = I (R) = 0 . \end{matrix}$ Then, by elementary computations, we can see that there exists a sequence ${λ_{n}}_{n \in N} \subset R^{+}$ such that $I (λ_{n} u_{n}) = 0$ and $λ_{n} \to 1$ . Next, define the sequence $f_{n} (x) = λ_{n} u_{n} (x)$ . It is clear that $‖ f_{n} ‖_{L^{2}}^{2} \to ‖ R ‖_{L^{2}}^{2}$ and $I (f_{n}) = 0$ for any $n \in N$ . Therefore, ${f_{n}}_{n \in N}$ is a minimizing sequence for d. Thus, by Theorem 2.4, up to a subsequence, there exist ${y_{n}}_{n \in N} \subset R^{N}$ and ${θ_{n}}_{n \in N} \subset R$ such that $e^{- i θ_{n}} f_{n} (\cdot + y_{n}) \to R$ strongly in $W (R^{N})$ . Since $‖ f_{n} - u_{n} ‖_{W (R^{N})}^{} \to 0$ as $n \to \infty$ , the assertion follows. □

Now we give the proof of Proposition 2.2. Proof of Proposition 2.2.
The result is proved by contradiction. Assume that there exist $η > 0$ and a sequence ${ϕ_{n}}_{n \in N} \subset Σ (R^{N})$ , such that $‖ ϕ_{n} ‖_{L^{2}}^{2} = ‖ R ‖_{L^{2}}^{2}$ and $\begin{array}{l} (2.8) & E (ϕ_{n}) - E (R) < \frac{1}{n}, \\ (2.9) & inf_{θ \in R} inf_{y \in R^{N}} {‖ ϕ_{n} - e^{i θ} R (\cdot - y) ‖}_{H^{1}} ⩾ η, for any n \in N . \end{array}$ Since $E (ϕ_{n}) ⩾ E (R)$ , from formula (2.8), it follows that $E (ϕ_{n}) \to E (R)$ as $n \to \infty$ . That is, ${u_{n}}_{n \in N}$ is a minimizing sequence for $ℓ_{R}$ . By Lemma 2.3, up to a subsequence, there exist ${y_{n}}_{n \in N} \subset R^{N}$ and ${θ_{n}}_{n \in N} \subset R$ such that $\begin{matrix} {‖ ϕ_{n} - e^{i θ_{n}} R (\cdot - y_{n}) ‖}_{H^{1}} \to 0 as n \to \infty, \end{matrix}$ which is a contradiction with (2.9). This ends the proof. □

3. Dynamics of the Gausson

The main aim of this section is to prove Theorem 1.2. Let $u_{ε}$ be a solution of the Cauchy problem (1.1) with initial data (1.2). We define the momentum as a function $p_{ε} : R^{N} \times R \to R^{N}$ by setting $\begin{matrix} p_{ε} (x, t) : = \frac{1}{ε^{N - 1}} Im (\overline{u_{ε} (x, t)} \nabla u_{ε} (x, t)), \end{matrix}$ where $Im (z)$ denotes the imaginary part of z and $\overline{u_{ε}}$ is the complex conjugate of $u_{ε}$ . Note that, by assumption, there exists $ω > 0$ such that $V (x) + ω ⩾ 0$ for all $x \in R^{N}$ . In particular, we can assume $V ⩾ 0$ . Indeed, if $u_{ε}$ is a solution to (1.1)–(1.2), then $e^{- i ω t / ε} u_{ε}$ is a solution of (1.1)-(1.2) with the potential $V (x) + ω ⩾ 0$ instead of V.

We have the following result.

Lemma 3.1.
Let $u_{ε}$ be the solution to problem ( 1.1 ) corresponding to the initial data ( 1.2 ). Then there exists a positive constant C, independent of $ε > 0$ , such that $\begin{matrix} sup_{t \in R} {‖ \nabla u_{ε} (t) ‖}_{L^{2}}^{2} ⩽ C ε^{N - 2} . \end{matrix}$ Moreover, there exists a positive constant $C_{1}$ such that $\begin{matrix} sup_{t \in R} | \int_{R^{N}} p_{ε} (x, t) d x | ⩽ C_{1} . \end{matrix}$
Proof.
First, notice that $E_{ε} (u_{ε} (x, t)) = E_{ε} (u_{ε} (0, t))$ . Moreover, by Lemma 3.3 below, there exists a constant $C > 0$ , independent of $ε > 0$ , such that $E_{ε} (u_{ε} (x, t)) ⩽ C$ . Set $v_{ε} (x) : = u_{ε} (ε x)$ . Since V is nonnegative, we see that $\begin{matrix} (3.1) & \frac{1}{2} ‖ \nabla v_{ε} ‖_{L^{2}}^{2} - \int_{R^{N}} | v_{ε} |^{2} Log | v_{ε} |^{2} d x ⩽ C . \end{matrix}$ Now, applying the logarithmic Sobolev inequality (see [21, Theorem 8.14]) we have $\begin{matrix} \int_{R^{N}} | v_{ε} |^{2} Log | v_{ε} |^{2} d x ⩽ \frac{α^{2}}{π} ‖ \nabla v_{ε} ‖_{L^{2}}^{2} + (Log ‖ v_{ε} ‖_{L^{2}}^{2} - N (1 + Log α)) ‖ v_{ε} ‖_{L^{2}}^{2}, \end{matrix}$ for any $α > 0$ . By conservation of the mass, we obtain that $‖ v_{ε} ‖_{L^{2}}^{2} = m : = ‖ R ‖_{L^{2}}^{2}$ . Therefore, $\begin{matrix} (\frac{1}{2} - \frac{α^{2}}{π}) ‖ \nabla v_{ε} ‖_{L^{2}}^{2} ⩽ C + (Log m - N (1 + Log α)) m . \end{matrix}$ Taking $α > 0$ sufficiently small, the first assertion of lemma follows by rescaling. On the other hand, by Hölder inequality, the mass conservation law and the first assertion of lemma we see that $\begin{array}{l} | \int_{R^{N}} p_{ε} (x, t) d x | ⩽ \int_{R^{N}} | p_{ε} (x, t) | d x ⩽ \frac{1}{ε^{N / 2}} {‖ u_{ε} (\cdot, t) ‖}_{L^{2}}^{} \frac{1}{ε^{N / 2 - 1}} {‖ \nabla u_{ε} (\cdot, t) ‖}_{L^{2}}^{} ⩽ C \end{array}$ for every $t \in R$ , which completes the proof. □

The following lemma will be useful later. For a proof see [19, Lemma 3.3].
Lemma 3.2.
Let $f \in C^{2} (R^{N})$ be such that $‖ f ‖_{C^{2} (R^{N})} < \infty$ . Then for every $y \in R^{N}$ fixed $\begin{matrix} \int_{R^{N}} f (ε x + y) R^{2} (x) d x = f (y) \int_{R^{N}} R^{2} (x) d x + O (ε^{2}), as ε ↘ 0, \end{matrix}$ where $O (ε^{2})$ is independent of y.
Lemma 3.3.
For every $t \in R^{+}$ we have $\begin{matrix} E_{ε} (u_{ε} (t)) = E (R) + m H (t) + O (ε^{2}) as ε ↘ 0 . \end{matrix}$
Proof.
Since R is real, it follows by the energy conservation law $\begin{matrix} E_{ε} (u_{ε} (t)) = E_{ε} (u_{ε} (0)) = m \frac{| v_{0} |^{2}}{2} + E (R) + \int_{R^{N}} V (ε x + x_{0}) R^{2} (x) d x . \end{matrix}$ Next, from Lemma 3.2 we see that $\begin{matrix} \int_{R^{N}} V (ε x + x_{0}) R^{2} (x) d x = m V (x (0)) + O (ε^{2}), as ε ↘ 0 . \end{matrix}$ But then, by the conservation law of the function $t \mapsto H (t)$ , we obtain $\begin{array}{l} E_{ε} (u_{ε} (t)) & = E (R) + m H (0) + O (ε^{2}) \\ = E (R) + m H (t) + O (ε^{2}), as ε ↘ 0 . \end{array}$ This completes the proof. □

