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On the Cubic Lowest Landau Level Equation

We study dynamical properties of the cubic lowest Landau level equation, which is used in the modeling of fast rotating Bose-Einstein condensates. We obtain bounds on the decay of general stationary solutions. We then provide a classification of stationary waves with a finite number of zeros. Finally, we are able to establish which of these stationary waves are stable, through a variational analysis.

The cubic lowest Landau level equation is induced by the energy given the standard symplectic form ω(u, v) = Im C uv dL on E. It reads i∂ t u = Π(|u| 2 u), (t, z) ∈ R × C, u(0, z) = u 0 (z).
1.2. Derivation of (LLL). This equation arises as a limiting problem in a number of situations.
1.2.1. Rotating Bose-Einstein condensates. Following [3,24], consider a Bose-Einstein condensate confined by a harmonic field, and rotating at a high velocity. In appropriate coordinates, and for constants ǫ and G, its Hamiltonian reads C − [∇ − iA]ψ 2 + ǫ 2 |z| 2 |ψ| 2 + G|ψ| 4 dL(z) For ǫ, G ≪ 1, the first term is dominant, and, in order to minimize H, one can consider that ψ ∈ E. This leaves us with the Hamiltonian C ǫ 2 |z| 2 |ψ| 2 + G|ψ| 4 dL(z), on E. Notice that the corresponding dynamics are the same as that given by H, which is the case ǫ = 0 (we will see later that the term |z| 2 |ψ| 2 simply corresponds to rotations, and is therefore harmless).
It is conjectured from physical [1,9] and numerical [2] observations that, as ǫ → 0, the minimizers (for fixed L 2 norm) of the above functional have a very specific structure: within a ball, u is close to a theta function (in particular, its zeros coincide with an Abrikosov lattice); and away from this ball it decays fast. See [4] for a mathematical approach to this conjecture.
1.2.2. Superconductivity. A parallel derivation can be followed for a superconducting material submitted to an exterior magnetic potential: we refer to [5]. The completely resonant system, which approximates the evolution of u as ǫ → 0 is given (after time rescaling) by It is derived and studied in [19,20]. A striking property of (CR) is that it agrees with (LLL) if its data are chosen in the Bargmann-Fock space E.
The equation (CR) can also be derived from the nonlinear Schrödinger equation on the torus [12] or in the presence of a magnetic potential [13].

1.3.
Comparison with similar equations. The formulation (LLL) of the cubic lowest Landau level equation is similar to the cubic Szegő equation, introduced by the first author and S. Grellier in [16], and identified in [17] as the completely resonant system associated to the cubic half-wave equation on the circle -see also Pocovnicu [26,27] concerning the cubic Szegő equation on the line. An important feature of the cubic Szegő equation is that it admits integrability properties through a Lax pair structure satisfied by Hankel operators. Using this structure, traveling wave solutions were classified in [16] for the circle and in [26] for the line, and growth of high Sobolev norms was established in [27] for the line and in [18] for the circle.
Though the Lax pair structure for Hankel operators does not seem to extend to (LLL), it is therefore natural to study similar questions for equation (LLL). A review of our results in this direction is the purpose of the next paragraph.
Finally, let us mention that the completely resonant system associated to the conformally invariant cubic wave equation on the three-dimensional sphere was recently introduced in [8]. 1.4. Main results. In this paragraph, we briefly describe the main results of this paper. We recall that E = u(z) = e − |z| 2 2 f (z) , f entire holomorphic ∩ L 2 (C).
1.4.1. The initial value problem and long time Sobolev bounds. The well-posedness of (LLL) was studied by F. Nier [29,Proposition 3.1] (see Remark 2.1), and the following statement holds true.
More generally, if, for some s > 0, z s u 0 ∈ L 2 (C), then z s u(t) ∈ L 2 (C) for every t ∈ R.
Our next result concerns the long time bounds for Sobolev norms, which are equivalent to weighted norms z k u L 2 -see Lemma C.1 below. Theorem 1.2. With the notation of Theorem 1.1, assume z k u 0 ∈ L 2 (C) for some integer k ≥ 1. Then z k u(t) L 2 (C) ≤ C k (1 + |t|) k− 1 2 .
Notice that Theorem 1.2 is in strong contrast with the results of [18] for the cubic Szegő equation on the circle, where superpolynomial growth of Sobolev norms is established to be generic in the Baire sense. On the other hand, we mention in this paper two results improving the above polynomial rate for a perturbation of (LLL) under generic Hermite multipliers. Theorem 4.4 is a direct consequence of normal form theory for semilinear quantum harmonic oscillators [21] and states that, for any exponent r and for a full measure set of Hermite multipliers of any given algebraic decay, solutions having an initial data of order ǫ in a big Sobolev space conserve the same size on a time of length ǫ −r . Theorem 4.3 is a direct application of KAM theory [22] to this context and allows to find small quasiperiodic solutions -hence uniformly small in any Sobolev spacefor the perturbation of (LLL) by a subset of Hermite multipliers of asymptotically full measure.

