Corrigendum: Stationary problem related to the nonlinear Schr ö dinger equation on the unit ball (2012 Nonlinearity 25 2271 – 301)

In this paper, we study the stability of standing waves for the nonlinear Schr¨odinger equation on the unit ball in R N with Dirichlet boundary condition. We generalize the result of Fibich and Merle (2001 Physica D 155 132–58), which proves the orbital stability of the least-energy solution with the cubic power nonlinearity in two space dimension. We also obtain several results concerning the excited states in one space dimension. Speciﬁcally, we show the linear stability of the ﬁrst three excited states and we give a proof of the orbital stability of the k th excited state, restricting ourselves to the perturbation of the same symmetry as the k th excited state. Finally, our numerical simulations on the stability of the k th excited state are presented.

The statement of proposition 9 (i) in [2] is incorrect. The space (1) ( ) ∈ − x 1, 1 . Here, a(n) = 0 if n is odd, a(n) = −1 if n is even, and χ is a characteristic function, i.e. for any Borel set R ⊂ A , A Then the statement of proposition 9 (i) in [2] is corrected as follows.
A correct version of proposition 9 (i) in [2]. Let any N ∈ k be fixed. The k-th standing , taking the corresponding ϕ, e.g. as the ground state ( )  (1) is in fact coincident with (12) of [2].
The error arises in the proof of the well-posedness of Cauchy problem in section 6, page 2290 of [2]. In this corrigendum, we give a precise proof of the local well-posedness in X k . We note that even if we replace H k 0, 1 by X k in section 6 of [2], the statement of proposition 26 and the proof of proposition 9 (i) in [2] still work without any other modification: in fact, the argument in the above Remark applies similarly to all eigenfunctions of the linearized operator, i.e. each eigenfunction, which is unique, may be written in the form of (1), thus belongs to X k . The computation of the L 2 norm of ω Q k, does not change thanks to the above Remark. Thus, we obtain the stability of the k-th standing wave ( ) be an open, bounded interval. Consider, for p > 1, Remark that the local well-posedness of (2) in ( ) H I 0 1 is established in theorems 3.3.5 and 3.3.9 of [1]. For the later use, we introduce the result here. Before showing proposition 2, let us explain why the symmetry defined in X k is conserved by the flow of (2). First, we introduce some properties of the gluing operator G whose verification can be found at the end of this corrigendum.
Using lemma 1, we may check the following fact.

Lemma 2.
Let N ∈ k and p > 1 be fixed. Let be a solution of (2) with I = (0, 1) for some T > 0. Then is of class and satisfies By the definition of X k , for ∈ u X k 0 , there exists ( ) ϕ ∈ H 0, 1 . Lemma 2 implies that if ϕ is smooth enough, and if the evolution starting from ϕ satisfies (2) with I = (0, 1) and remains smooth, the symmetry of X k is conserved through the evolution (2) with I = (−1, 1) starting from u 0 . Thus, we need a regularization procedure to apply lemma 2. By density, for Applying lemma 2 to ( ) where v m is constructed in corollary 1, we see that G[v m ] is a solution of (2) in H −1 (−1, 1). It thus suffices to show the conver- where v m and v are the corresponding solutions to ϕ m and ϕ respectively. Also, by the As a consequence, for any Taking the limit → ∞ m , by the dominated convergence theorem, But here we know already , and thus this equa- In conclusion, for any ∈ u X k 0 , there exists a so- with u(0) = u 0 . On the other side, by proposition 1, the solutions of (2) is unique in It is not difficult to see that this is a maximal time in the sense that We will give a proof of lemmas 1 and 2 hereafter. We use the notation f g f x g x x , d, Proof of lemma 1. By density, it is enough to check (3) and (4) for n k a n k w x x a n w n k a n w n k k x n k a n k w x x k k x n k a n k w x x k k x n k a n k x w x x 2 , 2 ,  for any n = 1, ..., k. Here, we have used the integration by parts in the fourth equality. This implies (3). Above computation implies that for any Again, by integration by parts, the last quantity is calculated as follows.
x w x x k a n w n k a n w n k k k x n k a n k w x x k k x n k a n k w x x   2  1  2  1  2  1  2  d   2  1  1  1  2  1  1  2  1   1  2  1  2  1  2  1  2  d   2  1  1  2  1  2  1 − a n w n k a n w n k w a k w