In our analysis, we use the following property of the functional $δ_{x}$ defined on the space $C^{2} (R^{N})$ endowed with the standard $C^{2}$ norm: there exist three constants $K_{0}, K_{1} > 0$ and $K_{2} > 1$ such that if $‖ δ_{y} - δ_{z} ‖_{C^{2 }} ⩽ K_{0}$ then $\begin{matrix} (3.2) & K_{1} | y - z | ⩽ ‖ δ_{y} - δ_{z} ‖_{C^{2 }} ⩽ K_{2} | y - z | . \end{matrix}$ Here, $C^{2 }$ is the dual space of $C^{2} (R^{N})$ . For a proof of such statement, see [19, Lemma 3.2].

Let ρ be a positive constant defined by $\begin{matrix} (3.3) & ρ : = K_{2} sup_{t \in [0, T]} | x (t) | + K_{0}, \end{matrix}$ where $T > 0$ , $x (t)$ is defined in (1.3), $K_{0}$ and $K_{2}$ are as in (3.2). Observe that, as $K_{2} > 1$ , we have $| x (t) | ⩽ ρ$ for every $t \in [0, T]$ . Furthermore, let $χ \in C^{\infty} (R^{N})$ be a function such that $\begin{matrix} (3.4) & χ (x) = 1 if | x | < ρ, χ (x) = 0 if | x | > 2 ρ . \end{matrix}$

Then we have the following.
Lemma 3.4.
Let* $u_{ε}$ be the family of solutions to problem ( 1.1 ) with initial data ( 1.2 ) and consider the functions $σ_{ε} : R \to R^{N}$ and $λ_{ε} : R \to R$ defined by $\begin{array}{l} σ_{ε} (t) & = \int_{R^{N}} p_{ε} (x, t) d x - m ν (t), \\ λ_{ε} (t) & = m V (x (t)) - \frac{1}{ε^{N}} \int_{R^{N}} χ (x) V (x) {| u_{ε} (x, t) |}^{2} d x, \end{array}$ where $ν (t)$ is defined in ( 1.3 ) and $m = ‖ R ‖_{L^{2}}^{2}$ . Then $σ_{ε} (t)$ and $λ_{ε} (t)$ are continuous on $R$ and satisfy $σ_{ε} (0) = 0$ , $| λ_{ε} (0) | = O (ε^{2})$ as $ε ↘ 0$ .
Proof.
The continuity of $σ_{ε}$ and $λ_{ε}$ follow from the regularity properties of the solution $u_{ε}$ . Since R is a real function, it follows easily that $σ_{ε} (0) = 0$ . Finally, it is not hard to prove, using the Lemma 3.2, that $| λ_{ε} (0) | = O (ε^{2})$ as ε goes to zero. See, e.g., [22, Lemma 3.7] for more details. □

Define now $\begin{matrix} (3.5) & ψ_{ε} (x, t) : = e^{- \frac{i}{ε} ν (t) \cdot [ε x + x (t)]} u_{ε} (ε x + x (t), t), \end{matrix}$ where $(x (t), ν (t))$ is the solution to problem (1.3). Notice that $ψ_{ε} \in Σ (R^{N})$ for every $t \in R$ and $ε > 0$ . Moreover, the mass of $ψ_{ε}$ is conserved. Indeed, by a change of variable we see that $\begin{matrix} (3.6) & {‖ ψ_{ε} (t) ‖}_{L^{2}}^{2} = {‖ u_{ε} (ε x + x (t), t) ‖}_{L^{2}}^{2} = \frac{1}{ε^{N}} {‖ u_{ε} (t) ‖}_{L^{2}}^{2} = ‖ R ‖_{L^{2}}^{2} . \end{matrix}$ Lemma 3.5.
For every $t \in R$ and $ε > 0$ , $\begin{matrix} 0 ⩽ E (ψ_{ε} (t)) - E (R) ⩽ | ν (t) | | σ_{ε} (t) | + | λ_{ε} (t) | + O (ε^{2}), \end{matrix}$ where $ψ_{ε}$ is defined in ( 3.5 ).
Proof.
We recall that $p_{ε} (x, t) = Im (\overline{u_{ε} (x, t)} \nabla u_{ε} (x, t)) / ε^{N - 1}$ . By a change of variable, it follows $\begin{array}{l} E (ψ_{ε} (t)) & = \frac{1}{2 ε^{N - 2}} \int_{R^{N}} | \nabla u_{ε} |^{2} d x + \frac{1}{2} m {| ν (t) |}^{2} - \frac{1}{ε^{N}} \int_{R^{N}} | u_{ε} |^{2} Log | u_{ε} |^{2} d x \\ - ν (t) \cdot \int_{R^{N}} p_{ε} (x, t) d x . \end{array}$ Thus, taking into account the definition of $E_{ε}$ , we have $\begin{array}{l} E (ψ_{ε} (t)) = E_{ε} (u_{ε} (t)) - \frac{1}{ε^{N}} \int_{R^{N}} V (x) {| u_{ε} (t, x) |}^{2} d x + \frac{1}{2} m {| ν (t) |}^{2} - ν (t) \cdot \int_{R^{N}} p_{ε} (x, t) d x . \end{array}$ Then, from Lemma 3.3 we get $\begin{array}{l} E (ψ_{ε} (t)) - E (R) & = m H (t) - \frac{1}{ε^{N}} \int_{R^{N}} V (x) {| u_{ε} (t, x) |}^{2} d x \\ + \frac{1}{2} m {| ν (t) |}^{2} - ν (t) \cdot \int_{R^{N}} p_{ε} (x, t) d x + O (ε^{2}) \\ ⩽ | ν (t) | | σ_{ε} (t) | + | λ_{ε} (t) | - \frac{1}{ε^{N}} \int_{R^{N}} (1 - χ (x)) V (x) {| u_{ε} (t, x) |}^{2} d x + O (ε^{2}) . \end{array}$ Since $V ⩾ 0$ , it follows that $\begin{matrix} 0 ⩽ E (ψ_{ε} (t)) - E (R) ⩽ | ν (t) | | σ_{ε} (t) | + | λ_{ε} (t) | + O (ε^{2}), \end{matrix}$ which concludes the proof of lemma. □
Proof of Theorem 1.2.
Set $β_{ε} (t) : = | ν (t) | | σ_{ε} (t) | + | λ_{ε} (t) |$ for $t \in R$ . From Lemma 3.4, since ${sup}_{t \in R} | ν (t) | < + \infty$ , it follows that the function ${t \to β_{ε} (t)}$ is continuous and $| β_{ε} (0) | = O (ε^{2})$ as $ε ↘ 0$ . Let $η > 0$ . Let us fix a time $T_{0} > 0$ . Let $h > 0$ , depending on η, be as in Proposition 2.2. Introduce the number $\begin{matrix} (3.7) & T_{ε}^{} : = sup {t \in [0, T_{0}] : β_{ε} (τ) ⩽ \frac{h}{2}, for all τ \in [0, t]} . \end{matrix}$ Since $| β_{ε} (0) | = O (ε^{2})$ it follows that $T_{ε}^{} > 0$ , for any $ε > 0$ small. By choosing ε sufficiently small, from Lemma 3.5, we get for all $t \in [0, T_{ε}^{})$ $\begin{matrix} E (ψ_{ε} (t)) - E (R) ⩽ β_{ε} (t) + O (ε^{2}) < h . \end{matrix}$ Since $ψ_{ε} \in Σ (R^{N})$ and $‖ ψ_{ε} (t) ‖_{L^{2}}^{2} = ‖ R ‖_{L^{2}}^{2}$ , by Proposition 2.2 there exist two families of uniformly bounded functions $θ_{ε}^{} : R \to R$ and $z_{ε} : R \to R^{N}$ such that $\begin{matrix} {‖ e^{- \frac{i}{ε} ν (t) \cdot [ε x + x (t)]} u_{ε} (ε x + x (t), t) - e^{i θ_{ε}^{} (t)} R (x + z_{ε} (t)) ‖}_{H^{1}}^{2} < η^{2} \end{matrix}$ for every $t \in [0, T_{ε}^{})$ . Finally, by rescaling and setting $\begin{matrix} θ_{ε} (t) : = ε θ_{ε}^{*} (t), y_{ε} (t) : = x (t) - ε z_{ε} (t), \end{matrix}$ we get formula (1.7). The proof of Theorem 1.2 is complete. □