Stationary waves.
In view of the two dimensional invariance by phase rotations and space rotations, it is natural to define stationary waves for equation (LLL) as solutions of the form for some (λ, µ) ∈ R 2 . Equivalently, the corresponding initial condition u 0 , also called a stationary wave, satisfies λu 0 + µΛu = Π(|u| 2 u) , Λ := z∂ z − z∂ z .
(1.2) We obtain several results about these special solutions. Firstly, we provide a priori bounds on the growth at infinity of any stationary wave. the following estimate holds, ∀z ∈ C , |u 0 (z)| ≤ C η e |z| η − 1 2 |z| 2 . As a consequence, if N (R) = # z ∈ C such that u(z) = 0 and |z| < R , then for any η > η 0 , N (R) R η −→ 0 as R → ∞. Secondly, we classify stationary waves with a finite number of zeros and we study their orbital stability in E and in L 2,1 E := {u ∈ E : zu ∈ L 2 (C)} . Theorem 1.4. Up to multiplicative factors, phase rotations and space rotations, the stationary waves in E having only a finite number of zeros are Furthermore, u α 0 and u α 1 are orbitally stable in L 2,1 E for phase rotations, v b is orbitally stable in L 2,1 E for phase and space rotations, and u α n , n ≥ 2, are not orbitally stable. Finally, the set Our third class of results about stationary waves concern existence of stationary waves with an infinite number of zeros. We construct these objects using three different methods. Firstly, by bifurcation from u 0 n -see Proposition 6.3. Secondly, by a minimization argument combined with the classification result of stationary waves having only a finite number of zeros-see Proposition 7.7. Finally, by explicit formulae we construct stationary waves having zeros on sets γZ and γZ ∪ iπ kγ Z, where γ = 0 is an arbitrary complex number, and k = 0 is an arbitrary integer-see Appendix A. In the first case, the growth at infinity is at most e c|z|−|z| 2 /2 , while this growth is optimal in the third case. We have not been able to find stationary waves with a faster growth at infinity.

1.4.3.
Number of zeros of the minimizer. We now turn to the question of the number of zeros (in particular, finite or not) of minimizers of a physically relevant variational problem involving the conserved quantities of the equation. In order to describe the results obtained in this respect, we switch to semi-classical coordinates, which are most commonly used in this context. Let 0 < h < 1 be a small parameter and denote by Then every local or global minimizer of (1.4) has an infinite number of zeros. Then is the unique global minimizer of (1.4) and As mentioned earlier, a question of great interest is the localization of the zeros of the minimizer in the case (1.5). The result of Theorem 1.3 gives some information about the distribution of the zeros, and we refer to [2] for a study of minimizing sequences whose zeros are localized on lattices.
1.5. Organization of the paper. Section 2 is devoted to general background about the equation (LLL). Local and global well-posedness results are established in Section 3. In Section 4, we prove Theorem 1.2, and two results improving this polynomial bound for perturbations of (LLL) under generic Hermite multipliers. The last three sections are devoted to stationary waves for (LLL). In Section 5, we prove general a priori bounds on the growth of stationary waves at infinity. In Section 6, we classify stationary waves with a finite number of zeros, and show how to construct others by perturbation from some of them. Section 7 deals with various variational problems leading to stationary waves, with applications to stability theory of stationary with a finite number of zeros. Finally, three appendices are devoted to some explicit stationary waves, a dictionary through Bargmann transform, and an elementary characterization of Sobolev spaces at the lowest Landau level.

