Stability of ground states for logarithmic Schr\"{o}dinger equation with a $\delta^{\prime}$-interaction

In this paper we study the one-dimensional logarithmic Schr\"odinger equation perturbed by an attractive $\delta^{\prime}$-interaction \[ i\partial_{t}u+\partial^{2}_{x}u+ \gamma\delta^{\prime}(x)u+u\, \mbox{Log}\left|u\right|^{2}=0, \quad (x,t)\in\mathbb{R}\times\mathbb{R}, \] where $\gamma>0$. We establish the existence and uniqueness of the solutions of the associated Cauchy problem in a suitable functional framework. In the attractive $\delta^{\prime}$-interaction case, the set of the ground state is completely determined. More precisely: if $0<\gamma\leq 2$, then there is a single ground state and it is an odd function; if $\gamma>2$, then there exist two non-symmetric ground states. Finally, we show that the ground states are orbitally stable via a variational approach.

i∂tu + ∂ 2 x u + γδ ′ (x)u + u Log |u| 2 = 0, (x, t) ∈ R × R, where γ > 0. We establish the existence and uniqueness of the solutions of the associated Cauchy problem in a suitable functional framework. In the attractive δ ′ -interaction case, the set of the ground state is completely determined. More precisely: if 0 < γ ≤ 2, then there is a single ground state and it is an odd function; if γ > 2, then there exist two non-symmetric ground states. Finally, we show that the ground states are orbitally stable via a variational approach.
1. Introduction. In this paper we consider the following nonlinear Schrödinger equation with a delta prime potential: where u = u(x, t) is a complex-valued function of (x, t) ∈ R × R. The parameter γ is real; when positive, the potential is called attractive, otherwise repulsive. The formal expression −∂ 2 x − γδ ′ (x) which appear in (1) admit a precise interpretation as self-adjoint operator H γ on L 2 (R) associated with the quadratic form t γ (see [1,23]), We note that H γ can also be defined via theory of self-adjoint extensions of symmetric operators (see [6]). Moreover, the following spectral properties of H γ are known: σ ess (H γ ) = [0, ∞); if γ ≤ 0, then σ p (H γ ) = ∅; if γ > 0, then σ p (H γ ) = −4/γ 2 .
In the absence of the point interaction, this equation is applied in many branches of physics, e.g., quantum optics, nuclear physics, fluid dynamics, geophysics, plasma physics and Bose-Einstein condensation (see, e.g. [18,29] and references therein). The study of stability and instability for one-dimensional singularly perturbed nonlinear Schrödinger equations started only a few years ago and is currently regarded as one of the most interesting research topics in modern nonlinear wave theory (see, e.g. [1,2,3,4,5,7,14,15,20,24] and references therein). In particular, the nonlinear Schrödinger equation with a power nonlinearity |u| p−1 u and a δ ′ -interaction have been studied extensively. Among such works, let us mention [1,2,3,4,5].
The nonlinear Schrödinger equation (1) is formally associated with the energy functional E define by Unfortunately, due to the singularity of the logarithm at the origin, the functional fails to be finite as well of class C 1 on dom(t γ ) = H 1 (R \ {0}). Due to this loss of smoothness, it is convenient to work in a suitable Banach space endowed with a Luxemburg type norm in order to make functional E well defined and C 1 smooth. Indeed, we consider the reflexive Banach space (see Section 2 below) Then, by Proposition 3 in Section 2 we have that the energy functional E is welldefined and of class C 1 on W (Ṙ). Moreover, from Lemma 2.1, we have that the operator W (Ṙ) → W ′ (Ṙ) u → H γ u − u Log |u| 2 is continuous and bounded. Here, W ′ (Ṙ) is the dual space of W (Ṙ). Therefore, if u ∈ C(R, W (Ṙ)) ∩ C 1 (R, W ′ (Ṙ)), then equation (1) makes sense in W ′ (Ṙ). The following proposition is concerned with the well-posedness of the Cauchy problem for (1) in the energy space W (Ṙ). Proposition 1. For any u 0 ∈ W (Ṙ), there is a unique maximal solution u ∈ C(R, W (Ṙ))∩C 1 (R, W ′ (Ṙ)) of (1) such that u(0) = u 0 and sup t∈R u(t) W (Ṙ) < ∞. Furthermore, the conservation of energy and charge hold; that is, E(u(t)) = E(u 0 ) and u(t) 2 for all t ∈ R.
