On the normalized ground states for the Kawahara equation and a fourth order NLS

We consider the Kawahara model and two fourth order semi-linear Schr\"odinger equations in any spatial dimension. We construct the corresponding normalized ground states, which we rigorously show to be spectrally stable. For the Kawahara model, our results provide a significant extension in parameter space of the current rigorous results. At the same time, we verify and clarify recent numerical simulations of the stability of these solitons. For the fourth order NLS models, we improve upon recent results on stability of very special, explicit solutions in the one dimensional case. Our multidimensional results for fourth order NLS seem to be the first of its kind. Of particular interest is a new paradigm that we discover herein. Namely, all else being equal, the form of the second order derivatives (mixed second derivatives vs. pure Laplacian) has implications on the range of existence and stability of the normalized waves.


INTRODUCTION
We consider several dispersive models in one and multiple space dimensions. Our main motivating example will be the (generalized) Kawahara equation, which is a fifth order generalized KdV equation, which allows for third order dispersion effects as well. Namely, we set (1.1) u t + u xxxxx + bu xxx − (|u| p−1 u) x = 0, x ∈ R, t ≥ 0, p > 1 This is a model that appears in the study of plasma and capillary waves, where the third order dispersion is considered to be weak. In fact, Kawahara studied the quadratic case 1 [26] and he argued that the inclusion of a fifth order derivative is necessary for capillary-gravity waves, for values of the Bond number close to the critical one. Craig and Groves, [8] offered some further generalizations. Kichenassamy and Olver, [28] have studied the cases where explicit waves exist, see also Hunter-Scheurle, [16] for existence of solitary waves. Another model, which is important in the applications, is the non-linear Schrödinger equation with fourth order dispersion. We consider two versions of it, which will turn out to be qualitatively different, from a the point of view of the stability of their standing waves. More precisely, i u t + ∆ 2 u + ǫ(〈 b, ∇〉) 2 u − |u| p−1 u = 0, (t , x) ∈ R × R d , (1.2) i u t + ∆ 2 u + b∆u − |u| p−1 u = 0, (t , x) ∈ R × R d , (1.3) where d ≥ 1, p > 1, ǫ = ±1. These have been much studied, both in the NLS as well as Klein-Gordon context, since the early 90's, see for example [1,2].
For both models, we will be interested in the existence of solitons, and the corresponding close to soliton dynamics, in particular spectral stability. For the Kawahara, the relevant objects are traveling waves, in the form u(x, t ) = φ(x + ωt ), where φ is dying off at infinity. These satisfy profile equation of the form (1.4) φ ′′′′ + bφ ′′ + ωφ − |φ| p−1 φ = 0.
Similarly, standing wave solutions in the form u = e −i ωt φ, ω > 0, with real-valued φ for the fourth order NLS (1.2) and (1.3) solve the elliptic profile equations Constructing solutions to (1.4), and more generally (1.5) and (1.6), is not straightforward task. In fact, it depends on the parameter p, the sign of the parameter b, as well as the dimension d ≥ 1. Here, it is worth noting the works of Albert, [1] and Andrade-Cristofani-Natali, [2] in which the authors have mostly studied the stability of some explicitly available solutions in one spatial dimension.
We proceed differently, by means of variational methods. More specifically, we employ the constrained minimization method, which minimizes total energy with respect to a fixed particle number, or L 2 mass. In addition to being the most physically relevant, the waves constructed this way (which we refer to henceforth as normalized waves) have good stability properties.
This brings us to the second important goal of the paper. Namely, we wish to examine the spectral stability of waves arising as solutions of (1.4) and (1.5). Our constructions will not yield explicit waves 2 . Thus, we need to decide about their stability, based on their construction and properties.