Symmetries, conserved quantities and special coordinates
In this whole section, we present the structure of (LLL) while remaining at a formal level.
2.1. Hamiltonian structure. Define first the symplectic form on E: The equation (LLL) corresponds to the Hamiltonian flow for the Hamiltonian In other words, the solution of (LLL) with data equal to u 0 can be written u(t) = ϕ H (t)u 0 . Observe that the Hamiltonian H is left invariant by the following symmetries: phase rotations for ϕ ∈ T, and magnetic translations for α ∈ C. (2.1) These symmetries are via Noether theorem related to quantities which are invariant by the flow of (LLL): the mass M , angular momentum P , and magnetic momentum Q which are given, for u ∈ E, by where Λ is the angular momentum operator, defined by It is clear that M , P and Q are left invariant by phase and space rotation; magnetic rotations R α leave M invariant but act on Q and P as follows, The following table recapitulates for each quantity conserved by the flow of (LLL) the corresponding symplectic gradient and the generated symmetry.

Conserved quantity
Symplectic gradient Symmetry Mass Notice that the phase rotation T γ obviously commutes with all the other symmetries, but this is not the case for L ϕ , R α and R iβ , for γ, ϕ, α, β ∈ R.
Remark 2.1. The equation i∂ t v − Λv = Π(|v| 2 v) which derives from the Hamiltonian H(u) + 2P (u) was studied by Nier [29]. Since the Hamiltonian flows generated by H and P commute, it is equivalent to (LLL).

2.2.
The basis (ϕ n ). Denote by (ϕ n ) n≥0 the family of the special Hermite functions given by By [32, Proposition 2.1], the family (ϕ n ) n≥0 forms a Hilbertian basis of E, and we can check that they are the eigenfunctions in E of H, Λ and of the Fourier transform 1 F Hϕ n = 2(n + 1)ϕ n , Λϕ n = nϕ n , Fϕ n = i n ϕ n .
Observe that the Fourier transform will not play any particular role, since F = L π 2 . Incidentally, this implies the invariance of the equation under F.
The kernel of the projector Π is given by √ n + 1c n c n+1 , see (7.1) and [19], while (LLL) reads 2.3. Tempered distributions. Sometimes we will need to work in the following enlarged lowest Landau level space, One can easily establish that elements ofẼ are series of the form where the sequence (c n ) has at most a polynomial growth in n.
3. Well-posedness 3.1. Local well-posedness in z coordinates. For p ∈ [1, ∞], the weighted L p space L p,α is given by the norm , f entire holomorphic ∩ L p,α (C). These are Banach spaces when endowed with their natural norms. A classical estimate [32] gives the embedding of L p E in L q E for p < q; the inequality with the optimal constant reads [10], for all u ∈ E In order to discuss (LLL) in L p E , we need to extend Π to L p ; this is easily achieved. Proposition 3.1. For any p ∈ [1, ∞] and α ≥ 0, the projector Π has a unique bounded extension to L p and L p,α , which is given by the kernel 1 π e − |z| 2 2 − |w| 2 2 +wz .
With this extension of Π to L p,α for any p, the meaning of (LLL) for u ∈ L ∞ ([0, T ], L p,α ) is now clear. Proof. Local well-posedness is obtained from the theory of ordinary differential equations, by observing that the vector field u → Π(|u| 2 u) is smooth on the spaces L p,α , 1 ≤ p ≤ ∞, α ≥ 0, with a differential bounded on bounded subsets. This observation uses successively the boundedness of Π, and the L p -L q estimate (3.1), Remark 3.3. The space L ∞ is the endpoint space as far as local well-posedness is concerned. Smaller data spaces, such as L p , with p < ∞, or L ∞,α , with α > 0, enjoy stronger properties: there exists a solution u ∈ L ∞ ([0, T ], L p ) such that u k (t) converges weakly in L p to u(t). The proofs are immediate and we omit them.