In this paper, we consider the case γ > 0, and study the variational structure and orbital stability of the standing wave solution of (1) of the form u(x, t) = e iωt ϕ(x) where ω ∈ R and ϕ is a real valued function which has to solve the following stationary problem More precisely, the main aim of this paper is to analyse the existence and stability of ground states for one-dimensional logarithmic Schrödinger equation (1). It is important to note that our work was inspired by the recent interesting results of R. Adami and D. Noja [3]. Indeed, the techniques used here are similar in many respects to those used in [3]. Definition 1.1. For γ > 0 and ω ∈ R, we define the following functionals of class C 1 on W (Ṙ): Note that (3) is equivalent to S ′ ω,γ (ϕ) = 0, and I ω,γ (u) = S ′ ω,γ (u), u is the so-called Nehari functional.
From the physical point viewpoint, an important role is played by the ground state solution of (3). We recall that a solution ϕ ∈ W (Ṙ) of (3) is termed as a ground state if it has some minimal action among all solutions of (3). To be more specific, we consider the minimization problem and define the set of ground states by The set u ∈ W (Ṙ) \ {0} , I ω,γ (u) = 0 is called the Nehari manifold. Notice that the above set contains all stationary point of S ω,γ . For γ > 0, the existence of minimizers for (4) is obtained through variational argument. We will show the following theorem in Section 4. Theorem 1.2. Let γ > 0. There exists a minimizer of d γ (ω) for any ω ∈ R; that is, the set of ground states N ω,γ is not empty. Remark 1. Let u ∈ N ω,γ . Then, there exists a Lagrange multiplier Λ ∈ R such that S ′ ω,γ (u) = ΛI ′ ω,γ (u). Thus, we have S ′ ω,γ (u), u = Λ I ′ ω,γ (u), u . The fact that S ′ ω,γ (u), u = I ω,γ (u) = 0 and I ′ ω,γ (u), u = −2 u 2 L 2 < 0, implies Λ = 0; that is, S ′ ω,γ (u) = 0. Therefore, u is a solution of the stationary problem (3). Before proceeding to our main results, we state the following proposition.
Proposition 2. Let γ > 0 and ω ∈ R. Any function that belongs to the set of ground states N ω,γ has the form e iθ φ t1,t2 and the couple (t 1 , t 2 ) ∈ R + × R + solves the system Notice that in order to identify the ground states we must find the solutions of system (6); this solutions will be explicitly calculated in Section 5. More precisely, for every 0 < γ ≤ 2 the system (6) has exactly one solution given by t 1 = 2γ −1 , t 2 = 2γ −1 . At γ = 2 two new solutions arise. Indeed, for γ > 2 the system (6) has exactly three solutions and one of them is given by t 1 = 2γ −1 , t 2 = 2γ −1 . See Proposition 5 for more details. Now we are ready to state our first main result. Theorem 1.3. Let γ > 0 and ω ∈ R. Then the following assertions hold.
A careful consideration of this theorem reveals the presence of a branch of ground states, that, at the critical value γ = 2, bifurcates in two branches; correspondingly, parity symmetry is broken. The occurrence of bifurcation and spontaneous symmetry breaking phenomenon in the ground state has been investigated in [19] and more recently in [22,26,16].
The next step in the study of ground states to (3) is to understand their stability. The basic symmetry associated to equation (1) is the phase-invariance (while the translation invariance does not hold due to the defect). Thus, the definition of stability takes into account only this type of symmetry and is formulated as follows.
Definition 1.4. We say that a standing wave solution u(x, t) = e iωt φ(x) of (1) is orbitally stable in W (Ṙ) if for any ǫ > 0 there exist η > 0 such that if u 0 ∈ W (Ṙ) and u 0 − ϕ W (Ṙ) < η, then the solution u(t) of (1) with u(0) = u 0 exist for all t ∈ R and satisfies sup Otherwise, the standing wave e iωt φ(x) is said to be unstable in W (Ṙ).
Our second main result shows that the ground states are orbitally stable for every ω ∈ R. The proof of Theorem 1.5 is based on the variational characterization of the stationary solutions ϕ for (3) as minimizers of the action S ω,γ on the Nehari manifold (see Theorem 1.3) and from the compactness of the minimizing sequences (see Lemma 6.1 below) for d γ (ω). We remark that nothing is known about orbital stability of the first excited state arising from the ground state from the bifurcation point, which exist for every γ > 2. It is a conjecture that excited states are unstable, but we not have a proof of this fact.