Previous results.
1.1.1. The Kawahara model. We would like to review the history of the problem for existence and stability of the traveling waves. We concentrate mostly on some recent results in the last twenty years or so, which we feel are most pertinent to our results. We would like to emphasize an important point, namely that since uniqueness results are generally lacking 3 , it is hard to compare different results about waves obtained by different methods, as they may be different in shape and stability properties.
In [12], [21], the authors have shown that certain waves of depression (i.e. b < 0) are stable. In [21], the author establishes an important, Vakhitov-Kolokolov type criteria for certain waves, but it appears that it is hard to verify outside of a few explicit examples. In [6], Bridges and Derks, have studied a Kawahara-type model, with more general nonlinearity. They have employed the Evans function method to locate the point spectrum (and hence the stability) of the corresponding linearizations. The results of their work are mostly computationally aided. 2 although some do exist, for very specific values of the parameter b and d = 1, more on this below 3 both as minimizers of constrained variational problem and as solutions of the PDE Levandosky, [29] has studied the problem for existence of such waves via an energy -momentum type argument and concentration compactness. Groves, [11] has shown the existence of multi-bump solitary waves for certain homogeneous nonlinearities. Haragus-Lombardi-Scheel, [15] have considered spatially periodic solutions and solitary waves, which are asymptotic to them at infinity. They showed spectral stability for such small amplitude solutions. We should also mention the work [2], in which the authors consider the orbital stability for explicit periodic solutions of the Kawahara problem, subjected to a quadratic nonlinearity.
The paper of Angulo, [3] gives some sufficient conditions for instability of such waves, both for the cases b > 0 and b < 0. Levandosky, [30] nicely summarizes the results in the literature 4 and offers rigorous analysis for stability/instability close to bifurcation points. Furthermore, his paper provides an useful, numerically aided, classification of solitary waves of the Kawahara model, based on the type of non-linearity (i.e. the power p) and the parameters of the problem b, ω. The exhaustive tables on p. 164, [30] provided a good starting point for our investigation. We should mention that the waves considered in [30] are produced as the constrained minimizers of the following variational problem We take different approach below, by constructing the normalized waves. These are the waves that precisely minimize energy, when one constrains the L 2 norm, see Section 3.1.
An important point we would like to make however is that the procedure outlined by (1.7) provides waves for a considerably wider range of p, than the ones produced in Section 3.1.
Namely, the minimizers of (1.7) exist for p ∈ (1, p max ), with p max (d )  [25], where it plays an important role in modeling the propagation of intense laser beams in a bulk medium with Kerr nonlinearity. Moreover, the equation was also used in nonlinear fiber optics and the theory of optical solitons in gyro tropic media. The problem for the existence and the stability of the waves arising in (1.5) has been the subject of investigations of a few recent works, the results of which we summarize below. For the case of d = 1, p = 3 (and in fact only for the special value of ǫ = −1, b = 1 and ω = 4 25 ), the elliptic problem (1.4) (or equivalently (1.5)) was considered by Albert, [1] in relation to soliton solutions to related approximate water wave models. The explicit soliton, φ 0 (x) = , was studied in detail in [1]. Important properties of the corresponding linearized operators were established. These properties allowed Natali and Pastor, [33] to establish the orbital stability of this wave, see also [13] for alternative approach and extensions to Klein-Gordon solitons. One of the central difficulties that the authors faced is that this solution is only available explicitly for an isolated value of 5 ω = 4 25 . Additionally, the problem for stability of the equation (1.2) in d = 1, ǫ = −1, b = 1 and general p were addressed in the works [22] and [23]. The numerically generated waves were shown to exists for every p > 1, but they are stable 4 but he considers more general non-linearities, containing powers of derivatives as well 5 which precludes one from differentiating with respect to the parameter ω as is customary in these types of arguments only for p ∈ (1, 5). Further (mostly numerical) investigations regarding this model are available in the papers [24], [25].
Finally, it is important to discuss the recent work [5], as it has significant overlap with ours. In it, the authors have studied (1.3) in great detail, including the stability of the waves. They have constructed the waves in a similar manner, in fact the existence part of our Theorem 4 is similar in nature 6 . In addition, they discuss some cases, in which they can show the important non-degeneracy property, that is K er [L + ] = span [∇φ]. This is rigorously verified in two cases only: • for any dimension d ≥ 2, but with b < 0 and |b| sufficiently large, Concerning stability of the waves, the authors of [5] do not actually establish stability for any given example. On the other hand, they show that orbital stability holds, once one can verify non-degeneracy and the index condition 〈L −1 + φ, φ〉 < 0. The concrete details of these results are provided in [5], although this is a more general theorem, see for example Theorem 5.2.11, [19]. The non-degeneracy was already discussed, while the verification of 〈L −1 + φ, φ〉 < 0 is left as an open problem in [5]. This last condition however is essentially equivalent, modulo some easy to establish technical assumptions, to the spectral stability, see Corollary 1 below.
In this work, we actually do show 〈L −1 + φ λ , φ λ 〉 ≤ 0 for all waves produced in Theorems 1, 3, 4, thus answering the open problem in [5]. With the exception of the case 〈L −1 + φ λ , φ λ 〉 = 0 (which is a non-degeneracy condition of sort, that we cannot rule out), our results provide rigorously for spectral stability for all waves constructed therein -in all dimensions d ≥ 1, for all allowed values of b : d = 1, b ∈ R and d ≥ 2, b < 0. This, in combination with the results of [5], shows orbital stability, for all normalized waves enjoying the non-degeneracy property of the wave as well as the property 〈L −1 + ϕ λ , ϕ λ 〉 = 0.

Main results: Kawahara waves.
It is easy to informally summarize our results -all normalized waves, whenever they exist, turn out to be spectrally stable. This is an interesting paradigm, which is currently under investigation in a variety of models. Our hope is that the approach here will shed further light on this interesting phenomena in a much more general setting. As we have alluded to above, our focus will be the Kawahara problem, (1.1), for both positive and negative values of b.
where one could take φ in the Schwartz class, in order to make I [φ] meaningful. Introduce the scalar function which plays a prominent form in the subsequent arguments. Let us emphasize that it is not a priori clear whether the problem (1.8) is well-posed (i.e. m b (λ) > −∞) for all λ. We have the following existence result.

Remark:
The Lagrange multiplier ω may depend on the normalized wave φ. In particular, we can not rule out the existence of two constrained minimizers of (1.8), φ λ ,φ λ , with ω(λ, φ λ ) = ω(λ,φ λ ). This is of course related to the uniqueness problem for the minimizers of (1.8) (and it should be a much simpler one), but it is open at the moment.
1.2.2. Kawahara waves: stability. We now discuss our results concerning the stability of the waves produced in Theorem 1 -we employ the standard definition of spectral stability, see Definition 2 in Section 2.3 below. Before we give the formal statements, we need to state an important property of the waves φ constructed in Theorem 1. Namely, upon introducing the self-adjoint linearized operator In particular, L −1 + φ λ is well-defined. Theorem 2. Let λ > 0 and p satisfy the requirements of Theorem 1, and φ λ is any minimizer constructed therein. Then, φ λ is weakly non-degenerate. If in addition, the condition 〈L −1 + φ λ , φ λ 〉 = 0 is satisfied, then the wave φ λ is spectrally stable, as a solution to the Kawahara problem (1.1), in the sense of Definition 2 below.