3.2.
Local well-posedness in (c k ) coordinates. Let α ≥ 0 and λ > 0. Denote ℓ ∞,α and C λ the Banach spaces of sequences given by the norms Remark 3.5. The spaces ℓ ∞,1/4 and C λ are of particular relevance, as will become clear in the remainder in this article. Roughly speaking, they are, in (c n ) coordinates, the largest and the smallest space for which local well-posedness holds.

Proof. (i) Recall that the equation (LLL) written in (c n ) coordinates reads
We need some bounds on the interaction coefficients: Stirling's formula gives the inequality One checks that ψ(x) takes values in (0, 1), is equal to 1 only if x = 1 2 , and satisfies the bound |ψ(x)| ≤ e −C(x− 1 2 ) 2 . In order to prove the proposition, it suffices to show that T maps (ℓ ∞,α ) 3 → ℓ ∞,α , which would follow from the inequality In order to prove that this inequality holds, we first consider the sum over m: It remains to sum over S: where the last inequality follows from α ≥ 1 4 .
(ii) Proceeding as in the previous point, it suffices to show that Setting p = k + ℓ, this can also be written, using the binomial identity, sup k≥0 p≥k n≤p hence the result.
The following lemma shows how the critical spaces in z space (L ∞ ) and (c n ) space (ℓ ∞,1/4 ) are related.

3.3.
Global well-posedness. The conservation of M and H combined with the local well-posedness in L 2 E and L 4 E easily leads to Proposition 3.7. Assume that 2 ≤ p ≤ 4. The equation (LLL) is globally well-posed for data in L p E and such data lead to solutions in C ∞ R, L p E , depending smoothly on the initial data.
Proof. We already know local well-posedness from Proposition 3.2. Furthermore, using successively the boundedness of Π (Proposition 3.1), Hölder's inequality, and (3.1), The previous inequality shows that the lifespan of the solution only depends on the L 4 norm which is preserved, hence we get global well-posedness.
Let us now prove the bound (3.4). We write We take the L 2 -norm and get with the help of (3.1) ). Therefore, by the conservation of the energy, we obtain v(t) L 2 (C) ≤ Ct which is the second bound. The first bound follows from interpolation with the energy.

Bounds of Sobolev norms.
Recall that the L 2,k E -norm is equivalent to the Sobolev H k (C)norm (see Lemma C.1). Then we have the following bounds on the growth of such norms.
Proof. The global wellposedness in L 2,k E easily follows from the global wellposedness in L 2 E . To get the bound, we compute Next, by Proposition 3.1 where the last line was obtained by the Carlen inequality (3.1) (using crucially that u ∈Ẽ implies z j u ∈Ẽ). Therefore, since z u L 2 (C) is uniformly bounded by conservation of M and P , we get by interpolating that Then by a classical argument, (4.2) implies z k u(t) L 2 (C) ≤ C(1 + |t|) k−1 2 , which in turn implies (4.1) by Lemma C.1.

Remark 4.2. It is interesting to compare this result to the bounds for the 2D cubic Schrödinger
It is likely that with the method developed in [28] one gets a bound (1 + |t|) k−1 .

Long time results for linear perturbations of the LLL equation.
Here we state some results concerning linear perturbations of the LLL equation which show, under generic assumptions, close-to-linear dynamics. In this setting, the resonant structure of LLL is destroyed.