4. Spectral analysis of linearization

The goal of this section is to prove Proposition 1.5. Before giving the proof, we need to establish some preliminary lemmas.

Lemma 4.1.
Let R be the Gausson ( 1.2 ). Then there exists a positive constant δ such that for every $w \in Σ (R^{N})$ satisfying $\begin{matrix} (4.1) & {(w, R)}_{L^{2}} = {(w, i R)}_{L^{2}} = {(w, \frac{\partial R}{\partial x_{j}})}_{L^{2}} = 0, for all j = 1, \dots, N \end{matrix}$ we have $\begin{matrix} ⟨ S^{″} (R) w, w ⟩ ⩾ δ ‖ ω ‖_{Σ (R^{N})}^{2}, \end{matrix}$ where the functional S is defined in ( 2.5 ).

We set $w = u + i v$ for real valued functions u, $v \in Σ (R^{N}, R)$ . Then it is not hard to show that $S^{″} (R)$ can be separated into a real and imaginary part $L_{+}$ and $L_{-}$ such that $\begin{matrix} ⟨ S^{″} (R) w, w ⟩ = ⟨ L_{+} u, u ⟩ + ⟨ L_{-} v, v ⟩, \end{matrix}$ where $L_{+}$ and $L_{-}$ are two bounded operator on $Σ (R^{N})$ with values in $Σ^{'} (R^{N})$ and given by $\begin{array}{l} L_{+} u = - \frac{1}{2} Δ u + 2 | x |^{2} u - (N + 2) u, \\ L_{-} v = - \frac{1}{2} Δ v + 2 | x |^{2} v - N v . \end{array}$ Indeed, let $f \in Σ (R^{N})$ . Then we see that $\begin{array}{l} S^{″} (f) w & = - \frac{1}{2} Δ w + w - Log | f |^{2} w - 2 \frac{f}{| f |^{2}} Re (f \overline{w}) . \end{array}$ Now recalling the definition of R given in (1.2) we get $\begin{array}{l} S^{″} (R) w & = - \frac{1}{2} Δ w + w - Log | R |^{2} w - 2 Re (\overline{w}) \\ = - \frac{1}{2} Δ w - N w + 2 | x |^{2} w - 2 Re (\overline{w}) \\ = L_{+} u + i L_{-} v . \end{array}$ The above lemma follows immediately from the two following lemmas.
Lemma 4.2.
There exists a positive constant $δ_{1}$ such that for every $v \in Σ (R^{N})$ satisfying $\begin{matrix} {(v, R)}_{L^{2}} = 0 \end{matrix}$ we have $⟨ L_{-} v, v ⟩ ⩾ δ_{1} ‖ v ‖_{Σ (R^{N})}^{2}$ .
Lemma 4.3.
There exists a positive constant $δ_{2}$ such that for every $u \in Σ (R^{N})$ satisfying $\begin{matrix} {(u, R)}_{L^{2}} = {(u, \frac{\partial R}{\partial x_{j}})}_{L^{2}} = 0 for all j = 1, \dots, N \end{matrix}$ we have $⟨ L_{+} u, u ⟩ ⩾ δ_{2} ‖ u ‖_{Σ (R^{N})}^{2}$ .

Before giving the proof of Lemmas 4.2 and 4.3, we discuss some spectral properties of $L_{-}$ and $L_{+}$ . First, since $| x |^{2} \to + \infty$ as $| x |$ goes to $+ \infty$ , the operators $L_{-}$ and $L_{+}$ have only discrete spectrum, i.e. $σ (L_{\pm}) = σ_{p} (L_{\pm}) = {λ_{i}^{\pm}}_{i \in N}$ . Moreover, the corresponding eigenvalues $λ_{i}^{\pm}$ tending to $+ \infty$ and the eigenfunctions are exponentially decaying as $| x | \to + \infty$ .

Now we give the proof of Lemma 4.2.
Proof of Lemma 4.2.
We claim that there exists $κ > 0$ such that for every $v \in Σ (R^{N}) ∖ {0}$ satisfying ${(v, R)}_{L^{2}}^{} = 0$ , we have $\begin{matrix} (4.2) & ⟨ L_{-} v, v ⟩ ⩾ κ ‖ v ‖_{L^{2}}^{2} . \end{matrix}$ Indeed, it is not hard to show that $L_{-} (R) = 0$ . Since $R > 0$ , it follows that 0 is the first simple eigenvalue of $L_{-}$ (see [3, Chapter 3]). Next let $v \in Σ (R^{N}) ∖ {0}$ with ${(v, R)}_{L^{2}}^{} = 0$ . From the min-max characterization of eigenvalues (see [3, Supplement 1]) there exist $κ > 0$ such that $⟨ L_{-} v, v ⟩ ⩾ κ ‖ v ‖_{L^{2}}^{2}$ . Notice that κ is the second eigenvalue of $L_{-}$ . This proves our claim.

Now, let us set $\begin{matrix} τ = inf {⟨ L_{-} v, v ⟩ : ‖ v ‖_{Σ (R^{N})}^{2} = 1, {(v, R)}_{L^{2}} = 0}, \end{matrix}$ and assume by contradiction that $τ = 0$ . Let ${v_{n}}_{n \in N}$ be a minimizing sequence for τ. Then since $‖ v_{n} ‖_{Σ (R^{N})} = 1$ , we can assume that the sequence converges weakly in $Σ (R^{N})$ to some v. Furthermore, since the embedding $Σ (R^{N}) ↪ L^{2} (R^{N})$ is compact, it follows that $v_{n} \to v$ strongly in $L^{2} (R^{N})$ as $n \to \infty$ . In particular, ${(v, R)}_{L^{2}}^{} = 0$ and $\begin{matrix} 0 ⩽ ⟨ L_{-} v, v ⟩ ⩽ \underset{n \to \infty}{lim inf} ⟨ L_{-} v_{n}, v_{n} ⟩ = 0 . \end{matrix}$ This implies that $⟨ L_{-} v, v ⟩ = 0$ and, since ${(v, R)}_{L^{2}}^{} = 0$ , it follows from (4.2) that $v \equiv 0$ . On the other hand, $\begin{array}{l} lim_{n \to \infty} ‖ v_{n} ‖_{L^{2}}^{2} & = lim_{n \to \infty} {‖ v_{n} ‖_{L^{2}}^{2} + ⟨ L_{-} v_{n}, v_{n} ⟩} ⩾ lim_{n \to \infty} {\frac{1}{2} ‖ v_{n} ‖_{Σ (R^{N})}^{2} - N ‖ v_{n} ‖_{L^{2}}^{2}} \\ = \frac{1}{2} - N lim_{n \to \infty} ‖ v_{n} ‖_{L^{2}}^{2} . \end{array}$ Therefore, $‖ v ‖_{L^{2}}^{2} ⩾ 1 / (2 N + 2)$ , which is a contradiction to the fact that $v \equiv 0$ . This completes of proof of lemma. □

We now turn our attention to $L_{+}$ . In order to prove Lemma 4.3, we first establish the following.
Lemma 4.4.
The following assertions hold.
The operator $L_{+}$ has only one negative eigenvalue. The unique negative simple eigenvalue equals $- 2$ , and R is the corresponding eigenfunction.