Proof. For any
n k a n a n XX p n In particular,

Introduction and main result
We consider the nonlinear Schrödinger equation on the unit ball in R N with Dirichlet boundary condition: where u = u(t, x) is an unknown complex-valued function for t ∈ R, x ∈ B 1 = {x ∈ R N | |x| < 1}. We always assume N 1, and 1 < p 2 * − 1. Here 2 * is the usual Sobolev critical exponent, that is, 2 * = 2N/(N − 2) if N 3 and 2 * = ∞ if N = 1 or 2. The nonlinear Schrödinger equation models the propagation of a laser beam in an optical fibre (see, e.g., Agrawal [1]). Equation (1) on the unit ball with Dirichlet boundary condition, with N = 2 and p = 3, describes the propagation of the laser beam in the hollow-core fibre. A hollow waveguide filled with a noble gas is used as an effective technique for extending the interaction length between nonlinear optical materials and high-energy laser pulses.
In this technique, the electric field outside the core is negligible because the reflective index difference between the hollow-core and the cladding is sufficiently large, and almost all of the laser beam reflects on the interface between the core and the cladding. This implies the Dirichlet boundary condition (see [19,20,30,50] for details). Note that the use of a bulk medium, whose model corresponds to equation (1) in R N , causes the self-focusing of the laser beam easier than the case of the hollow-core fibre. The authors in [39,40] have found some novel self-focusing dynamics due to the boundary condition, which do not occur in a bulk medium.
In this paper we are interested in the influence of this boundary condition on the standing wave solutions. A standing wave solution is a solution of equation (1) with the form u(t, x) = e iωt Q ω (x), where ω ∈ R and Q ω is a solution of the stationary problem: Few attempts to understand the influence of the boundary have been made except for Fibich and Merle [20]. The authors [20] found that the boundary has a stabilizing effect by showing that in the case where N = 2 and p = 3, there exist stable least-energy solutions although all least-energy solutions in R N are unstable. In this paper, we investigate how the boundary has an effect on the stability, and will find some stabilizing effects for more general power p and general dimension N.
Moreover, we discuss the stability of the excited states, which have never been studied well so far. Surprisingly, we will see that there are some cases where the excited states are stable due to the boundary condition, while the excited states in R N are believed to be unstable in any case (see, e.g., [51, sections 3 and 4]).
We introduce several notations. Let be a domain in R N and K represents a field. In this paper we will take K for R or C. We denote by L 2 ( , K) the space of all K-valued L 2 -functions on with the scalar product for u, v ∈ L 2 ( , K). We denote by C ∞ 0 ( , K) the function space of K-valued infinitely many differentiable functions with compact support in . We define a function space H 1 0 ( , K) as a closure of C ∞ 0 ( , K) with the norm We denote by H −1 ( , K) the dual space of H 1 0 ( , K). The duality pairing between H −1 ( , K) and H 1 0 ( , K) is denoted by u, v for u ∈ H −1 ( , K) and v ∈ H 1 ( , K). If there is no fear of confusion, we write H 1 0 ( ) or L 2 ( ), for short. For the sake of definiteness, here we consider only the case of = B 1 = {x ∈ R N ||x| 1}. We may generalize some parts of our results to more general domains (see remark 6).
Throughout this paper, we will assume the local well-posedness of the Cauchy problem for equation (1) in the energy space H 1 0 (B 1 , C). (1) with u(0) = u 0 . Moreover, the solution satisfies the following conservation laws:
The stability of standing waves is defined as follows.

Definition 2.
We say that the standing waves e iωt Q ω of equation (1)

are orbitally stable in
Otherwise, e iωt Q ω is said to be orbitally unstable. In particular, e iωt Q ω is said to be strongly unstable if there exists an initial datum It is well-known that the least-energy solutions in R N are stable when 1 < p < 1 + 4/N and strongly unstable when 1 + 4/N p < 2 * − 1 (see [11] for the stability and [5,52] for the strong instability).
Concerning the stability of the least-energy solution on the unit ball in R N , we obtain the following results.

Theorem 3. Let N
1 and Q ω be the least-energy solution of equation (2). Suppose that assumption (A) holds.
The case where N = 2 and p = 3 has already been investigated by Fibich and Merle [20, lemma 10]. The authors proved that the least-energy solution is orbitally stable for sufficiently large ω > 0, or ω sufficiently close to −λ 1 . We generalize this result to more general nonlinearity powers, adding some detailed arguments. We also remove the condition on the frequency ω and prove the orbital stability for all ω when 1 < p 1 + 4/N, in case of N = 1. More precisely, we have the following theorem.  (2). Then, the standing wave e iωt Q ω is orbitally stable for all ω ∈ (−λ 1 , ∞).
To prove theorems 3 and 4, we invoke the general theory in [28, theorem 3], which gives a sufficient condition for the orbital stability of solitary waves to the abstract Hamiltonian system. If we apply it to our equation, linearizing around the least-energy solution Q ω , that sufficient condition may be interpreted as follows.