KAM results for a perturbed equation.
In the sequel, we consider the (non-local) perturbation of the (LLL) equation where ν, ǫ > 0 are small and where M is the (Hermite) multiplier, defined by Mϕ j = ξ j ϕ j with −1 ≤ ξ j ≤ 1.
Notice that M and H commute and that we have the following conservation laws : which are the L 2 and L 2,1 norms as well the Hamiltonian (there are other conservation laws).
Using the commutation of M and H, as well as the relation which can be obtained by testing on u j = ϕ j , we see that (4.3) is equivalent to the equation The abstract KAM result [22,Theorem 2.3] can directly be applied to the equation (4.4) and hence (4.3) (see also [22,Theorem 6.6] for a similar statement for the Schrödinger equation with harmonic potential).
Notice that one already knew that the equation (4.3) is globally well-posed for such initial conditions.

Control of Sobolev norms for a perturbed equation.
We define the Hermite multiplier M by Mϕ j = m j ϕ j , where (m j ) j∈N is a bounded sequence of real numbers chosen in the following classes: for any k ≥ 1, we define the class which is endowed with the product Lebesgue (probability) measure. Consider the problem (4.7) The following almost global existence result is proved in [21, Theorem 1.1].
the equation (4.7) with initial datum u 0 has a unique global solution u ∈ C ∞ R, L 2,s E and it satisfies To prove this result, we apply [21, Theorem 1.1] to the equation i∂ t v + Hv + Mv = Π(|v| 2 v), obtained with the change of unknown v = e itH u.
By the result of Lemma C.1, Theorem 4.4 shows that if the initial condition is strongly localised in space, then the corresponding solution also remains localised for large times.
An M P -stationary wave is a solution of (LLL) of the form The concept of M and M P -stationary waves can immediately be extended to the space E. Note that M and M P -stationary waves are given, respectively, by the solutions of By conservation of Q(u), this implies that Q(u) = 0.
(i) Assume that u = +∞ n=0 c n ϕ n ∈ E is an M P -stationary wave such that |c n | r n for some r < 1. Then, for any there holds |c n | n −γn . (ii) Assume that u(z) ∈ E is an M P -stationary wave such that |u(z)| ∈ L ∞ and u(z) → 0 as z → ∞. Then for any These examples show that some of the conditions of the theorem are optimal; but they also suggest that stationary waves of finite energy might in general enjoy the bound sup |z|=ρ |u(z)| e − ρ 2 2 +rρ for some r.
Corollary 5.5. Let u be an M P -stationary wave in L ∞ such that u(z) → 0 as |z| → ∞, and let N (R) = # z ∈ C such that u(z) = 0 and |z| < R .
Then for any η > η 0 , Remark 5.6. Let v ∈ E, then with the same proof one obtains N (R) R 2 . This bound is sharp as shown by the two following examples : The zeros are located on the real and imaginary axes.
• Let 0 < α < 1. The Weierstrass σ α -function associated to the lattice is an entire function. Denote {a k } the zeros of f . Assuming for simplicity that f (0) = 0, and provided that f does not vanish on ∂B(0, R), Jensen's formula gives Denoting N ′ (R) = #{z ∈ C such that u(z) = 0 and 0 < |z| < R 2 }, the above clearly implies that By Theorem 5.3, for any η > η 0 , |f (z)| η e |z| η . Combining this with the above inequality gives which leads to the desired result.