The second eigenvalue of $L_{+}$ is 0 and $\begin{matrix} ker (L_{+}) = span {\frac{\partial R}{\partial x_{j}} : j = 1, \dots, N} . \end{matrix}$

Proof.
We remark that $L_{+} (R) = - 2 R$ . But then, since $R > 0$ , it follows that $- 2$ is the first simple eigenvalue of $L_{+}$ . From [25, Example 4.5 and Section 4.2], we see that $- 2$ is the only negative eigenvalue. On the other hand, it is clear that 0 is the next eigenvalue. Indeed, since $R \in C^{\infty} (R^{N})$ , an easy calculation shows that $L_{+} (\partial R / \partial x_{j}) = 0$ . Thus, 0 is an eigenvalue of $L_{+}$ and $\begin{matrix} span {\frac{\partial R}{\partial x_{j}} : j = 1, \dots, N} \subseteq ker (L_{+}) . \end{matrix}$ The reverse inclusion is shown in [14, Theorem 1.3]. The proof is completed. □
Proof of Lemma 4.3.
From the spectral decomposition theorem [18, p. 177] and Lemma 4.3, the space $Σ (R^{N})$ can be decomposed into $Σ (R^{N}) = span {R / ‖ R ‖_{L^{2}}^{}} \oplus ker (L_{+}) \oplus E_{+}$ , where $E_{+}$ is the image of the spectral projection to the positive part of the spectrum of $L_{+}$ . In particular, if $ξ \in E_{+}$ , then $⟨ L_{+} ξ, ξ ⟩ > 0$ . Then for every $u \in Σ (R^{N}) ∖ {0}$ with $\begin{matrix} {(u, R)}_{L^{2}} = {(u, \frac{\partial R}{\partial x_{j}})}_{L^{2}} = 0, for all j = 1, \dots, N \end{matrix}$ we get $⟨ L_{+} u, u ⟩ > 0$ . The remainder of the argument is a literal repetition of the proof of Lemma 4.2. We omit the details. □
Lemma 4.5.
Let $ψ \in Σ (R^{N})$ such that $\begin{matrix} inf_{y \in R^{N}, θ \in [0, 2 π)} {‖ ψ - e^{- i θ} R (\cdot + y) ‖}_{L^{2}} ⩽ ‖ R ‖_{L^{2}} \end{matrix}$ and $‖ ψ ‖_{L^{2}}^{2} = ‖ R ‖_{L^{2}}^{2}$ . Then $\begin{matrix} inf_{y \in R^{N}, θ \in [0, 2 π)} {‖ ψ - e^{- i θ} R (\cdot + y) ‖}_{L^{2}}^{2} \end{matrix}$ is achieved at some $y_{0} \in R^{N}$ and $θ_{0} \in [0, 2 π)$ .
Proof.
Consider the auxiliary function $Γ : R^{N} \times [0, 2 π) \to R$ defined by $\begin{matrix} Γ (y, θ) : = {‖ ψ - e^{- i θ} R (\cdot + y) ‖}_{L^{2}} . \end{matrix}$ It is clear that Γ is a continuous function. Now, since $‖ ψ ‖_{L^{2}}^{2} = ‖ R ‖_{L^{2}}^{2}$ we see that $\begin{matrix} Γ^{2} (y, θ) = 2 ‖ R ‖_{L^{2}}^{2} - 2 Re (e^{- i θ} \int_{R^{N}} R (x + y) \overline{ψ (x)} d x) . \end{matrix}$ Since $R (\cdot + y)$ decay exponentially to zero as $| y | \to + \infty$ , we have $R (\cdot + y) ⇀ 0$ in $L^{2} (R^{N})$ as $| y |$ goes to $+ \infty$ . Thus, we have $\begin{matrix} lim_{| y | \to \infty} Γ^{2} (y, θ) = 2 ‖ R ‖_{L^{2}}^{2} . \end{matrix}$ By the first assumption on the function ψ, for every $δ > 0$ , we see that there exist points $y^{} \in R^{N}$ and $θ^{} \in [0, 2 π)$ such that $Γ (y^{}, θ^{}) ⩽ ‖ R ‖_{L^{2}} + δ$ . As a consequence, $Γ (y, θ)$ attains its infimum over the compact set ${\overline{B}}_{ϱ} (0) \times [0, 2 π]$ , for a suitable $ϱ > 0$ , which finishes the proof. □

We define now the tubular neighbourhood of R of size $ε \in (0, ‖ R ‖_{L^{2}}^{2})$ by $\begin{matrix} U_{ε} (R) : = {u \in Σ (R^{N}) : ‖ u ‖_{L^{2}}^{2} = ‖ R ‖_{L^{2}}^{2}, inf_{y \in R^{N}, θ \in [0, 2 π)} {‖ e^{i θ} u (\cdot - y) - R ‖}_{H^{1}}^{2} < ε} . \end{matrix}$ By Lemma 4.5, there exist $σ : U_{ε} (R) \to R$ and $Y : U_{ε} (R) \to R^{N}$ such that, for all $u \in U_{ε} (R)$ , $\begin{matrix} (4.3) & {‖ e^{i σ (u)} u (\cdot - Y (u)) - R ‖}_{L^{2}}^{2} = inf_{y \in R^{N}, θ \in [0, 2 π)} {‖ e^{i θ} u (\cdot - y) - R ‖}_{L^{2}}^{2} . \end{matrix}$ We claim that the function $w : = e^{i σ (u)} u (\cdot - Y (u))$ satisfies the orthogonality conditions $\begin{matrix} (4.4) & {(w, i R)}_{L^{2}} = {(w, \frac{\partial R}{\partial x_{j}})}_{L^{2}} = 0 for all j = 1, \dots, N . \end{matrix}$ Indeed, by differentiating (4.3) with respect to θ we see that $\begin{array}{l} {(w, i R)}_{L^{2}} & = - Re \int_{R^{N}} \overline{i e^{i σ (u)} u (\cdot - Y (u))} R (x) d x \\ = {(i e^{i σ (u)} u (\cdot - Y (u)), e^{i σ (u)} u (\cdot - Y (u)) - R)}_{L^{2}} = 0 . \end{array}$ On the other hand, by differentiating (4.3) with respect to $y_{j}$ we get $\begin{array}{l} {(w, \frac{\partial R}{\partial x_{j}})}_{L^{2}} & = Re \int_{R^{N}} \overline{e^{i σ (u)} u (\cdot - Y (u))} \frac{\partial R (x)}{\partial x_{j}} d x - \int_{R^{N}} R (x) \frac{\partial R (x)}{\partial x_{j}} d x \\ = {(u - e^{- i σ (u)} R (\cdot + Y (u)), e^{- i σ (u)} \frac{\partial R}{\partial x_{j}} (\cdot + Y (u)))}_{L^{2}} = 0 . \end{array}$