the least-energy solution of equation (2). Suppose that assumption (A) holds. Assume that (i) the positive spectrum of S ω (Q ω ) is bounded away from zero; (ii) the kernel is spanned by
(iii) the operator S ω (Q ω ) has only one negative simple eigenvalue.
All spectra of the operator S ω (Q ω ) are discrete, and a modification of the proof in [31, theorem 0.2] shows that the linearized operator S ω (Q ω ) satisfies the properties (ii) and (iii).
Accordingly, in order to show the orbital stability, it is enough to investigate the slope condition, i.e. the sign of ∂ ω Q ω [45, theorem 18] for the proof).
We see that v ω ∈ H 1 0 (B 1 ) satisfies the following equation : In the case of equation (1) in R N , the slope condition is explicitly written as a function of ω, making use of the scale invariance of equation (1). However, we cannot expect such a scaling property in the present case, and thus we try to obtain the information by a scaling limit in ω of solutions. In fact, this idea was already used in the works by Esteban and Strauss [18], the first author and Ohta [22,23], Fibich and Merle [20] to analyse the orbital stability for some equations that do not possess the scaling invariance.
We rescale the solutions as follows.
ThenQ ω andṽ ω satisfy the following equations respectively: We extend both functionsQ ω andṽ ω to the entire space by definingQ In fact, as in [22,23], the asymptotics of Q ω (orQ ω ) with respect to ω is enough for the study of the case p = 1 + 4/N . However, in the case where p = 1 + 4/N , we need more details about the asymptotics. With this aim, we use the second approximation v ω . As a side effect, we can give a simpler proof than those of [22,23] also for the case of p = 1 + 4/N .
We briefly mention how to check the slope condition if p = 1 + 4/N. Note that by the Pohozaev identity, we have E(Q ω ) = |∂ r Q ω (1)| 2 /4 when p = 1 + 4/N , as was seen in [20]. Moreover, in this paper, we use the relation ω∂ ω Q ω Since ∂ r Q ω (1) < 0, our interest is attracted towards the sign of ∂ r v ω (1). If N = 1, we can analyse the behaviour of v ω by an ODE approach, more precisely than the case of N 2, which yields theorem 4. Remark 6. Theorem 3 would be applicable for more general bounded domains , so long as it is smooth and star-shaped.
Next, we turn our attention to the excited states in one space dimension. For k ∈ N\{0} let λ k ∈ R and χ k ∈ H 1 0 ((−1, 1), R) be the kth eigenvalue and kth eigenfunction of the operator −d 2 /dx 2 in (−1, 1). Then we can write λ k ∈ R and χ k ∈ H 1 0 ((−1, 1), R) explicitly as follows λ k = (kπ ) 2 /4 and χ k = sin(kπ(x − 1)/2). Moreover, it is known that the eigenvalue λ k is simple for each k ∈ N \{0}. Therefore, we can apply the local bifurcation theorem by Crandall and Rabinowitz [14, theorem 1.7] and we see that each k ∈ N \ {0} there exists s 0 > 0 and the unique solution Q k,ω(s) ∈ H 1 ((−1, 1), R) of equation (2) such that (12) below that Q k,ω is an even function if k is odd and an odd function if k is even for all ω > −λ k . Note that the least-energy solution Q 1,ω has been denoted by Q ω in the above theorems.
The linearized equation of (1) around the kth standing wave e iω(s)t Q k,ω(s) is written as follows.