Remark 2.
A similar analysis is carried out in [7], in the case of logarithmic Schrödinger equation with attractive delta potential. Indeed, it was shown in [7] that there exists a unique positive (up to a phase) ground state and it is orbitally stable.
The rest of the paper is organized as follows. In Section 2, we analyse the structure of the energy spaceẆ (R). In Section 3, we give an idea of the proof of Proposition 1. In Section 4 we prove, by variational techniques, the existence of a minimizer for d γ (ω). In Section 5, we explicitly compute the ground states (Theorem 1.3). The Section 6 is devoted to the proof of Theorem 1.5. In the Appendix we list some properties of the Orlicz space L A (R) defined in Section 2.

LOGARITHMIC NLS EQUATION WITH
Notation. The space L 2 (R, C) will be denoted by L 2 (R) and its norm by · L 2 . This space will be endowed with the real scalar product The space H 1 (R, C) will be denoted by H 1 (R), its norm by · H 1 (R) . We write H 1 rad (R) for the space of radial (even) function on H 1 (R). The space H 1 (R\ {0} , C) is equipped with their usual real inner product, it will be denoted by Σ and its norm by · Σ . We denote by C ∞ 0 (R \ {0}) the set of C ∞ functions from R \ {0} to C with compact support. ·, · is the duality pairing between E ′ and E, where E is a Hilbert (more generally, Banach space) and E ′ is its dual. Characteristic function on R + = (0, +∞) (resp. R − = (−∞, 0)) will be denoted by χ + (resp. χ − ). Throughout this paper, the letter C will denote positive constants.

2.
Preliminaries. The purpose of this section is to describe the structure of space W (Ṙ). Also, we will show that the energy functional E is of class C 1 on W (Ṙ).
We need to introduce some notation. Define and as in [11], we define the functions A, B on [0, ∞) by Furthermore, let be functions a, b, defined by Notice that we have b(z) − a(z) = z Log |z| 2 . It follows that A is a nonnegative convex and increasing function, and A ∈ C 1 ([0, +∞)) ∩ C 2 ((0, +∞)). The Orlicz space L A (R) corresponding to A is defined by Here as usual L 1 loc (R) is the space of all locally Lebesgue integrable functions. It is proved in [11, Lemma 2.1] that A is a Young-function which is ∆ 2 -regular and L A (R), · L A is a separable reflexive Banach space.
Next, we consider the reflexive Banach space (2)). It is easy to see that  [11]).
Proof. Notice that, as usual, the operator H γ is naturally extended to H γ : On the other hand, one easily verifies that for ǫ > 0 there exist C ǫ such that which combined with Hölder inequality and Sobolev embedding gives Thus, we obtain that u → b(u) is continuous and bounded from Σ to L 2 (R), then The following proposition shows that E ∈ C 1 (W (Ṙ), R). More exactly, we obtain the following result.

derivative of E in u exists and it is given by
Proof. We first show that E is continuous. Notice that The first term in the right-hand side of (9) is continuous Σ → R, and it follows from Proposition 6(i) in Appendix that the second term is continuous L A (R) → R. Moreover, by (28) below, we get that the third term right-hand side of (9) is 3. The Cauchy problem. In this section we sketch the proof of the global wellposedness of the Cauchy Problem for (1) in the energy spaceẆ (R). The proof of Proposition 1 is an adaptation of the proof of [12, Theorem 9.3.4] (see also [7]). So, we will approximate the logarithmic nonlinearity by a smooth nonlinearity, and as a consequence we construct a sequence of global solutions of the regularized Cauchy problem in C(R, Σ) ∩ C 1 (R, Σ ′ ), then we pass to the limit using standard compactness results, extract a subsequence which converges to the solution of the limiting equation (1). Before proceeding to the proof of Proposition 1, we first need some preliminary remarks. Let us recall that Moreover, it is known that for any function u ∈ Σ there exists a unique couple of As a consequence (see [1,3]), Next, we regularize the logarithmic nonlinearity near the origin. For z ∈ C and m ∈ N, we define the functions a m and b m by For the proof of Proposition 1, we will use the following two results.