Step 1: a closer look at the (c n ) equation. The equation satisfied by M P -becomes, in (c n ) coordinates For simplicity, and since this does not affect estimates, we shall take µ = 0 in the following. The above can also be written By Stirling's formula, we can bound It will be important that ψ(x) takes values in (2 −1/2 , 1], and is equal to 1 only if x = 1 2 . Assuming that |c n | r n for some r < 1, the above immediately implies that Step 2: the bootstrap argument. Here we assume first that |c k | ≤ C r r k , for some r < 1 to be determined and aim at obtaining a bound of the type |c k | ≤ C ρ ρ k , where ρ depends on r and C ρ on C r . We fix κ ∈ 1 √ 2 , 1 and let ǫ ∈ 0, 1 2 be such that ψ(ǫ) = κ. Observe that Splitting the sum above estimating |c k |, we get We now assume that r is such that the second term in the above right-hand side dominates the first one, which corresponds to r ≤ κ 1−ǫ 2ǫ . (5.1) Notice that, given r < 1, (5.1) is satisfied if κ < 1 is close enough to 1. Choosing furthermore any κ ′ ∈ (κ, 1), this gives |c k | (κ ′ r) k . Thus we found that, for r < 1 satisfying (5.1), κ ′ ∈ (κ, 1), and for a constant A > 0, Iterating this implication gives that This implies in particular that, for any n, k, Choosing n = log k log 3 + 1, this gives the bound In particular, this implies |c k | r k for any r ∈ (0, 1). This means that (5.1) is satisfied for every κ ∈ (2 −1/2 , 1). Applying again the same bootstrap argument, we obtain |c k | k −γk for any γ < γ 0 = log 2 2 log 3 .
Step 1: establishing Gaussian decay for M -stationary waves in z coordinates. Without loss of generality, start with u, a function in L ∞ going to zero at infinity, solving Using first the Gaussian bound on the kernel of Π, and then elementary estimates, we get for κ ∈ (0, 1) for a constant C 0 . We now claim that M n < Ae −ǫκ −2n for n > n 0 , where n 0 , A and ǫ are positive constants to be determined. This will follow by induction if we can make sure that In order to make sure that these inequalities are satisfied, we choose κ, n 0 , A and ǫ as follows.
. This ensures that the third inequality in (5.2) holds.
Thus the claim holds, and we get that |u(z)| e −σ|z| 2 for some σ > 0.
In this case, the equation is equivalent to Let us compute the kernel of (Λ + α) −1 . For all n ∈ N, (Λ + α) −1 ϕ n = (n + α) −1 ϕ n , then for u ∈ F 2 , We claim that there exists A ≥ 0 such that Let n 0 be the smallest integer such that n 0 + α > 0. Then In the sequel we assume |wz| ≥ 1. Then In the first integral, we make the change of variables s = t|wz| and get I 1 ≤ C. For the second, we get I 2 ≤ C(1 + |zw| 1−α )(e Re(zw) + |zw| n 0 −1 + 1). We also have the bound As a conclusion, from (5.5) and the previous estimates we get (5.4).
For n ≥ n 0 + 1 we write (n − n 0 ) −1 = Similarly, there exists A > 0 such that In the sequel, we assume that |z| ≥ 1. We set v = Π(|u| 2 u). Then, by Step 1, Then thanks to (5.3) and (5.6) The term J 3 is the contribution of the mode n 0 in the case α = −n 0 , and we have J 3 ≤ Ce − |z| 2 3 . Then we clearly have J 1 ≤ Ce − |z| 2 3 . We write This implies that We set M n = sup |w|>κ −n |w| A |u(w)| and N n = sup We are now able to conclude as in Step 1 by induction (here we need to initialize M n 0 and M n 0 +1 ).
Step 2: bootstrapping in(c n ) coordinates. Since |u(z)| e −σ|z| 2 for some σ > 0, we can bound the coordinates (c n ) of u by where Γ is Euler's Gamma function. By Stirling's formula, This means that |c n | r n for some r ∈ (0, 1). By (i) in Theorem 5.3, we obtain that, for any γ < γ 0 , |c n | n −γn .
Step 3: back to z coordinates. Using that |c n | n −γn , for γ < γ 0 , we get by Stirling's formula that By Young's inequality, which is the desired result.
They satisfy (ii) Besides the ϕ α n , M P -stationary waves in E with a finite number of zeros and unit mass are given, modulo space and phase rotation, by ψ b (e −iµt z)e −iλt , where .