Remark 8.
If there exists λ ∈ σ (J L k,ω(s) ) such that Re λ = 0, then we see that the standing wave e iω(s)t Q k,ω(s) is orbitally unstable by the argument in [27, appendix]. When k 5, it would be possible that some linearly unstable bound states exist. This number k 5 is derived from the appearance of some eigenvalues with high geometric multiplicity (see remark 24). On the other hand, in the case of the cubic nonlinear Schrödinger equation with a harmonic potential in R where we have a 'similar' spectrum distribution, the excited states for k = 3 and 4 are already observed to be linearly unstable in [55]. This difference arises in the fact that the eigenvalue distribution is equidistant in the case of harmonic potential, and as a result an eigenvalue with high geometric multiplicity takes place for k 2. The stability issue for k = 2, benefiting from an additional symmetry of harmonic potential, is also discussed in [33].
Note that the linear stability (theorem 7 in the case of k = 2, 3, 4) does not imply the nonlinear (orbital) stability. However, we can see rigorously that the kth excited state e iωt Q k,ω is orbitally stable if we restrict ourselves to the perturbation of the same symmetry as Q k,ω for any k ∈ N, under certain conditions on ω and p. A similar result but only for the first excited state is obtained by Kurth [34] for the nonlinear Schrödinger equation with the harmonic potential. Proposition 9. Let N = 1. Then the kth (k 2) bound states Q k,ω (x) of equation (2) can be written as follows. For ω > −λ k , Here, we extend the least-energy solution (ii) Let N = 1 and p > 5. Then the kth (k 1) standing wave e iωt Q k,ω is orbitally unstable for sufficiently large ω > 0.
As a result of proposition 9 we can say that the stability of the kth excited state is due to the Dirichlet boundary condition compared with R N -case even if it is limited to the H 1 0,k -symmetric perturbation. However, the stability of the excited states under non-H 1 0,ksymmetric perturbation is not completely covered yet. Thus, using a numerical simulation, we examine the orbital stability of the excited states Q 2,ω and Q 3,ω more closely.
Our numerical approach is similar to those in [21,37]. We numerically observe the profile of the solution of equation (1) with the following initial conditions: for each k = 2, 3, where δ p , δ c > 0 and θ ∈ C ∞ (−1, 1) is defined by The positive constant δ p is used to compute the uniform multiplicative perturbation (14) of the excited states Q k,ω , perturbs the power, that is, L 2 -norm of the excited state. It preserves its symmetry with respect to x = 0. On the other hand, the positive constant δ c perturbs the centre of the standing waves while it preserves the L 2 -norm. This analysis provides not only the stability property but also the information on the character of the dynamics in the stable and unstable cases.
Here, we state briefly what our simulations suggest.
It follows from observations (i) and (iii) that the nature of the instability in proposition 9 (ii) would be in the sense of blow-up. Observation (ii) does not give any information about the case of large frequencies ω > 0. This is because our simulation does not indicate clearly whether Q k,ω remains orbitally unstable or not for very large ω (for the details see section 7.2.2). According to observation (iii), even if p 5 and under the non-symmetric perturbation (15), the excited states are stable for ω sufficiently close to −λ k . Remark 10. The fact that there are some cases where the excited states are stable was weakly recognized in section 7.1 of Fibich and Merle [20]. Precisely, letting N c = R 1 2 L 2 , the authors gave a remark that the excited states whose power is sufficiently below N c appear to be numerically stable for quite a long time, and those whose power is above N c are strongly unstable. We give more precise observations and in the case where p = 5 and N = 1, we find that there exists the unique thresholdω c,k which separate the stable excited states from the unstable excited states. Our numerical calculation indicates thatω c,2 = −8.2 is different from the frequency (=−4.8) of the excited state having the squared L 2 norm whose value equals to N c . However, we have to be careful with the situation such that we are in one-dimensional case, and the thresholdω c,k arises only by the non-symmetric perturbation (15) unlikely to the case of Fibich and Merle [20]. This paper is organized as follows. Rigorous discussions will be found from section 2 to section 6. Section 7 is dedicated to the observations by numerical simulations: in section 2, we study the asymptotic behaviour of the least-energy solution Q ω and its derivative v ω with respect to ω. Using the convergence properties of Q ω and v ω proved in section 2, we investigate the slope condition and show the stability of the least-energy solution for all ω which are very large or sufficiently close to −λ 1 in general dimension N 1 in section 3. This stability result will be extended for any ω for the case N = 1 in section 4. Section 5 is devoted to the theoretical study on the linear stability of some excited states. In section 6, we show rigorously that the k-excited state e iωt Q k,ω is orbitally stable if we restrict ourselves to H 1 0,k -perturbation. The (nonlinear) stability of the excited states is observed by numerical simulations in section 7. Numerically, we study the first and second excited states in one space dimension in sections 7.2 and 7.3. Following [21,37], the profiles of the solutions are investigated in each case classifying the instability type and the dynamics of the stability.