Furthermore, the conservation of charge and energy hold; that is, for all t ∈ R, Proof. We use the argument in [15, Proposition 3] and we apply Theorem 3.7.1 in [12]. First, we note that In addition, it is not difficult to show that the norm , is equivalent to the usual Σ-norm. Moreover, it is easy to see that the conditions (3.7.1), (3.7.3)-(3.7.6) in [12, Section 3.7] hold choosing r = ρ = 2, since we are in one dimensional case. Also, the condition (3.7.2) with p = 2 follows easily from the self-adjointness of A. We remark that only the case p = 2 in (3.7.2) is needed for our case since we can take r = ρ = 2. Notice that the uniqueness of solutions follows from Gronwall's lemma (see [12,Corollary 3.3.11]). Finally, from [12, Corollary 3.5.2] we see that the solution is global and uniformly bounded in Σ.
For k > 0, we introduce the Hilbert space |x| < k} and the function ζ is defined in (10). It follows in particular that the inclusion map Σ k ֒→ Σ is continuous.
Then there exists a subsequence, which we still denote by {u m } m∈N , and there exist u ∈ L ∞ (R, Σ) ∩ W 1,∞ (R, Σ ′ k ) for every k ∈ N, such that the following properties hold: Proof. We just sketch the proof since it follows the same ideas as the proof of Lemma 9.3.6 in [12]. In fact, fix k ∈ N. Note that {u m } is a bounded sequence of . Therefore, by [12, Proposition 1.1.2] there exists a subsequence, which we still denote by {u m } m∈N , and there exist u ∈ L ∞ (B k , Σ) such that u m (t) ⇀ u(t) in Σ as m → ∞ for every t ∈ B k . Thus, considering a diagonal sequence, we see that u m (t) ⇀ u(t) in Σ as m → ∞ for every t ∈ R and u ∈ L ∞ (R, Σ), and (i) follows. In addition, by [12, Remark 1.3.13(ii)] and (i), we have that u ∈ W 1,∞ (R, Σ ′ k ) for every k ∈ N. The remainder of the proof follows similarly to the remainder of the proof of [12, Lemma 9.3.6].
Proof of Proposition 1. We only discuss the modifications that are not sufficiently clear. Applying Proposition 4, we see that for every m ∈ N there exists a unique global solution u m ∈ C(R, Σ) ∩ C 1 (R, Σ ′ ) of (12), which satisfies where and thus from (12) we see that {u m } m∈N is bounded in W 1,∞ (R, Σ ′ k ) for every k ∈ N. Therefore, we have that {u m } m∈N satisfies the assumptions of Lemma 3.1. Let u be its limit. It follows from (12), that u m satisfies for every ψ ∈ Σ k and every φ ∈ C ∞ c (R). This means that It follows from property (i) of Lemma 3.1 that Next, let h m (x, t) = g m (u m )ψ(x)φ(t). One can easily see that, by the dominated convergence theorem, h m → (u Log |u| 2 )ψφ in L 1 (R × R). Moreover, using (14) we obtain Since u ∈ L ∞ (R, L A (R)) (see proof of Step 3 of [12, Theorem 9.3.4]) we see that u ∈ L ∞ (R, W (Ṙ)), so that u ∈ W 1,∞ (R, W ′ (Ṙ)) by (15) and Lemma 2.1. In particular, it follows from (15), that for all t ∈ R, In addition, u(0) = u 0 by property (i) of Lemma 3.1. Thus, we obtain that there is a solution u ∈ L ∞ (R, W (Ṙ)) ∩ W 1,∞ (R, W ′ (Ṙ)) of (1) with u(0) = u 0 . Moreover, arguing in the same way as in the proof of the Step 3 of [12, Theorem 9.3.4] we deduce that On the other hand, let u and v be two solutions of (1) in that class. On taking the difference of the two equations and taking the W (Ṙ) − W ′ (Ṙ) duality product with i(u − u), we see that Thus, from [12, Lemma 9.3.5] we obtain Therefore, the uniqueness of the solution follows by Gronwall's Lemma. In particular, by uniqueness of solution, we deduce the conservation of energy. Later statements can be proved along the same lines as in the proof of Step 4 of [12, Theorem 9.3.4]. Finally, the inclusion u ∈ C(R, W (Ṙ)) ∩ C 1 (R, W ′ (Ṙ)) follows from conservation laws.

4.
Existence of a ground state. This section is devoted to the proof of Theorem 1.2.
We have divided the proof into a sequence of lemmas. Firstly we give a lemma that extends the one-dimensional logarithmic Sobolev inequality to the space Σ.