Now we give the proof of Proposition 1.5. Proof of Proposition 1.5.
Our proof is inspired by the one contained in [20, Lemma 6.3]. First, we claim that there exist $ε > 0$ and $C > 0$ such that for all $v \in U_{ε} (R)$ we have $\begin{matrix} E (v) - E (R) ⩾ C inf_{y \in R^{N}, θ \in R} {‖ e^{i θ} v (\cdot - y) - R ‖}_{H^{1}}^{2} . \end{matrix}$ Indeed, for ε small enough, let $w : = e^{i σ (v)} v (\cdot - Y (v))$ be as in (4.3). Let $λ \in R$ and $z \in Σ (R^{N})$ be such that $w = R + λ R + z$ with ${(z, R)}_{L^{2}} = 0$ . From (4.4) we see that $\begin{matrix} {(z, \partial R / \partial x_{j})}_{L^{2}} = 0, {(z, i R)}_{L^{2}} = 0 . \end{matrix}$ Then z satisfies the conditions (4.1) in Lemma 4.1. Hence, there exists $δ > 0$ such that $\begin{matrix} (4.5) & ⟨ S^{″} (R) z, z ⟩ ⩾ δ ‖ z ‖_{Σ (R^{N})}^{2} ⩾ δ ‖ z ‖_{H^{1}}^{2} . \end{matrix}$ On the other hand, since $S^{'} (R) = 0$ , under the hypothesis of Proposition 1.5 and by virtue of Taylor formula we get $\begin{array}{l} S (v) - S (R) & = S (w) - S (R) \\ (4.6) & = \frac{1}{2} ⟨ S^{″} (R) (w - R), w - R ⟩ + o (‖ w - R ‖_{H^{1}}^{2}) . \end{array}$ Moreover, it is not hard to show that $λ = o (‖ w - R ‖_{H^{1}})$ and (see [20, Lemma 6.3]) $\begin{matrix} ⟨ S^{″} (R) (w - R), w - R ⟩ = ⟨ S^{″} (R) z, z ⟩ + o (‖ w - R ‖_{H^{1}}^{2}) . \end{matrix}$ Notice that $‖ z ‖_{H^{1}}^{2} ⩾ ‖ w - R ‖_{H^{1}}^{2} + o (‖ w - R ‖_{H^{1}}^{2})$ . Thus, since $‖ v ‖_{L^{2}}^{2} = ‖ R ‖_{L^{2}}^{2}$ , from (4.5) and (4.6) we obtain $\begin{matrix} E (v) - E (R) ⩾ \frac{δ}{2} ‖ w - R ‖_{H^{1}}^{2} + o (‖ w - R ‖_{H^{1}}^{2}) . \end{matrix}$ Then choosing ε small enough and recalling the definition of w in (4.3), it follows that $\begin{matrix} E (v) - E (R) ⩾ \frac{δ}{4} inf_{y \in R^{N}, θ \in R} {‖ e^{i θ} v (\cdot - y) - R ‖}_{H^{1}}^{2}, \end{matrix}$ for every $v \in U_{ε} (R)$ . This concludes the proof of the claim.

Finally, from Proposition 2.2 we see that for every $ε > 0$ , there exists $h > 0$ such that if $E (v) - E (R) < h$ and $‖ v ‖_{L^{2}}^{2} = ‖ R ‖_{L^{2}}^{2}$ , then $v \in U_{ε} (R)$ . Then, choosing h small enough, Proposition 1.5 follows. □

5. Proof of Theorem 1.6

In this section we will show the main steps of the proof of Theorem 1.6. We essentially follow the argument of [19], which is based upon the original paper by Bronski and Jerrard [7]. Using the variational structure of (1.1) and by the regularity of solutions (see the paragraph after Proposition 1.1), it is not difficult to show that the solution $u_{ε}$ satisfies the identities $\begin{array}{l} (5.1) & \frac{1}{ε^{N}} \frac{\partial | u_{ε} |^{2}}{\partial t} (x, t) = - {div}_{x} p_{ε} (x, t), (x, t) \in R^{N} \times R^{}, \\ (5.2) & \int_{R^{N}} \frac{\partial p_{ε}}{\partial t} (x, t) d x = - \int_{R^{N}} \nabla V (x) \frac{| u_{ε} (x, t) |^{2}}{ε^{N}}, x \in R^{N} . \end{array}$ In light of Proposition 1.5, the result in Theorem 1.2 can be improved. More precisely, combining Proposition 1.5 and Lemma 3.5 and following the same argument as Theorem 1.2 we have the following

Proposition 5.1.

If $u_{ε} \in Σ (R^{N})$ is the family of solutions to the Cauchy problem ( 1.1 ) with initial data ( 1.2 ), then there exist $ε_{0} > 0$ , a time $T_{ε}^{*} > 0$ , families of bounded functions $θ_{ε} : R \to [0, 2 π)$ , $y_{ε} : R \to R^{N}$ such that $\begin{matrix} u^{ε} (x, t) = e^{\frac{i}{ε} (ν (t) \cdot x + θ_{ε} (t))} e^{\frac{1 + N}{2}} e^{- \frac{1}{ε^{2}} | x - y_{ε} (t) |^{2}} + ω_{ε} (x, t), \end{matrix}$ where $\begin{matrix} {‖ ω_{ε} (t) ‖}_{H_{ε}^{1}}^{2} ⩽ C (| ν (t) | | σ_{ε} (t) | + | λ_{ε} (t) |) + O (ε^{2}) for all ε \in (0, ε_{0}) and t \in [0, T_{ε}^{*}) . \end{matrix}$

Let $ε_{0} > 0$ , $T_{ε}^{*} > 0$ and $y_{ε} (t)$ be as in Proposition 5.1. Then we have the following

Lemma 5.2.

There exists a constant $C > 0$ such that $\begin{array}{l} {‖ \frac{| u_{ε} (x, t) |^{2}}{ε^{N}} d x - m δ_{y_{ε} (t)} ‖}_{{(C^{2})}^{*}} + {‖ p_{ε} (x, t) d x - m ν (t) δ_{y_{ε} (t)} ‖}_{{(C^{2})}^{*}} \\ ⩽ C | σ_{ε} (t) | + C | λ_{ε} (t) | + O (ε^{2}) \end{array}$ for all $ε \in (0, ε_{0})$ and $t \in [0, T_{ε}^{*})$ .

Proof.