Remarks:
• The condition 〈L −1 + φ λ , φ λ 〉 = 0 appears frequently as a non-degeneracy condition in the literature, [19]. It is worth noting that such a condition has a clear physical spectral meaning, namely that the eigenvalue at zero for ∂ x L + , generated by the translational invariance, has an associated Jordan cell of order exactly two. Physically, such an eigenvalue is expected to be of algebraic multiplicity exactly two and geometric multiplicity one, as this is the only invariance in the system, so this must hold generically. We do not have a rigorous proof of this fact at the moment.
• The results of Theorem 2 present rigorous sufficient conditions for stability of traveling waves in much wider range than previously available. In fact, our results confirm 8 the available numerical simulations by Levandosky, [30]. For example, it is quite obvious that the bifurcation point is at 9 p = 5. More precisely, for powers p < 5 all waves are stable 10 , while for p > 5, some unstable waves start to appear (which are of course not normalized). For p ≥ 9, Levandosky observed a very small set of stable waves, again none of them normalized, but rather generated as minimizers of (1.7). 7 Here, for all given p ∈ [5,9), for both b > 0, b < 0, there is a specific valueλ b,p and we assume that λ > λ b,p 8 With the usual caveat, that since there is no uniqueness, it is possible that the waves considered in [30] are different than ours! 9 corresponds to the case p = 6 in the notations of [30] 10 except at p = 4 (p = 5 in the notations of [30]) -for a small region in the parameter space, an instability is observed numerically. This must be a fluke of the computations in [30], because as we see from Theorem 1, the stable region is up to p < 5 • The Cauchy problem for the particular version of the Kawahara problem (1.1) considered herein, has not been studied methodically, to the best of our knowledge. Based on the results of the standard NLS though, one might conjecture that the problem is globally well-posed for all values 1 < p < 9. An important related issue is the conservation of Hamiltonian, momentum and L 2 mass along the evolution of solutions emanating from sufficiently nice data. • In the presence of satisfactory well-posedness theory, as outlined above, nonlinear (or strong orbital) stability of the wave φ(x + ωt ) follows from our arguments, once one can establish that the linearized operator L + has one dimensional kernel, namely K er . This is in essence standard, but it does not follow directly within the Grillakis-Shatah-Strauss formalism, [14], since this approach would require the smoothness of the mapping λ → φ λ , which is currently unknown. In particular, we refer to a method pioneered by T. B. Benjamin in [4], for the stability of the KdV waves, which has since been refined and improved by other authors. On the other hand, we refer to the arguments for the NLS case to [5]. • The non-degeneracy K er [L + ] = span[φ ′ ] appears to be a hard problem in the theory.
An easier version would be to establish such a non-degeneracy of the kernel, if φ is a minimizer of (1.8). A harder problem would be to do so, knowing that φ is just a solution to the PDE (1.4). In both cases, the non-degeneracy is directly relevant to the uniqueness of the ground state, which is even harder open problem in the area. See [9] for discussion about these and related issues.
1.3. Main results: fourth order NLS waves. We start with the existence result for the models.
1.3.1. Existence of normalized waves for fourth order NLS models. Before we state the results for the fourth order NLS models, we need to make an obvious reduction of the equation (1.2). Namely, picking a matrix A ∈ SU (n), so that b = | b|A e 1 , we can clearly reduce matters (both the existence of the solutions of the profile equation (1.5) and its stability analysis), by the transformationû(ξ) →û(A * ξ), to the following problem: and its associated elliptic profile equation That is, the existence of solutions to (1.10) is equivalent to the existence of solutions to (1.5) (under the appropriate transformation) and their stability is equivalent to the stability of their counterparts. Thus, it suffices to discuss the fourth order NLS problem (1.9), with its solitons satisfying (1.10). Our variational setup in the anisotropic case is as follows Then, there exists φ ∈ H 4 (R d ) ∩ L p+1 (R d ) satisfying (1.10), with an appropriate ω = ω(λ, φ). The wave φ λ is constructed as constrained minimizer of (1.11), with φ λ 2 L 2 = λ. Assuming in addition the condition 〈L −1 + φ λ , φ λ 〉 = 0, then e −i ω λ t φ λ (x) is a spectrally stable solution of (1.9), in the sense of Definition 2 below.

Remark:
The case ǫ = 1, in the higher dimensions d ≥ 2, while undoubtedly interesting in the applications, is much more subtle, and it cannot be analyzed with the methods of this paper. We will address some aspects of it in a forthcoming publication [27].
Despite the obvious similarities with (1.5), the fourth order NLS with pure Laplacian, (1.3) and its associated profile equation (1.6), turn out quite different -even at the level of the existence of the waves and their stability. We introduce the relevant variational problem

Theorem 4. (Stability of the normalized waves for the fourth order NLS: pure Laplacian case)
Remarks: • The results extend the stability results of Albert, [1] for the one dimensional cubic case p = 3. • The results here also extend the NLS related results of [13] (namely, stability for p < 1+ 8 d and instability otherwise), which apply to the case b = 0.
• Both results, Theorem 3 and 4 of course coincide for d = 1, but are different for d ≥ 2. We do not have a good physical explanation as to why the range of existence and stability of standing waves for the models (1.9) vis a vis (1.3) differ. In particular, the mixed derivative model, (1.9) seems to support all stable normalized waves in the wider range p ∈ (1, 1 . This topic clearly merits further investigations.
The rest of the paper is organized as follows. In Section 2, we show that distributional solutions of the elliptic problems are in fact strong solutions. We also set up the relevant eigenvalue problems, and in regards to that, we review the relevant instability index counting theories and some useful corollaries. Finally, we present the Pohozaev identities, which imply some necessary conditions for the existence of the waves. We also note that better necessary conditions (which are closer to what we conjecture are the optimal ones) are possible, under a natural spectral condition. In Section 3, we develop the existence theory in the one dimensional problem -this already contains all the difficulties, that one encounters in the higher dimensional situation as well. In particular, we discuss the well-posedness of the constrained minimization problem, the compensated compactness step, as well as the derivation of the Euler-Lagrange equation and various spectral properties of the linearized operators, which are useful in the sequel. In Section 4, we indicate the main steps in the variational construction for the waves in the higher dimensional case. In Section 5, we provide a general framework for spectral stability, based on the index counting formula, which is easily applicable in our setting.