Proof. The lemma follows immediately from the standard logarithmic Sobolev inequality on H 1 (R) (see [25,Theorem 8.14]) and the decomposition in (11).
Before stating our next lemma we recall a well-known result on the logarithmic Schrödinger equation in the absence of the delta prime potential: namely, the set of solutions of stationary problem is given by e iθ φ ω (· − y); θ ∈ R, y ∈ R (see e.g. [9, Appendix D]), where In addition, φ ω (x) is the only minimizer (modulo translation and phase) of the problem where Moreover d(ω) = e ω+1 √ π/2. For the proof of this result we refer to A. H. Ardila is given by e iθ χ + φ ω , e iθ χ − φ ω : θ ∈ R , where φ ω is defined in (21).
Proof. We use the argument in [3, Lemma 4]. First, we remark that the following variational problem is equivalent to d 0 (ω):

LOGARITHMIC NLS EQUATION WITH A δ ′ -INTERACTION 11
Arguing as in Lemma 4.2 we can show that the quantity d 1 (ω) is positive. Now, let u ∈ H 1 rad (R) ∩ L A (R) be such that I ω (χ + u) ≤ 0. Then, since u is even, we have I ω (u) = 2 I ω (χ + u) ≤ 0. Thus, from (22), we see that and I ω (χ + φ ω ) = 0. Therefore, χ + φ ω is a minimizer of L 2 (R)-norm among the functions of W (Ṙ), supported on R + and satisfying I ω ≤ 0. We observe that the equality in (24) is satisfied if and only if u(x) = φ ω (x) for all x ∈ R (modulo phase). Indeed, suppose we have the equality in (24). Since u satisfies I ω (u) ≤ 0, we have , and thus u(x) = φ ω (x − y) for some y ∈ R. Moreover, since u is even, we have that y = 0.
The proof of the following lemma can be found in [7,Lemma 4.10], and is presented here for the sake of completeness. Proof. We first recall that, by (7), |z| 2 Log |z| 2 = A(|z|) − B(|z|) for every z ∈ C. By the weak-lower semicontinuity of the L 2 (R)-norm and Fatou lemma we have u ∈ W (Ṙ). It is clear that the sequence {u n } is bounded in L A (R). Since A is convex and increasing function with A(0) = 0, it is follows from Brézis-Lieb lemma On the other hand, thanks to the continuous embedding W (Ṙ) ֒→ Σ, we have that the sequence {u n } is also bounded in Σ. An easy computations shows that the function B is convex, increasing and nonnegative with B(0) = 0. Furthermore, by Hölder and Sobolev inequalities, for any u, v ∈ Σ we have that (see [11,Lemma 1 Thus the result follows from (27) and (29).
Indeed, from W (Ṙ) ֒→ Σ we have that u n ⇀ ϕ weakly in Σ. Since in addition we have the compact embedding H 1 (0, 1) ֒→ C[0, 1], we get (30). Now, suppose that ϕ ≡ 0. Since u n satisfies I ω,γ (u n ) = 0, it follows from (30) that Define the sequence v n (x) = λ n u n (x) with where exp(x) represent the exponential function. Then, it follows from (31) that lim n→∞ λ n = 1. Moreover, an easy calculation shows that I ω (v n ) = 0 for any n ∈ N. Thus, by the definition of d 0 (ω) and Lemma 4.3 leads to that it is contrary to (26) and therefore we conclude that ϕ is nontrivial.

5.
Characterizations of ground states. The aim of this section is to prove Theorem 1.3. Some preparation is needed.
Then there exist θ ∈ R and c ∈ R such that The same conclusion can be reached if we replace R + by R − .
, ω ∈ R and v ∈ W (Ṙ) be a solution of (3). Then, v verifies the following: Proof. The proof of item (39) follow by a standard bootstrap argument using test functions ξ ∈ C ∞ 0 (R \ {0}) (see e.g. [12,Chapter 8]). Indeed, from (3) applied with ξ v we deduce that Thus, from the fact that W (Ṙ) is dense in Σ, we have Now, we recall that the form t γ is associated with the operator self-adjoint H γ . Then the theory of representation of forms by operators [27,Theorem 10.7] implies that v ∈ dom(H γ ); that is, v satisfies the two conditions in (41). Finally, the proof of (42) is contained in Lemma 5.1. This concludes the proof.