First, notice that for any $v \in H^{1} (R^{N})$ , we have $| \nabla | v | |^{2} = | \nabla v |^{2} - \frac{| Im (\overline{v} \nabla v) |^{2}}{| v |^{2}}$ . Furthermore, it is not hard to show that $\begin{array}{l} \int_{R^{N}} \frac{| Im (\overline{ψ_{ε}} \nabla ψ_{ε}) |^{2}}{| ψ_{ε} |^{2}} d x & = ε^{N} \int_{R^{N}} \frac{| p_{ε} (x, t) |^{2}}{| u_{ε} (x, t) |^{2}} d x + m {| ν (t) |}^{2} - 2 ν (t) \cdot \int_{R^{N}} p_{ε} (x, t) d x \\ = \int_{R^{N}} {| ε^{N / 2} \frac{p_{ε} (x, t)}{| u_{ε} (x, t) |} - \frac{1}{m} (\int_{R^{N}} p_{ε} (x, t) d x) \frac{| u_{ε} (x, t) |}{ε^{N / 2}} |}^{2} d x \\ (5.3) & + m {| ν (t) - \frac{\int_{R^{N}} p_{ε} (x, t) d x}{m} |}^{2}, \end{array}$ where $ψ_{ε} (x, t)$ is the function defined in (3.5). By Lemma 3.5 we see that $\begin{array}{l} 0 & ⩽ E (| ψ_{ε} |) - E (R) + \frac{1}{2} \int_{R^{N}} \frac{| Im (\overline{ψ_{ε}} \nabla ψ_{ε}) |^{2}}{| ψ_{ε} |^{2}} d x \\ ⩽ | ν (t) | | σ_{ε} (t) | + | λ_{ε} (t) | + O (ε^{2}) \end{array}$ for every $t \in [0, T_{ε}^{*})$ and $ε \in (0, ε_{0})$ . Since $E (| ψ_{ε} |) - E (R) ⩾ 0$ , it follows from (5.3), $\begin{array}{l} \int_{R^{N}} {| ε^{N / 2} \frac{p_{ε} (x, t)}{| u_{ε} (x, t) |} - \frac{1}{m} (\int_{R^{N}} p_{ε} (x, t) d x) \frac{| u_{ε} (x, t) |}{ε^{N / 2}} |}^{2} d x + m {| ν (t) - \frac{\int_{R^{N}} p_{ε} (x, t) d x}{m} |}^{2} \\ (5.4) & ⩽ C | ν (t) | | σ_{ε} (t) | + C | λ_{ε} (t) | + O (ε^{2}) \end{array}$ for every $t \in [0, T_{ε}^{*})$ and $ε \in (0, ε_{0})$ . Now, to prove the assertion, we need to estimate $Γ (t)$ , where $\begin{array}{l} Γ (t) : & = | \int_{R^{N}} f (x) \frac{| u_{ε} (x, t) |^{2}}{ε^{N}} d x - m f (y_{ε}) | + | \int_{R^{N}} p_{ε} (x, t) f (x) d x - m ν (t) f (y_{ε}) | \end{array}$ for every function f in $C^{2} (R^{N})$ with $‖ f ‖_{C^{2}} ⩽ 1$ . By simple computations we see that $\begin{array}{l} | \int_{R^{N}} p_{ε} (x, t) f (x) d x - m ν (t) f (y_{ε}) | \\ ⩽ \frac{1}{m} | \int_{R^{N}} p_{ε} (x, t) d x | | \int_{R^{N}} \frac{f (x) | u_{ε} (x, t) |^{2}}{ε^{N}} d x - m f (y_{ε} (t)) | \\ + | \int_{R^{N}} f (x) {p_{ε} (x, t) - \frac{1}{m} (\int_{R^{N}} p_{ε} (x, t) d x) \frac{| u_{ε} (x, t) |^{2}}{ε^{N}}} d x | + C σ_{ε} (t), \end{array}$ for every $t \in [0, T_{ε}^{*})$ and $ε \in (0, ε_{0})$ . Here, the function $σ_{ε} (t)$ is defined in Lemma 3.4. Set $f_{♯} (x) : = f (x) - f (y_{ε} (t))$ . Since $\int_{R^{N}} p_{ε} (x, t) d x$ is bounded (see Lemma 3.1) and $\begin{array}{l} \int_{R^{N}} {p_{ε} (x, t) - \frac{1}{m} (\int_{R^{N}} p_{ε} (x, t) d x) \frac{| u_{ε} (x, t) |^{2}}{ε^{N}}} d x = 0, \end{array}$ it follows that $\begin{array}{l} Γ (t) & ⩽ \int_{R^{N}} f_{♯} (x) | p_{ε} (x, t) - \frac{1}{m} (\int_{R^{N}} p_{ε} (x, t) d x) \frac{| u_{ε} (x, t) |^{2}}{ε^{N}} | d x \\ + \int_{R^{N}} | f_{♯} (x) | \frac{| u_{ε} (x, t) |^{2}}{ε^{N}} d x + C \int_{R^{N}} | f_{♯} (x) | \frac{| u_{ε} (x, t) |^{2}}{ε^{N}} d x + C σ_{ε} (t) . \end{array}$ Using the inequality $a b ⩽ a^{2} + b^{2}$ and (5.4) we obtain $\begin{matrix} Γ (t) & ⩽ \frac{1}{2} \int_{R^{N}} {| ε^{N / 2} \frac{p_{ε} (x, t)}{| u_{ε} (x, t) |} - \frac{1}{m} (\int_{R^{N}} p_{ε} (x, t) d x) \frac{| u_{ε} (x, t) |^{}}{ε^{N / 2}} |}^{2} d x \\ + \int_{R^{N}} {C | f_{♯} (x) | + \frac{1}{2} {| f_{♯} (x) |}^{2}} \frac{| u_{ε} (x, t) |^{2}}{ε^{N}} d x + C σ_{ε} (t) \\ ⩽ \int_{R^{N}} {C | f_{♯} (x) | + \frac{1}{2} {| f_{♯} (x) |}^{2}} \frac{| u_{ε} (x, t) |^{2}}{ε^{N}} d x + C | σ_{ε} (t) | + C | λ_{ε} (t) | + O (ε^{2}) . \end{matrix}$ Finally, in view of the elementary inequality $a^{2} ⩽ 2 b^{2} + 2 {(a - b)}^{2}$ with $\begin{array}{l} a = \frac{| u_{ε} (x, t) |}{ε^{N / 2}}, b = \frac{1}{ε^{N / 2}} R (\frac{x - y_{ε} (t)}{ε}), \end{array}$ since $f_{♯} (y_{ε} (t)) = 0$ , it follows from Lemma 3.2 and Proposition 5.1, $\begin{array}{l} Γ (t) & ⩽ \frac{C}{ε^{N}} \int_{R^{N}} {| f_{♯} (x) | + {| f_{♯} (x) |}^{2}} R^{2} (\frac{x - y_{ε} (t)}{ε}) d x \\ + \frac{C}{ε^{N}} \int_{R^{N}} {| u_{ε} (x, t) - R (\frac{x - y_{ε} (t)}{ε}) |}^{2} d x + C | σ_{ε} (t) | + C | λ_{ε} (t) | + O (ε^{2}) \\ ⩽ C | σ_{ε} (t) | + C | λ_{ε} (t) | + O (ε^{2}), \end{array}$ for every $t \in [0, T_{ε}^{*})$ and $ε \in (0, ε_{0})$ , which concluded the proof. □

We now turn to estimate the distance $| x (t) - y_{ε} (t) |$ , where the function $y_{ε} (t)$ is given in Proposition 5.1 and $x (t)$ is the solution of the classical Hamiltonian system (1.3).

Lemma 5.3.

Let $u_{ε}$ be the family of solutions to problem ( 1.1 ) with initial data ( 1.2 ). Consider the function $γ_{ε}$ defined by $\begin{array}{l} γ_{ε} (t) = m x (t) - \frac{1}{ε^{N}} \int_{R^{N}} x χ (x) {| u_{ε} (x, t) |}^{2} d x, \end{array}$ where $χ (x)$ is defined in ( 3.4 ). Then $γ_{ε} (t)$ is a continuous function on $R$ and satisfy $| γ_{ε} (0) | = O (ε^{2})$ as ε goes to zero.

Proof.

The proof easily follows from Lemma 3.2, and the properties of the functions $u_{ε} (x, t)$ and $χ (x)$ . □

Lemma 5.4.