PRELIMINARIES
We first introduce some notations and standard inequalities. We will frequently use the notation f g , when f , g are positive quantities/functions and there is a constant C , independent on the parameters so that f ≤ C g .

Function spaces and GNS inequalities.
Next, we need some Fourier analysis basics. Fourier transform and its inverse are defined viâ Note that for non-integer values of s, the norm on the right-hand side is defined via In addition, we shall make use of the Gagliardo-Nirenberg-Sobolev (GNS) inequality, which combines the Sobolev estimate with the well-known log-convexity of the complex interpola- For example, the following estimate proves useful in the sequel We record the formula for the Green function of (−∆ + 1) −1 , that isQ(ξ) = (1 + 4π 2 |ξ| 2 ) −1 (see [11], p. 418) Note that Q > 0, radial and radially decreasing. Also, In fact, there are the following classical estimates for it, p. 418, [11], In particular, Q ∈ L q (R n ), whenever q < n n−2 (or q < ∞, when n = 2).

Distributional vs strong solutions of the Euler-Lagrange equation.
In particular, the weak solutions of (2.5) in fact satisfy (2.5) as L 2 functions.
Proof. Note that by the restrictions on b, ω, we have that the operator ( In addition, for every test function h, we have Clearly, q 0 ≥ p + 1, by assumption. We will show first that q 0 = ∞. Assume not. By Sobolev embedding, we have In particular, we can take q as close to ∞ (and hence q 0 = ∞), if d ≤ 4. So, assume d ≥ 5. It follows that 1 d . Take any q 0 < q < ∞. We have, by Sobolev embedding q ≥ 1, we can take r = p+1 p and we have a contradiction right away, since the left-hand side of (2.6) is unbounded (by the definition of q 0 ), while the right-hand is bounded. For the remainder, take r : 1 r = 4 d + 1 q . Clearly, if r p < q 0 , this would be a contradiction, because the left-hand side is supposed to be unbounded (by the definition of q 0 ), while the right-hand side clearly is. We claim that this is the case, under our restrictions for p ∈ (1, 1 q 0 , we will have achieved the contradiction, as we can take q very close to q 0 . Indeed, by the inequality for 1 q 0 , we have leads to the solution 1 < p < 1 + 8 d−4 , which of course contains the set (1, 1 + 8 d ), so it is true for all p in the set that we are interested in. We have reached a contradiction, with q 0 < ∞.
Thus, q 0 = ∞. This does not mean yet that g ∈ L ∞ (R d ), but this follows easily by Sobolev embedding, once we know that g ∈ ∩ 2≤q<∞ L q (R d ). Furthermore, we see that the same type of arguments imply g ∈ H 5 (R d ) and that for every p < ∞ and for every ǫ > 0, g ∈ W 4−ǫ,p (R d ).
For our next step, we shall need a representation of the Green's function of the operator (∆ 2 + b∆ + ω) −1 as follows. We have are positive numbers, so clearly the corre- As far as the case b 2 − 4ω < 0 is concerned, it is not hard to see, in the same way, that the Green's function G has decay rate e −k ω |x| , where In both cases, the Green's function enjoys exponential rate of decay.
For p ≥ 2, we can actually conclude that g ∈ L 1 (R d ) since by the Hardy-Littlewood-Sobolev inequality Our claim is that q 0 = 1. Assume for a contradiction that q 0 > 1. We will show that for every q > q 0 , we have that g ∈ L q p (R d ), which would be a contradiction with q 0 > 1. Indeed, by Hardy-Littlewood-Sobolev g .
This establishes the contradiction with q 0 > 1, hence g ∈ ∩ 1<q L q (R d ).
2.3. Linearized problems and spectral stability. We next discuss the linearized problems and the stability of the waves. For solutions φ of (1.4), we introduce the traveling wave ansatz, . Plugging this back in (1.1) and ignoring all terms O(v 2 ), we obtain the following linearized problem We proceed similarly with the linearization of the NLS problem (1.2). Consider solutions φ of (1.10) and then perturbations of the solution u(t , x) = e −i ωt φ of (1.9) in the form u = e −i ωt [φ + z 1 + i z 2 ]. Plugging this ansatz into (1.2), retaining only the linear in z terms and taking real and imaginary parts leads us to the system Thus, we introduce the scalar self-adjoint operators L ± (note L + < L − ) so that the eigenvalue problem associated with (2.9) and the assignment z → e µt z, takes the form where Finally, for solutions φ of (1.6), the linearized problem appears in the form This is again in the form (2.10), with We are now ready to give the definition of spectral stability. Note that the essential spectrum is, by Weyl's theorem, is the range of the function ξ ∈ R d → |ξ| 4 − b|ξ| 2 + ω. Clearly, this is the