Proof of Proposition 2. Let u ∈ N ω,γ . By Remark 1, we see that u satisfies the stationary problem (3). From Lemma 5.2 and the characterization give by Lemma 5.1, we see that all possible solutions to (3) must be given by where θ 1 , θ 2 ∈ R and the couple (t 1 , t 2 ) ∈ R × R. Notice that, by Lemma 5.2, the solution v must satisfy the boundary conditions (41). Now, since u is a minimizer of S ω,γ under the constraint I ω,γ (u) = 0, we have that e iθ1 = −e iθ2 . Indeed, once fixed t 1 and t 2 , it is clear that such condition minimizes the quadratic form t γ , while the other terms in the functional S ω,γ are the same. This explains the negative sign in (5). Taking into account the phase invariance of the problem we can choose θ 1 = 0 and θ 2 = π.
Finally, from (41) and (45), the boundary conditions for v can be converted to the system (6) for the unknowns θ 1 and θ 1 . We remark that, by the first equation of the system (6), t 1 and t 1 must have the same sign. Thus, since γ > 0, by the second equation of the system (6), we have that t 1 > 0 and t 2 > 0. This concludes the proof.
In order to identify the ground states we must find the solutions of system (6). As a first step in that direction we have the following lemma.
has exactly one zero in the interval (1, ∞).
(ii) Let γ > 2. Then the system (6) has three solutions; one of them is given by Proof. We use the argument in [3,Theorem 5.3]. We set f (t) = te − 1 2 t 2 for all t ≥ 0. It is clear that the first equation of the system (6) is equivalent to f (t 1 ) = f (t 2 ). Notice that f (0) = 0, f (t) > 0 for all t > 0, and f (t) → 0 as t → ∞. Moreover, f has a unique critical point at t = 1, which is a maximum.
It is not hard to see that the set of the solutions of the first equation in (6) with t 1 ≤ t 2 consists of the union of the following curves: We remark that due to the regularity of f , I 2 is a regular curve; the curve I 1 ∪ I 2 is given in Figure 1.
Next, for each γ positive the second equation of system (6) is a hyperbola in the plane (t 1 , t 2 ). Thus, the solution of the system of equations (6) is the intersection of these hyperbolas with I 1 ∪ I 2 . In order to find all points of intersection of the two curves, it is convenient to prove that inf t1,t2∈I1 inf t1,t2∈I2 To show this, we use the Lagrange multiplier method. We note that (46) is obvious. z 0 in the interval (1, +∞). Thus, (z 0 + 1)γ −1 , z −1 0 (z 0 + 1)γ −1 is a solution to (6). Due to the symmetry of (6) under change of t 1 and t 2 , the third and last solution is given by z −1 0 (z 0 + 1)γ −1 , (z 0 + 1)γ −1 . This concludes the proof of (ii).
6. Stability of the ground states. The proof of Theorem 1.5 relies on the following compactness result. Lemma 6.1. Let {u n } ⊆ W (Ṙ) be a minimizing sequence for d γ (ω). Then, up to a subsequence, there exist ϕ ∈ N ω,γ such that u n → ϕ strongly in W (Ṙ).
Furthermore, tanks to (33), we have u n → ϕ in L 2 (R). Then, since the sequence {u n } is bounded in Σ, from (28) Since u n ⇀ ϕ weakly in Σ, it follows from (56) and (58) that u n → ϕ in Σ. Finally, by Proposition 6-ii) (Appendix below) and (59) we have u n → ϕ in L A (R). Thus, by definition of the W (Ṙ)-norm, we infer that u n → ϕ in W (Ṙ). Which concludes the proof.
Proof of Theorem 1.5. Our proof is inspired by the results of [2,7]. The proof of part (i) in theorem, the stability of the ground state φ t * ,t * ω for 0 < γ ≤ 2, follows along the same lines as [7, Theorem 1.2]. We omit the details.
Next we prove (ii) of theorem. Fix γ > 2. Now arguing by contradiction and suppose that e iωt φ t1,t2 ω is not stable in W (Ṙ), then there exist ǫ > 0, a sequence (u n,0 ) n∈N such that and a sequence (τ n ) n∈N such that inf θ∈R u n (τ n ) − e iθ φ t1,t2 where u n denotes the solution of the Cauchy problem (1) with initial data u n,0 . With no loss of generality, we assume Set v n (x) = u n (x, τ n ). By (60) and conservation laws, as n → ∞, In particular, it follows from (63) and (64) that, as n → ∞, Moreover, combining (63)