Let $T_{ε}^{*} > 0$ be the time introduced in ( 3.7 ). There exist positive constants $h_{0}$ and $ε_{0}$ , such that for a constant $C > 0$ , $\begin{array}{l} | x (t) - y_{ε} (t) | ⩽ C (| σ_{ε} (t) | + | λ_{ε} (t) | + | γ_{ε} (t) |) + O (ε^{2}) \end{array}$ for every $t \in [0, T_{ε}^{*}]$ .

Proof.

First, we claim that there exists $T_{0} > 0$ such that $| y_{ε} (t) | < ρ$ , for every $t \in [0, T_{ε}^{*})$ with $T_{ε}^{*} ⩽ T_{0}$ , where the constant ρ is defined in (3.3). Let us first prove that $\begin{array}{l} ‖ δ_{y_{ε} (t_{1})} - δ_{y_{ε} (t_{2})} ‖_{C^{2 *}} < ρ, for all t_{1}, t_{2} \in [0, T_{ε}^{*}] . \end{array}$ Let $f \in C^{2} (R^{N})$ with $‖ f ‖_{C^{2}} ⩽ 1$ and pick $t_{1}$ , $t_{2} \in [0, T_{ε}^{*})$ . From Lemma 3.1 and identity (5.1) we see that $\begin{array}{l} \frac{1}{ε^{N}} \int_{R^{N}} ({| u_{ε} (x, t_{2}) |}^{2} - {| u_{ε} (x, t_{1}) |}^{2}) f (x) d x & = \frac{1}{ε^{N}} \int_{R^{N}} \int_{t_{1}}^{t_{2}} \frac{\partial | u_{ε} |^{2}}{\partial t} (x, t) f (x) d t d x \\ = \int_{R^{N}} \int_{t_{1}}^{t_{2}} - f (x) {div}_{x} p_{ε} (x, t) d t d x \\ = \int_{t_{1}}^{t_{2}} \int_{R^{N}} \nabla f (x) \cdot p_{ε} (x, t) d x d t \\ ⩽ ‖ \nabla f ‖_{L^{\infty}} \int_{t_{1}}^{t_{2}} d t \int_{R^{N}} | p_{ε} (x, t) | d x \\ ⩽ C | t_{2} - t_{1} | . \end{array}$ Therefore, there exists a constant $C > 0$ such that $\begin{array}{l} {‖ \frac{| u_{ε} (x, t_{2}) |^{2}}{ε^{N}} d x - \frac{| u_{ε} (x, t_{1}) |^{2}}{ε^{N}} d x ‖}_{C^{2 *}} ⩽ C | t_{2} - t_{1} | ⩽ C T_{0} . \end{array}$ Now, from Lemma 5.2 we obtain $\begin{array}{l} m ‖ δ_{y_{ε} (t_{2})} - δ_{y_{ε} (t_{1})} ‖_{C^{2 *}} ⩽ C T_{0} + C | σ_{ε} (t) | + C | λ_{ε} (t) | + O (ε^{2}) ⩽ C (T_{0} + h / 2) + O (ε^{2}) . \end{array}$ Here we choose $T_{0}$ and then $ε_{0}$ , $h_{0}$ such that $C (T_{0} + h / 2) + O (ε^{2}) < min {m K_{0}, m K_{1} K_{0}}$ , where $K_{0}$ and $K_{1}$ are the constants defined in formula (3.2). Thus, from inequality (3.2) we get $| y_{ε} (t_{2}) - y_{ε} (t_{1}) | < K_{0}$ for every $t_{1}$ , $t_{2} \in [0, T_{ε}^{*})$ , and since $y_{ε} (0) = 0$ , this implies the claim. We now conclude the proof of lemma. By the definition of χ, it follows that $\begin{array}{l} | x (t) - y_{ε} (t) | = \frac{1}{m} | m x (t) - m y_{ε} (t) | ⩽ \frac{1}{m} | γ_{ε} (t) | + \frac{1}{m} | \int_{R^{N}} x χ (x) \frac{| u_{ε} (x, t) |^{2}}{ε^{N}} d x - m y_{ε} (t) | . \end{array}$ Notice that from the claim above and (3.4) we see that $χ (y_{ε} (t)) = 1$ for all $t \in [0, T_{ε}^{*})$ . In particular, there exists a constant $C > 0$ such that $\begin{array}{l} | x (t) - y_{ε} (t) | ⩽ C ‖ x χ ‖_{C^{2}} {‖ \frac{| u_{ε} (x, t) |^{2}}{ε^{N}} d x - m δ_{y ε} (t) ‖}_{C^{2 *}} + C | γ_{ε} (t) | . \end{array}$ Then the statement follows by Lemma 5.2. □

Using Lemmas 5.2 and 5.4 and inequality (3.2), one can prove the following result.

Lemma 5.5.

There exists a positive constant C such that $\begin{array}{l} {‖ \frac{| u_{ε} (x, t) |^{2}}{ε^{N}} d x - m δ_{x (t)} ‖}_{{(C^{2})}^{*}} + {‖ p_{ε} (x, t) d x - m ν (t) δ_{x (t)} ‖}_{{(C^{2})}^{*}} ⩽ C Ξ (t) + O (ε^{2}), \end{array}$ where $Ξ (t) : = | σ_{ε} (t) | + | λ_{ε} (t) | + | γ_{ε} (t) |$ , for all $ε \in (0, ε_{0})$ and $t \in [0, T_{ε}^{*})$ .

Proof.

The proof follows the same lines as Lemma 6.4 in [26]. □

In Lemma 5.4 we have fixed $T_{0}$ such that Proposition 5.1 and Lemmas 5.2 and 5.5 hold. With this in mind, now we give the proof of Theorem 1.6.

Proof of Theorem 1.6.

The proof follows the lines of the corresponding proof in [19, Theorem 1.1 and Lemma 3.6]. Let us give a brief sketch of the proof. First, we want to use a Gronwall inequality argument to show that $\begin{array}{l} (5.5) & | σ_{ε} (t) | + | λ_{ε} (t) | + | γ_{ε} (t) | ⩽ C (T_{0}) ε^{2} for every t \in [0, T_{ε}^{*}] . \end{array}$ Indeed, from identities (5.1), (5.2) and Lemmas 5.4 and 5.5, and repeating the steps of the proof of Lemma 3.6 in [19] we get for every $t \in [0, T_{ε}^{*}]$ , $\begin{array}{l} | \frac{d}{d t} σ_{ε} (t) | + | \frac{d}{d t} λ_{ε} (t) | + | \frac{d}{d t} γ_{ε} (t) | ⩽ C [| σ_{ε} (t) | + | λ_{ε} (t) | + | γ_{ε} (t) | + O (ε^{2})], \end{array}$ for some positive constant C. Moreover, by Lemmas 3.4 and 5.3 we see that $| σ_{ε} (0) | + | λ_{ε} (0) | + | γ_{ε} (0) | = O (ε^{2})$ , then (5.5) is a simple consequence of the Gronwall inequality. By the definition of $T_{ε}^{*}$ in formula (3.7) and due to the continuity of $σ_{ε}$ , $λ_{ε}$ and $γ_{ε}$ one gets $T_{ε}^{*} = T_{0}$ for ε small enough, $ε \in (0, ε_{0})$ . Next, in light of Proposition 5.1 there exist families of bounded functions $θ_{ε} : R \to [0, 2 π)$ , $y_{ε} : R \to R^{N}$ such that $\begin{matrix} {‖ u_{ε} (\cdot, t) - e^{\frac{1}{ε} (ν (t) \cdot x + θ_{ε} (t))} R (\frac{x - y_{ε} (t)}{ε}) ‖}_{H_{ε}^{1}}^{2} = O (ε^{2}), \end{matrix}$ for all $t \in [0, T_{0}]$ . Furthermore, from Lemma 5.4 and (5.5), it is clear that $| x (t) - y_{ε} (t) | ⩽ C ε^{2}$ for $t \in [0, T_{0}]$ and $ε \in (0, ε_{0})$ . Therefore, $\begin{matrix} {‖ R (\frac{x - y_{ε} (t)}{ε}) - R (\frac{x - x (t)}{ε}) ‖}_{H_{ε}^{1}}^{2} ⩽ C \frac{| x (t) - y_{ε} (t) |^{2}}{ε^{2}} = O (ε^{2}), \end{matrix}$ for every $t \in [0, T_{0}]$ and $ε \in (0, ε_{0})$ . Hence Theorem 1.6 holds in $[0, T_{0}]$ . Finally, taking as new data $x (T_{0})$ and $ν (T_{0})$ in system (1.3) and $\begin{matrix} u_{ε, 0} (x) : = e^{\frac{i}{ε} x \cdot ν (T_{0})} e^{\frac{1 + N}{2}} e^{- \frac{1}{ε^{2}} | x - x (T_{0}) |^{2}}, \end{matrix}$ as a new initial data in Cauchy problem (1.1), the statement is valid in $[T_{0}, 2 T_{0}]$ . Since $T_{0}$ only depends on the problem, we can achieve any finite time interval $[0, T]$ . This concludes the proof of Theorem 1.6. □