Stability of linearized systems and index counting theories.
We need a quick introduction of the instability index count theory, as developed in [17], [18], [34] (see also the book [19]) and more recently in [20], [31]. We will only consider appropriate representative corollaries, which serve our purposes. For the purposes of this paper, we will follow closely the approach and the notations in [31]. To that end, we consider an eigenvalue problem in the form 12 We need to introduce a a real Hilbert space, so that f ∈ X , its dual X * , so that L : X → X * , so that the bilinear form (u, v ) → 〈L u, v 〉 is a bounded symmetric bilinear form on X × X . Next, 11 Note that by the Hamiltonian symmetry of the problem µ → −µ, the existence of eigenvalues µ : ℜµ < 0 is equivalent to the existence of µ : ℜµ > 0 12 Before we embark on further details, let us once again emphasize that the examples that we will be interested in herein will be either in the form (2.8) (i.e. the KdV-like case) or in the form (2.10) (i.e. the NLS like case).
we shall need to assume that J has a domain D(J ) ⊂ X * , so that J : D(J ) → X , J * = −J . Furthermore, ssume that there is an L invariant decomposition of the base space in the form where (see Section 2.1, [31]), L | X − < 0, n(L ) := d i m(X − ) < ∞, d i m(K er [L ]) < ∞ and L | X + ≥ δ, for some δ > 0. In general, we will denote by n(M) the (finite) number of negative eigenvalues (counted with multiplicities) of a generic self-adjoint operator M. Next, consider the finite dimensional generalized eigenspace at the zero eigenvalue, defined as follows Under these general assumptions, it is proved in [31] (see Theorem 2.3 and also Theorem 1, [18] for the case where J has a bounded inverse) that By the index counting inequality (2.13) if n(L ) ≤ n(D), we can conclude that spectral stability holds true, since the right-hand side of (2.13) is non-positive, hence all the indices on the left are zero as well.
Next, we discuss g K er This means that the algebraic multiplicity of the zero eigenvalue is at least 2(d + 1), consisting of d + 1 eigenfunctions and d + 1 generalized eigenfunctions. One may wonder whether there is any more non-trivial elements in g K er [J L ]. The non-degeneracy condition 〈L −1 + φ, φ〉 = 0, which appears in the statement of the main result is necessary condition that the Jordan block associated to the eigenvector φ 0 is exactly two dimensional. To this end, assume that there is a third element, q : J L q = ψ 0 . This would mean, that there is q : L − q = L −1 + φ. By the self-adjointness of L − , the solvability condition is exactly 〈L −1 + φ, φ〉 = 0. Indeed, R(L − ) = K er (L − ) ⊥ = span{φ} ⊥ , so a third element in the Jordan cell for φ 0 does not exist exactly when 〈L −1 + φ, φ〉 = 0.

Kawahara-like problem.
For eigenvalues problem in the form (2.8) where we set up again X = H 2 (R), X * = H −2 (R), while , L = L + , J = ∂ x , J * = −J . This satisfies the requirements of the theory put forward in the beginning of this section. Next, regarding the generalized kernel of ∂ x L + , we clearly have that This means that the zero is multiplicity two eigenvalue for ∂ x L + , which is generated by the translational invariance. + φ, φ〉, whence since D 11 < 0, we can assert that the matrix D has at least one negative eigenvalue (since 〈De 1 , e 1 〉 = D 11 < 0, which would then imply stability. Thus, when we specify to the specific problems that we face, we can formulate the following sufficient condition for spectral stability.

Corollary 1.
For the spectral problems (2.8) and (2.10), spectral stability follows, provided Necessary conditions for existence of (1.5). We have the following Pohozaev identities.

Lemma 1. (Pohozaev's identities) Let some smooth and decaying φ satisfy
Proof. Multiplying (2.16) by φ and integrating over R d we get Also, multiplying (2.16) by x · ∇φ and integrating over R d we get Solving for A and B in terms of C and D we get which is (2.17) and (2.18). The formula (2.19) follows similarly.
Proof. If d = 1, 2, the first term on the right of (2.17) is negative, forcing the positivity of the second term, so ω > 0. Next, from the relation (2.18), we see that if ω > 0, ǫ = −1, then If b = 0, it is clear from (2.18) that either ω > 0 and p < p max or ω < 0 and p > p max (the second one being impossible immediately for d = 1, 2, 3, 4). For d ≥ 5, assume for a moment that ω < 0 and p > p max = d+4 d−4 . Let us look at (2.17). The second term is now negative, while for the first term, since p > p max > d+2 d−2 , we also conclude its negativity. It follows that the right hand side of (2.17) is negative a contradiction. Thus, ω > 0, p < p max .
As we see from the results of Corollary 2, the Pohozaev's identities are by themselves not strong enough to derive necessary conditions on ω, p that are close to the sufficient ones.
We believe that indeed, the necessary conditions are close to the ones required by [30] to construct solutions of the constrained minimization problem (1.7). Namely, we expect p < p max and ω > b 2 4 for b > 0 to be necessary for existence of localized and smooth solutions to (2.16) and (1.6). Let us show that in fact, these follow from a natural assumption on the spectrum for the operator L + , namely that zero cannot be an embedded eigenvalue in the continuous spectrum of L + . Let us note that while for second order Schrödinger operators H = −∆ + V , this is generally the case 13 under decay conditions on V , this is not the case for their fourth order counterparts, [10]. In physically relevant situations however (and the case of L + certainly merits this designation), embedded eigenvalues should not exist. If this is the case for L + , we see that since by Weyl's theorem Clearly, if zero is not embedded, it must be that ω satisfies ω ≥ b 2 4 b ≥ 0 0 b < 0 . If that holds, at least in the case b < 0, it follows from Corollary 2 that p < p max as well.

VARIATIONAL CONSTRUCTION IN THE ONE DIMENSIONAL CASE
We start with some preparatory results.