Convergence properties
In this section, we show several convergence properties ofQ ω andṽ ω . Recall that we consider only the least-energy solution solution Q ω in this section, andQ ω andṽ ω are defined in (4).
Moreover, it is well-known that R 1 ∈ H 1 (R N ) is the unique minimizer for the following minimization problem: . It is easily seen that Thus, by the variational characterization of R 1 , Now, we use a cut-off function η ∈ C ∞ 0 (R N ) such that and satisfies |∇η(x)| 4 for all x ∈ R N . Then we set η ω (x) = η(x/ √ ω). It follows that for any η > 1, We see that (µ 2 − µ p+1 ) R 1 p+1 L p+1 (R N ) < 0 and the other terms vanish if ω goes to infinity. This yields thatK ω (µη ω R 1 ) < 0 for sufficiently large ω > 0. It follows from the variational characterization ofQ ω that, for any η > 1, Using this, together with (17) On the other hand, we know that lim ω→∞ K ∞ (Q ω ) = 0 by (16). Let {ω j } ⊂ R + be any sequence such that ω j goes to ∞ as j goes to ∞. Thus, we have This yields that {Q ω j } ⊂ H 1 (R N ) is a minimizing sequence for d ∞ . Then by a standard variational argument, there exists a subsequence of {Q ω j } (we still denote it by the same letters) such thatQ ω j converges to R 1 strongly in H 1 (R N ) as ω j goes to ∞. This completes the proof.
Next, we give a proof of proposition 11 (ii). We prepare the following lemma, which will be needed later. Lemma 13. Let N 1 and 1 < p < 2 * − 1. There exist constants ω 0 > 0 and C 0 > 0 such that for any ω ∈ (ω 0 , ∞) we have Here the constant C 0 is independent of ω.
Proof. We show this lemma by contradiction. Suppose that (18) would not hold. Then, there exist sequences {ω j } ⊂ R + and {g j } ⊂ H 2 0,rad (B √ ω j ) such that ω j → ∞, and as j goes to infinity. We extend g j ∈ H 2 0,rad (B √ ω j ) to the entire space by defining g j ( from proposition 11 (i). This yields, using the elliptic bootstrap argument, that g ∞ ∈ Ker L ∞ | H 2 rad (R N ) . Hence it is seen that g ∞ = 0 since Ker L ∞ | H 2 rad (R N ) = {0} (see [38,  L ω j g j , g j + p Q p−1 ω j g 2 j dx. (21) SinceQ ω j converges to R 1 strongly in H 1 (R N ) andg j converges to 0 weakly in H 1 (R N ) as j goes ∞, we have lim j →∞ Q p−1 ω j g 2 j dx = 0. Therefore, the right-hand side of (21) tends to zero when j goes to ∞ because of (19), which is absurd. Thus, (18) holds.
We now give a proof for proposition 11 (ii). (6), proposition 11 (i) and lemma 13 that there exists a constant C 0 > 0 such that

Proof of proposition 11 (ii). It follows from equation
for sufficiently large ω > 0. Let {ω j } ⊂ R + such that ω j goes to ∞ as j goes to ∞. Since {ṽ ω j } is bounded, there exist a subsequence of {ṽ ω j } (we still denote it by the same letters) and V ∈ H 1 (R N ) such thatṽ ω j converges to V weakly in H 1 (R N ) as j goes to ∞. Suppose that V = 0. Let ϕ ∈ C ∞ 0 (R N ) be a real-valued, positive function on R N . Multiplying equation (3) by ϕ and integrating the resulting equation, we have By proposition 11 (i), we see that − R NQω j ϕ dx → − R N R 1 ϕ dx < 0 as j → ∞. On the other hand, since V = 0, we have where , the solution of equation (22) is unique in H 2 rad (R N ), and it is not difficult to check that R 1 /(p − 1) + x · ∇R 1 /2 ∈ H 2 rad (R N ) satisfies equation (22). Thus, we find that V = R 1 /(p − 1) + x · ∇R 1 /2. Moreover, we have, multiplying equation (6) byṽ ω j , and letting j go to ∞, by (22). This yields thatṽ ω j converges to V = R 1 /(p − 1) + x · ∇R 1 /2 strongly in H 1 (R N ) as j goes to ∞. This completes the proof.
Next, we consider the case where the frequency ω is sufficiently close to −λ 1 . Using a similar argument in [23, lemma 4.3], we obtain the following lemma.
be the least-energy solution of equation (2). Then the following assertion holds: In [23], the authors gave a proof only for the Sobolev subcritical case 1 < p < 2 * − 1, but we can extend it to the Sobolev critical case p = 2 * − 1 without any modification.
In order to prove proposition 12, we need the following lemma, which is straightforward from the standard elliptic regularity argument (see, e.g., [48, p 269]).

Calculation of ∂ ω Q ω 2
L 2 (B 1 ) for N 1 In this section, we calculate ∂ ω Q ω 2 L 2 (B 1 ) for general dimensions. First, we consider the case where p = 1 + 4/N and ω > 0 is large, or ω is sufficiently close to −λ 1 and for any p. Theorem 17. Let N 1, p > 1 and Q ω be the least-energy solution of equation (2). Then we have the following.
We begin by showing the following lemma.
Then we see that v ω * (r) > 0 if r > 0 is sufficiently close to 1.