Footnotes

Acknowledgements

The first author was partially supported by CNPq/Brazil, through grant No. 152672/2016-8. The second author is member of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni and of INdAM.

References

A.H.

Ardila, Orbital stability of Gausson solutions to logarithmic Schrödinger equations, Electron. J. Differential Equations 335 (2016), 1–9.

A.H.

Ardila, Existence and stability of standing waves for nonlinear fractional Schrödinger equation with logarithmic nonlinearity, Nonlinear Analysis 155 (2017), 52–64. doi:10.1016/j.na.2017.01.006.

F.A.

Berezin and

Shubin, The Schrödinger Equation, Kluwer Acad. Publ., Dordrecht, 1991.

Bialynicki-Birula and

Mycielski, Nonlinear wave mechanics, Ann. Phys 100 (1976), 62–93. doi:10.1016/0003-4916(76)90057-9.

Blanchard and

Stubbe, Stability of ground states for nonlinear classical field theories, in: Lecture Notes in Physics, Vol. 347, Springer, Heidelberg, 1989, pp. 19–35.

Blanchard,

Stubbe and

Vázquez, On the stability of solitary waves for classical scalar fields, Ann. Inst. Henri-Poncaré, Phys. Théor. 47 (1987), 309–336.

Bronski and

Jerrard, Soliton dynamics in a potential, Math. Res. Letters 7 (2000), 329–342. doi:10.4310/MRL.2000.v7.n3.a7.

Buljan,

Siber,

Soljacic,

Schwartz,

Segev and

D.N.

Christodoulides, Incoherent white light solitons in logarithmically saturable noninstantaneous nonlinear media, Phys. Rev. E 68 (2003), Article ID 036607. doi:10.1103/PhysRevE.68.036607.

Carles and

Gallagher, Universal dynamics for the defocusing logarithmic Schrödinger equation, 2016, preprint, https://hal.archives-ouvertes.fr/hal-01398526.

10.

Cazenave, Stable solutions of the logarithmic Schrödinger equation, Nonlinear. Anal. 7 (1983), 1127–1140. doi:10.1016/0362-546X(83)90022-6.

11.

Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, Vol. 10, American Mathematical Society, Courant Institute of Mathematical Sciences, 2003.

12.

Cazenave and

Haraux, Equations d’évolution avec non-linéarité logarithmique, Ann. Fac. Sci. Toulouse Math. 2 (1980), 21–51. doi:10.5802/afst.543.

13.

Cazenave and

P.L.

Lions, Orbital stability of standing waves for some nonlinear Schrödinger equations, Comm. Math. Phys. 85 (1982), 549–561. doi:10.1007/BF01403504.

14.

d’Avenia,

Montefusco and

Squassina, On the logarithmic Schrödinger equation, Commun. Contemp. Math. 16 (2014), Article ID 1350032. doi:10.1142/S0219199713500326.

15.

d’Avenia and

Squassina, Soliton dynamics for the Schrödinger–Newton system, Math. Models Methods Appl. Sci. 24 (2014), 553–572. doi:10.1142/S0218202513500590.

16.

De Martino,

Falanga,

Godano and

Lauro, Logarithmic Schrödinger-like equation as a model for magma transport, Europhys Lett. 63 (2003), 472–475. doi:10.1209/epl/i2003-00547-6.

17.

E.F.

Hefter, Application of the nonlinear Schrödinger equation with a logarithmic inhomogeneous term to nuclear physics, Phys. Rev. 32 (1985), 1201–1204. doi:10.1103/PhysRevA.32.1201.

18.

Kato, Perturbation Theory for Linear Operators, Springer-Verlag, Berlin, 1966.

19.

Keraani, Semiclassical limit for nonlinear Schrödinger equation with potential. II, Asymptotic Anal. 47 (2006), 171–186.

20.

Le Coz, Standing waves in nonlinear Schrödinger equations, in: Analytical and Numerical Aspects of Partial Differential Equations, Walter de Gruyter, Berlin, 2009, pp. 151–192.

21.

Lieb and

Loss, Analysis, 2nd edn, Graduate Studies in Mathematics, Vol. 14, American Mathematical Society, Providence, RI, 2001.

22.

Montefusco,

Pellacci and

Squassina, Soliton dynamics for CNLS systems with potentials, Asymptot. Anal. 66 (2010), 61–86.

23.

Secchi and

Squassina, Soliton dynamics for fractional Schrödinger equation, Appl. Anal. 93 (2014), 1702–1729. doi:10.1080/00036811.2013.844793.

24.

Selvitella, Asymptotic evolution for the semiclassical nonlinear Schrödinger equation in presence of electric and magnetic fields, J. Differential Equations 245 (2008), 2566–2584. doi:10.1016/j.jde.2008.05.012.

25.

M.A.

Shubin, Partial Differential Equations VII: Spectral Theory of Differential Operators, Encyclopaedia of Mathematical Sciences, Vol. 64, Springer-Verlag, Berlin, 1994. doi:10.1007/978-3-662-06719-2_1.

26.

Squassina, Soliton dynamics for the nonlinear Schrödinger equation with magnetic field, Manuscripta Math. 130 (2009), 461–494. doi:10.1007/s00229-009-0307-y.

27.

Weinstein, Modulation stability of ground state of nonlinear Schrödinger equations, SIAM J. Math. Anal. 16 (1985), 472–491. doi:10.1137/0516034.

28.

Weinstein, Lyapunov stability of ground states of nonlinear dispersive evolution equations, Comm. Pure. Appl. Math 39 (1986), 51–68. doi:10.1002/cpa.3160390103.

29.

Zhang, Stability of standing waves for nonlinear Schrödinger equations with unbounded potentials, Z. Angew. Math. Phys. 51 (2000), 498–503. doi:10.1007/PL00001512.

30.

K.G.

Zloshchastiev, Logarithmic nonlinearity in theories of quantum gravity: Origin of time and observational consequences, Grav. Cosmol. 16 (2010), 288–297. doi:10.1134/S0202289310040067.

31.

K.G.

Zloshchastiev, Spontaneous symmetry breaking and mass generation as built-in phenomena in logarithmic nonlinear quantum theory, Acta Physica Polonica B 42 (2011), 261–292. doi:10.5506/APhysPolB.42.261.