Variational problem: preliminary steps.
We now discuss the variational problem (1.8). It is certainly not a priori clear that for a given λ > 0, such a value is finite (that is m b (λ) > −∞) and non-trivial (i.e. m b (λ) < 0). In fact, in some cases, it is not finite, as we show below. Note that This is, clearly, a non-increasing function. In particular, is differentiable a.e. and so is m b (λ). Our considerations naturally split in two case, b > 0 and b < 0.
3.1.1. The case b < 0. In this section, we develop criteria (based on the parameters in the problem), which address the question for finiteness and non-triviality of m b (λ). The next lemma shows this for p ∈ (1, 5) and in addition, it establishes that m b (λ) = −∞ for p > 9.
Consider now the case p = 9. Clearly, for large λ, m b (λ) < 0, as it is evident from the formula (3.1). Assuming that m b (λ) ∈ (−∞, 0) for some λ, let φ be such that m b (λ) ≤ I [φ] < m b (λ) 2 . Using φ N as in the formula (3.2), we see that φ N 2 L 2 = λ, while for N ≥ 1, we have But then Our next lemma shows that for p ∈ [5,9), there is a threshold value λ p > 0, below which m b (λ) is trivial. Lemma 3. If b < 0 and p ∈ [5,9), then there exists a finite number λ p > 0 such that Proof. Take φ ε as in Lemma 2 with φ 2 2 = 1. We have which implies that m b (λ) ≤ 0. Now, we are going to show that for each p ∈ [5,9] there exists a constant c p > 0 such that Using the GNS inequality (2.1), we get the following estimates for the L p+1 norm: Note that for p ∈ [5,9), we have that 4 . Therefore, interpolating between estimates (3.6) and (3.7) we get Thus we have that for all φ ∈ H 2 with φ Observe that for a very large λ, the quantity Lemma 4. Suppose b < 0, 1 < p < 9 and −∞ < m b (λ) < 0. Let φ k be a minimizing sequence. Then, there exists a subsequence φ k such that: where L 1 > 0, L 2 > 0 and L 3 > 0. 14 which can be seen by fixing φ in the infimum and taking λ > λ(φ) Proof. We have already established in Lemma 2 that Since, φ k is minimizing, it follows that the sequence { R |φ ′′ k (x)| 2 d x} k is bounded. By GNS inequality, the sequences { R |φ ′ k (x)| 2 d x} k and R |φ k (x)| p+1 d x} k are bounded as well. Passing to a subsequence a couple of times we get a subsequence {φ k } such that all of the above sequences converge. We claim that L 3 cannot be zero. Indeed, otherwise, which is a contradiction with the fact that m b (λ) < 0. By Sobolev embedding, neither L 1 nor L 2 could be zero, as this would force L 3 = 0, which we have shown to be impossible.
2 ), so if we just take ε small enough, we see that m b (λ) < 0. Boundedness from below follows from (3.8).
Proof. First, by (3.8), the quantity R |φ ′′ k (x)| 2 d x is bounded. By Sobolev embedding so are the other two. By passing to a subsequence (denoted again φ k ), we can assume that they converge to three non-negative reals, L 1 , L 2 , L 3 .
Suppose first that L 3 = 0. Then, consider the following minimization problem Thus, φ k is minimizing forĨ as well and On the other hand, inf φ 2 2 =λĨ [φ] is easily seen to be − λb 2 8 . Indeed, for function φ : On the other hand, for any Schwartz function χ, consider which has φ 2 L 2 = λ and saturates the inequality (3.12) in the sense that holds for all φ with φ 2 2 = λ. Applying this to an arbitrary f and φ := λ f f L 2 , so that φ 2 L 2 = λ the following inequality holds for all f = 0. This last inequality however contradicts Lemma 6 -for every λ > 0, if p ∈ (1, 5) and for all large enough λ, if p ∈ [5,9). Thus L 3 = 0. Clearly, by Sobolev embedding L 1 > 0, L 2 > 0, otherwise L 3 must be zero, which previously lead to a contradiction.

Strict sub-additivity.
Lemma 8. Let 1 < p < 9 and λ > 0 Then for all α ∈ (0, λ) we have Proof. First, suppose that 1 < p < 5 and b < 0. Then where the last strict inequality holds because there exists a minimizing sequence for m b (α), which has the property lim k φ k p+1 > 0. This means that the function λ → m b (λ) λ is strictly decreasing. Assuming that α ∈ [ λ 2 , λ) (and otherwise we work with λ − α) we get where we have used This completes the case p ∈ (1, 5), b < 0. Let 5 ≤ p < 9 and b < 0. Note that in this case, m b (x) is zero for small x, by Lemma 3. So, there are three possibilities: In this case (3.13) trivially holds, since by assumption m b (λ) < 0.
Next, we consider the cases when b > 0. In this case for all 1 < p < 5 and all λ > 0 we have that −∞ < m b (λ) < 0. The proof is the same as in the case b < 0, p ∈ (1, 5), since we never develop the complication that m b (λ) = 0 for any λ > 0. The case p ∈ [5,9) and λ > λ b,p is similar as well.
Therefore, by passing to a further subsequence, by Lemma 4 and Lemma 7, we have By the concentration compactness lemma of P.L.Lions (see Lemma 1.1, [32]), there is a subsequence (denoted again by ρ k ), so that at least one of the following is satisfied: (1) Tightness. There exists y k ∈ R such that for any ε > 0 there exists R(ε) such that for all k (3) Dichotomy. There exists α ∈ (0, λ), such that for any ε > 0 there exist R, R k → ∞, y k and k 0 such that (3.14) We proceed to rule out the dichotomy and vanishing alternatives, which will leave us with tightness.