Calculation of ∂ ω Q ω 2 L 2 (B 1 ) for N = 1
This section is devoted to show the following: We prepare several lemmata, which will be needed later.
(ii) Suppose that the continuous function v ω has a zero in (0, 1). Then one of the following can occur: (Case 1) v ω changes the sign more than two times.
(Case 2) v ω changes the sign exactly one times.
(Case 3) v ω does not change sign.
In case 2, it suffices to show that ∂ r v ω (1) = 0 since v ω (0) > 0. The fact ∂ r v ω (1) = 0 immediately follows from the proof of lemma 19. Next, we show that case 3 does not occur. Suppose that case 3 occurs and let r 0 ∈ (0, 1) be the zero of v ω . Then since v ω is non-negative, we see that ∂ r v ω (r 0 ) = 0. It follows from (30) that This is a contradiction.
We continue to investigate some properties of v ω , but from now on, as a function of ω. Let us recall the results of proposition 12 (ii), and we define ω * as follows. −λ 1 , ω), then v ω (r) > 0 for all r ∈ (0, 1) .

Linear stability of the excited states
Following section 6.2 of [15], we give a proof of theorem 7. For this purpose, we consider the space which is identified with the space of C 2 -valued, square integrable functions, endowed with the inner product Here u, v is defined by We write the operator J L k,ω(s) as follows We set, for each k ∈ N, and ξ ± n,k = ±(λ n − λ k )i = ±(n 2 − k 2 )π 2 i/4, n = 1, 2, . . .. It may be seen that the spectrum of J H k consists only of the discrete eigenvalues, and the set of spectrum is expressed as σ (J H k ) = {ξ ± n,k | n ∈ N}. Moreover, if k 4, each eigenvalue ξ ± n,k for n = k is simple and the corresponding eigenfunction is given by and Remark 24. If k 5, we have at least, except n = k, two eigenvalues whose geometric multiplicities are two. Indeed, if we look for a pair (n 1 , n 2 ), n 1 , n 2 ∈ N satisfying the condition for high geometric multiplicity n 2 1 + n 2 2 = 2k 2 , n 1 = k, n 2 = k for fixed k ∈ N, we have only one pair for k 4, on the other hand we have more than three pairs for k 5, which cause the eigenvalues with geometric multiplicities.
Assume k 4. In order to consider the spectrum of the linearized operator J L k,ω(s) for ω sufficiently close to −λ k , we choose the following contour: and the projection ± n,k (J L k,ω(s) ) = Then we have the following lemma.
For any z ∈ ρ(J L k,ω(s) ) ∩ ρ(J H k ), using the Neumann series, we have By the spectrum theorem, we have Moreover, as in [15], for ω(s) sufficiently close to −λ k , since Q k,ω(s) L ∞ vanishes when ω tends to −λ k by (9) and Sobolev embedding. Hence, we obtain From lemma 25, we see that Proof of theorem 7. All eigenvalues of J H k are discrete, and on the imaginary axis. They are simple except the zero eigenvalue. It is known (see [27, has an eigenvalue with positive real part, then it has another eigenvalue being symmetric with respect to the real and imaginary axes. Hence, the perturbed eigenvalues for n = k should be on the imaginary axis, otherwise a contradiction to lemma 25 occurs. We may treat the case where n = k similarly. This completes the proof.

Proposition 26.
Let Q k,ω ∈ H 1 0,k be the kth bound state of equation (2). Let T = S ω (Q k,ω )| H 1 0,k . Assume that (i) the positive spectrum of T is bounded away from zero; (ii) the kernel is spanned by iQ k,ω , that is, Ker T = Span{iQ k,ω }; (iii) the operator T has exactly one negative simple eigenvalue.
Using proposition 26, we give a proof of proposition 9.
Proof of (i) of proposition 9. We first show the explicit form of (12). It was shown in [4, theorem 7] and [49, remark 3.1] that the kth bound state Q k,ω of equation (2) has (k − 1)zeros in (−1, 1) and is unique up to phase. In addition, we can easily see that the function satisfies equation (2) and has (k − 1)-zeros in (−1, 1) (see [13, section 1.2]). Therefore, (12) holds. Next, we shall show the stability in H 1 0,k (−1, 1). Recall that all spectra of the operator T are discrete. Moreover, it follows from [4, lemma 8] that Ker L + k,ω = {0} for all ω > −λ k . Moreover, we can easily see that the all eigenvalues of the operator L − k,ω are simple and L − k,ω Q k,ω = 0. Thus, we infer that the operator T satisfies the assumptions (i) and (ii) in proposition 26.
Finally, let us explain how we check the slope condition ∂ ω Q k,ω 2 L 2 > 0. It follows from the form (12) that for all k 2. Then by the result of theorem 17 (ii) and proposition 20, we see that Thus, (i) of proposition 9 holds.
Proof of (ii) of proposition 9. It follows from theorem 17 (i) that Then by a similar way in [27, theorem 3.2], we see that kth standing wave e iωt Q k,ω is linearly unstable, that is, there exists λ ∈ σ (J L k,ω ) such that Re λ = 0. This implies that the standing wave e iωt Q k,ω is also orbitally unstable from the argument in [27, appendix].

Numerical methods
In this section, numerically, we investigate the case of higher-order excited states for which few results are known not only from a theoretical point of view but also a numerical point of view (see [2,20,21,32,34,55]). Precisely, we treat the case of the first and second excited states Q 2,ω and Q 3,ω in one space dimension for the purpose of analysing the stability in the cases that are not proved in proposition 9.
Our approach here is similar to those in [21,37]. We solve equation (1) in one space dimension numerically using the symmetric Crank-Nicolson scheme with the initial datum (14) and (15).
First, we briefly explain the scheme for equation (1). We recall N = 1. Let t and x, respectively, stand for the time and space steps. We look for the approximate value, say u n j , of the solution at t n = n t and the x j = j x. Then, equation (1) can be discretized as follows Moreover, we add the condition u n j = 0 for all n 0 and j such that |j x| 1. This assumption ensures that the computed solution satisfies the Dirichlet boundary condition.
We employ this symmetric nonlinear Crank-Nicolson scheme when studying the stability of the standing waves even if it requires to perform a nonlinear algebraic inversion at each step, which we use a fixed-point algorithm (see [9, p 674]). The above form of the discretization of the nonlinear term ensures that the discrete L 2 -norm and the discrete energy are conserved as in the continuous case. This is an important property because the stability analysis is related to these conservation laws as seen in the previous sections. When we consider the solution with the initial datum (14), we make an additional procedure in the scheme in order to obtain a better approximation. Note that the initial datum u 0,p = (1+δ p )Q k,ω belongs to the function space H 1 0,k (−1, 1) for each k ∈ N and we can prove theoretically that the Cauchy problem for equation (1) is locally well-posed in H 1 0,k (−1, 1). This yields that the position of zeros of the solution u p preserves. Therefore, we may force the approximate solution to satisfy for n = 1, 2, . . . , k at each time step. Next, let us explain how we seek an approximate solution of equation (2) in one space dimension. We first consider the least-energy solution. Since the positive solution of equation (2) is radially symmetric, we can reduce equation (2) to the following ordinary differential equation: We note that equation (37) can be written as a general first-order system of the form with X(r) = (Q(r), Q (r)) ∈ R 2 . Here, F (r, X(r)) = (Q (r), ωQ(r) − |Q(r)| p−1 Q(r)).
Therefore, for each β > 0, we can solve equation (37) with the initial datum (38) using the fourth-order Runge-Kutta method. Then it is enough to choose β > 0 so that the obtained solution satisfies the boundary condition (39). This can be solved by employing the well-known shooting method (see, e.g., [16]). Consequently, we can find the profile of the least-energy solution, which enables us to check the slope condition. Once we obtain the profile of the least-energy solution, we can find that of the kth excited state from (12).
In order to check the agreement of the numerics with the rigorous stability theory, we need to observe that u(t, .) − Q k,ω H 1 remains 'small' or not. However, an increase in the norm u(t, .) − Q k,ω H 1 does not ensure us to know the dynamics of solution u. Therefore, instead of investigating the H 1 -norm, we plot the time evolution of the maximal amplitude and the L 2 -norm of the solution. In addition to the stability of standing waves, these two quantities provide us information about the dynamics of the solution u.