Dichotomy is not an option.
Assuming dichotomy, we have by (3.14) and Define φ k,1 and φ k,2 as follows: Clearly, for k large enough we have In fact, by taking a sequence ε k → 0, we can find subsequence of φ k,1 , φ k,2 (denoted again the same) and sequences 2 ]. Using (3.15) we get The error term E k , contains only terms having at least one derivative on the cutoff functions, therefore generating R −1 k . At the same time, there is at most one derivative falling on the φ k . So, we can estimate these terms away as follows Since sup k φ k L 2 , sup k φ ′′ k L 2 < ∞, we conclude that lim k E k = 0. For the next term, we have the positivity relation Thus, by Hölder's inequality Note that since R k → ∞ and on the other hand φ k H 2 is uniformly bounded in k, this term goes to zero, by the last estimate in (3.15). Finally, Since by GNS and φ ′′ k L 2 is uniformly bounded in k, we conclude that this term also goes to zero as k → ∞. It follows that (3.16) lim inf Note that a k , b k → 1. Using (3.16), there is β k : lim k β k = 0, so that where we have used that sup k φ k H 2 < ∞, the estimate |I (φ)−I (aφ)| ≤ C ( φ H 2 )|1−a| (which is a direct consequence of the definition of the functional I [·]) and the definition of m b (z). Taking limits in k, we see that which is a contradiction with the sub-additiivity of m b (·) established in Lemma 8. So, dichotomy cannot occur.

3.2.2.
Vanishing does not occur. Suppose vanishing occurs and ε > 0. Let φ ∈ C ∞ be such that Using GNS we have for all R and y ∈ R We can cover R with balls of radius 2 such that every point is contained in at most 3 balls, let it be {B(y j , 2)}. Moreover, we can choose these balls so that {B(y j , 1)} still covers R. Choose N ∈ N so large that for all k > N , for all y ∈ R. We can estimate the L p+1 (R) norm of φ k as follows So, we get that φ k p+1 L p+1 (R) → 0 as k → ∞ which is a contradiction. Therefore, the sequence ρ k = |φ k | 2 is tight.

3.2.3.
Existence of the minimizer. We have that there exists a sequence {y k } ∞ k=1 such that for all ε > 0 there exists R(ε) such that Define u k (x) := φ k (y k + x). The sequence {u k } ∞ k=1 ⊂ H 2 is bounded, therefore there exists a weakly convergent subsequence( renamed to {u k } ∞ k=1 ), say, to u ∈ H 2 . By the tightness and the compactness criterion on L 2 (R n ), the sequence {u k } ∞ k=1 has a strongly convergent subsequence in L 2 (R), say, to u ∈ H 2 . Since weak convergence on H 2 implies weak convergence on L 2 , we have that u = u by uniqueness of weak limits. In addition, u 2 L 2 = lim k u k 2 L 2 = λ, so u satisfies the constraint.
We also have that u k converges to u in L p+1 norm. Indeed, using GNS inequality we get Finally, by the lower semicontinuity of the L 2 norm with respect to weak convergence, we have lim inf k R |u ′′ k | 2 ≥ R |u ′′ | 2 . We conclude that lim inf whence we have that m b (λ) ≥ I [u], therefore I (u) = m b (λ) and u is a minimizer.
Proof. We have shown that minimizers for the constrained minimization problem exists in the two cases described above, for both b > 0 and b < 0.
Consider u δ = λ φ λ +δh φ λ +δh , where h is a test function. Note that u δ 2 L 2 = λ, so it satisfies the constraint. Expanding I [u δ ] in powers of δ we obtain Using only the first order in δ information and the fact that I [u δ ] ≥ m b (λ) for all δ ∈ R, we conclude that Since this is true for any test function h, we conclude that φ λ is a distributional solution of the Euler-Lagrange equation (3.17). According to Proposition 1, this turns out to be a solution in stronger sense, in particular φ λ ∈ H 4 (R). Now, using the fact that the function g h (δ) := I [u δ ] has a minimum at zero, we also conclude that g ′′ h (0) ≥ 0. This is of course valid for all h, but in order to simplify the expression, we only look at h : h = 1, which are orthogonal to the wave φ λ , i.e. 〈h, φ λ 〉 = 0. This implies that In other words, 〈L + h, h〉 ≥ 0, whenever h : h = 1, 〈h, φ λ 〉 = 0. This is exactly the claim that L + | {φ λ } ⊥ ≥ 0. In particular, this implies that the second smallest eigenvalue of L + is nonnegative or n(L + ) ≤ 1. On the other hand, since 〈L + φ λ , φ λ 〉 = −(p − 1) |φ λ (x)| p+1 d x < 0, it follows that there is a negative eigenvalue or n(L + ) = 1.

VARIATIONAL CONSTRUCTION IN HIGHER DIMENSIONS
In this section, we follow the approach and constructions from Section 3. Most, if not all, of the steps go through essentially unchanged, save for the numerology, which is of course impacted by the dimension d . Thus, we will be just indicating the main points, without providing full details, where the arguments follow closely the one dimensional case.
Recall that we work with the variational problem (1.11). Again, we introduce Note that since is non-increasing, we conclude that m b (λ) is differentiable a.e. As we have previously discussed, the case ǫ = 1 seems much more technically complicated, and it is to be addressed in a subsequent publication [27].
We concentrate on the case ǫ = −1. We have the following regarding m b,λ .