Stability of the first excited state
We now study the stability of the first excited state Q 2,ω in one space dimension. We recall that it is observed in [2, section 3.1.2] and [32, section 3.1] that the first excited state e iωt Q 2,ω is linearly stable for ω ∈ (−3, 5) in the case of the cubic nonlinear Schrödinger equation with a harmonic potential, but with a different numerical approach. Here, we study the problem of the orbital (nonlinear) stability to investigate the profile of the solution of equation (1) with the initial data (14) and (15) in case of k = 2, using the same approach as in [21]. We examine the general case p > 1 and ω > −λ 2 . (14). By proposition 9, we know that e iωt Q 2,ω is orbitally stable in H 1 0,2 (−1, 1) if the frequency ω is sufficiently close to −λ 2 = −π 2 ≈ −9.86 in the case of p > 5 and for all ω > −λ 2 in the case of 1 < p 5. Also we know that from (ii) of proposition 9, e iωt Q 2,ω is orbitally unstable for sufficiently large ω in the case of p > 5. Thus, we observe only the nature of the instability in the supercritical case p > 5 for large ω with the perturbation (14). Figure 1 shows that the profiles of solution u p with the initial datum (14) for p = 8, ω = 7 and δ p = 0.001. The obtained solution numerically blows up even for small δ p > 0. We also verified that the collapse occurs faster when we increase frequency ω. Thus, we expect that the standing wave e iωt Q 2,ω is orbitally unstable in the sense of blow-up for all large ω in as stated in observation (i).  (15). This section is concerned with the stability of the standing wave e iωt Q 2,ω under the perturbation (15). First, we study the standing wave e iωt Q 2,ω for ω close to −λ 2 in the subcritical case 1 < p < 5. Figure 2 shows the profile of the solution u c with the initial datum (15) for p = 3, δ c = 0.01 and ω = −9. The amplitude of the observed solution u c remains close to |Q 2,ω |. Furthermore, performing the same calculation with δ c = 0.001 implies that both curves |u c | and |Q 2,ω | are almost the same.

Stability under the perturbations
Let us examine the profile of the solution u c from another point of view. Figure 3 shows that the values of two quantities   Next, we consider the standing wave e iωt Q 2,ω for somehow large frequency ω in the subcritical case 1 < p < 5. Figure 4 shows the profile of obtained solution u c with the initial datum (15) figure 5). About 95% of the L 2 -norm is moving between the left and the right side by t = 4. As a result, we suggest the standing wave e iωt Q 2,ω is orbitally unstable if ω is large. The profile of solution u c shown in figure 4 suggests that the orbital instability holds for some frequencies around −3.8.
Finally, we focus on the stability of standing wave e iωt Q 2,ω in critical and supercritical cases p 5. Our simulations show that the same phenomenon as in the subcritical case (1 < p < 5) holds if the frequency ω is close to −λ 2 . For example, figure 6 shows the values of two quantities  solution with the initial datum (15) for p = 7, ω = −9.3 and δ c = 0.01. Figure 6 seems similar to figure 3. Moreover, we note that less than 3% of the L 2 -norm moves between the left and right side. Similar profiles of the solution with initial datum (15) are observed for any other ω which is close to −λ 2 .
Repeating the same simulation when varying the frequency ω, we see that the behaviour of the solution u c significantly changes at ω = −7 and we see that the strong instability of the standing wave e iωt Q 2,ω occurs. Moreover, the collapse occurs faster when we increase ω. Therefore, we expect the statements of (ii) with the case k = 2 of observation in section 1. (iii) with k = 2. It is natural to address the question of stability of the standing wave e iωt Q 2,ω for sufficiently large ω in the subcritical case 1 < p < 5. Figure 9 shows the profile of the solution u c with the initial datum (15) for p = 1.9, ω = 200 and δ c = 0.002. The obtained solution u c keeps almost the same profile except some oscillations until t = 63. On the other hand, figure 10 shows that the profile of the solution with the initial datum (15) for p = 1.9, ω = 10 and δ c = 0.0001 changes before t = 7. We may interpret this difference as follows. Since the L 2 -norm of the solution u c becomes very large in figure 9, the oscillations do not occur except for very large values of time which makes difficult the numerical observation of these phenomena and requires an unreachable time of calculation. That is why we are not sure whether Q 2,ω remains orbitally unstable under the perturbation (15) for sufficiently large ω.

Remark 27.
In [37,46], the authors observed the drift instability, that is, the centre of mass defined by x|u(t, x)| 2 / u(0, .) 2 2 dx drifts away from its initial location even for arbitrarily small perturbation. In their case, the principal amount of the L 2 -norm moves away to the right side if it is initially perturbed to the right direction (see figures 11, 13, and 16 in [37]). In our case, the centre of mass oscillates between −1 and 1. The principal amount of the L 2 -norm keeps moving between the left and right side.

Second excited state
Concerning the stability of the second excited state e iωt Q 3,ω , our numerical computation shows that the same results as the case k = 2 hold for all p > 1 and both perturbations (14) and (15). Thus, we refrain from explaining the details.