Lemma 9.
Let ǫ = −1. Then, Proof. The proof goes through the same steps as in Lemma 2.
Next, we present a technical lemma.
The next two lemmas are the generalizations of Lemma 3 and Lemma 4 to higher dimensions. • for all λ > λ p we have −∞ < m b (λ) < 0.
Proof. The inequality m(λ) ≤ 0 follows in the same way as in Lemma 3. Then, by Lemma 10, we have Thus, for all φ ∈ H 2 (R d ), we have which by (4.1) implies that for λ ≤ λ b,p := Since we always have the opposite inequality, this implies m b (λ) = 0, when λ is small enough. Note that for very large λ, the quantity in (4.1) is clearly negative, so this implies that λ b,p < ∞.
The next lemma is the generalization of Lemma 4 to the higher dimensional case. Its proof follows an identical arguments and it is thus omitted.
and λ > λ b,p . Let φ k be a minimizing sequence for the constrained minimization problem (1.11). Then, there exists a subsequence φ k such that: where L 1 > 0, L 2 > 0 and L 3 > 0.

Existence of minimizers.
Before we go ahead with the existence of minimizers, we need an analog of Lemma 8. Their proofs in the higher dimensional case goes in an identical manner. Lemma 13. Let 1 < p < 1 + 8 d and λ > 0. Then λ → m b,p (λ) is strictly subadditive. That is, for every α ∈ (0, λ), With the basic results in place, we can now proceed to establish the existence of the minimizers of (1.11). Supposing By eventually passing to a subsequence, we can without loss of generality assume, by using Lemma 12, where 15 L 1 > 0, L 2 > 0 and L 3 > 0. The next task is to show that this sequence does not split nor vanish. The absence of splitting is established in the same way as the first part of Section 3.2. 15 For conciseness, we use φ k , instead of φ n k Next, we rule out vanishing. The proof presented in Section 3.2 works for d = 1, 2, 3, 4, but breaks down in d ≥ 5, so let us present another one that works in all dimensions. More concretely, for all R > 0 and y ∈ R d and a cutoff function η introduced in Section 3.2.2, we have by the GNS inequality y,2R)) . So, if we assume that vanishing occurs, then for every ε > 0, we will be able to cover R d with balls of radius 1, say B(y j , 1), so that B(y j , 3) From here, it follows that the sequence ρ k = |φ k (x)| 2 is tight and the existence of the minimizer is done as in Section 3.2.3.
The Euler-Lagrange equation, together with the appropriate properties of the linearized operators is done similar to Proposition 2.
As we mentioned above, the proof goes along the lines of Proposition 2. The only new element are the statements about L − , which we now prove. Note that by direct inspection, L − [φ λ ] = 0, by (4.5), so zero is an eigenvalue. Assuming that there is a negative eigenvalue, say L − [ψ] = −σ 2 ψ, ψ = 1, we clearly would have ψ ⊥ φ λ . In addition, since 16 L + < L − , 〈L + ψ, ψ〉 < 〈L − ψ, ψ〉 = −σ 2 〈L + φ λ , φ λ 〉 < 0. 16 This is an obvious statement, once we realize that φ λ cannot vanish on an interval. Indeed, otherwise, since it solves the fourth order equation (4.5), it follows that φ λ is trivial, which it is not. This would force n(L + ) ≥ 2, a contradiction. Thus, L − ≥ 0. Finally, 0 is a simple eigenvalue of L − along the same line of reasoning. Indeed, take ψ : L − ψ = 0, ψ ⊥ φ λ . Again, we conclude n(L + ) ≥ 2, which leads to a contradiction. 4.2. Discussion of the proof of Theorem 4: existence of the waves. We do not provide an extensive review of the existence claims in Theorem 4 ,as this would be repetitious, but we would like to make a few notable points. We work with the variational problem (1.12), where we set up b = −1 for simplicity as this will not affect the calculations.
Our goal in this section is to clarify the range of indices in p. More concretely, we have the following analogue of Lemmas 10. The proof proceeds in a similar fashion, so we omit it. A combination of arguments in the flavor of the proofs for Lemma 9 and Lemma 11 leads us to the following variant of Lemma 11 and Lemma 12. In addition, assuming that −∞ < m b (λ) < 0, that is and λ > λ b,p . and φ k be a minimizing sequence for the constrained minimization problem (1.11), there exists a subsequence φ k such that: where L 1 > 0, L 2 > 0 and L 3 > 0.
With these tools at hand, the existence of the waves follows in the same manner as before, so we omit the details.

STABILITY OF THE NORMALIZED WAVES
Interestingly, the proof of the spectral stability proceeds by a common argument, both for the Kawahara and the fourth order NLS case. By Proposition 1, it suffices to show that n(L + ) = 1, L − ≥ 0, φ λ ⊥ K er [L + ] and to verify that the index 〈L −1 + φ λ , φ λ 〉 < 0. Indeed, the condition n(L + ) = 1 was already verified as part of the variational construction, see Proposition 2 and 3. Similarly, L − ≥ 0 was verified in the higher dimensional case in Proposition 3.
Our next result is a general lemma, which is of independent interest. where we have used the assumption 〈H ξ 0 , ξ 0 〉 ≤ 0. It follows that 〈H −1 ξ 0 , ξ 0 〉 ≤ 0, which is the claim.
Remark: Unfortunately, it is impossible to conclude that 〈H −1 ξ 0 , ξ 0 〉 < 0, based on the assumptions made in Lemma 17. It turns out that such a statement is in general false, that is it is in general impossible to rule out 〈H −1 ξ 0 , ξ 0 〉 = 0.
These arguments establish rigorously the spectral stability of the waves for the Kawahara made in Theorem 2 and in the high dimensional fourth order NLS problems in Theorem 3 and Theorem 4. 17 We owe this to a generous remark made by an anonymous referee in response to our initial claims to the contrary.