They satisfy
(iii) M -stationary waves in E \E with a finite number of zeros are given, modulo space and phase rotation, by (iv) Besides the previous example, M P -stationary waves in E \ E with a finite number of zeros are given, modulo space and phase rotation, by We postpone the proof of Theorem 6.1 to Paragraph 6.3, and refer to Section B for the expression of these stationary waves in different coordinates. 6.2. An invariant three-dimensional submanifold. As a consequence of identifying ψ b in Theorem 6.1 as a stationary wave, we prove that the three-dimensional manifold u(z) = (λz + µ) e αz− |z| 2 2 , λ ∈ C * , µ ∈ C , α ∈ C , (6.1) is invariant by the flow of (LLL). This allows to recover results of [7] which were obtained by a direct calculation.
Thus, up to the symmetries of the equation, every solution to (LLL) corresponding to an initial condition of the form (6.1), is a stationary wave.
Proof. It is clear that multiplication by c ∈ C * , action of L ϕ and of R a act on the manifold defined by (6.1). With an operator R a we can reduce to the case when C z|u(z)| 2 dL(z) = 0. Then the transform L ϕ allows to reduce to the case α ∈ R, and by multiplication by c we can assume that λ = 1. Hence, we are reduced to with α ∈ R -the calculation can be easily made using identity (6.4) below. We now claim that (6.2) is satisfied if and only if there exists b ∈ R such that α = b b 2 +1 and µ = − b(2+b 2 ) 1+b 2 , and this will complete the proof.

6.3.
Proof of the classification result. We will simply solve the equations λu = Π |u| 2 u and λu + µΛu = Π |u| 2 u , over λ, µ ∈ R, and u ∈ E. First we need a result describing functions in E with a finite number of zeros.
Step 1: functions u ∈ E with a finite number of zeros. Write u(z) = e − 1 2 |z| 2 f (z), let z 1 . . . z k be the zeros of f and define P (z) = k j=1 (z − z j ). Then f (z) P (z) is an entire function which does not vanish, thus it can be written f (z) P (z) = e Q(z) , where Q is an entire function. By the bounds on u, Q is such that ReQ(z) z 2 . The Borel-Caratheodory lemma implies that |Q(z)| enjoys the same bounds, namely |Q(z)| z 2 , which means, by the Liouville theorem, that Q is a polynomial of degree at most 2. As a conclusion, any function satisfying the hypotheses of the proposition is of the type u(z) = P (z)e Q(z)− 1 2 |z| 2 , where P and Q are polynomials, and the degree of Q is at most 2.
Step 2: Q of degree 1, µ = 0. We look for u of the form P (z)e αz− 1 2 |z| 2 , with α ∈ C, and P a polynomial, solving λu = Π|u| 2 u. Recall the Gaussian integral identity For any polynomial P in w, w, this implies Let n ≥ 0 be the degree of P ; the Taylor expansion of the polynomial P at point a/2 gives Observe that (∂ b − a 2 )e ab 2 = 0, then by (6.5) we get If u solves λu = Π(|u| 2 u), then the polynomial in z appearing in the r.h.s. must have degree n. This is the case if and only if P (k) (α) = 0 for all 0 ≤ k ≤ n−1, hence P takes the form P (z) = A(z −α) n , with A ∈ C. Conversely, with (6.6) we check that u(z) = A(z − α) n e αz− 1 2 |z| 2 is a stationary wave. There remains to normalize it to have mass one, giving ϕ α n .
Step 3: Q of degree 1, µ = 0. Proceeding as in the previous step, for u of the form P (z)e αz− 1 2 |z| 2 , the equation λu + µΛu = Π|u| 2 u is equivalent to the equality between polynomials If P has degree n, the polynomial on the l.h.s. has degree n + 1, so this must be the degree of the polynomial on the r.h.s. This is only possible if P (k) (α) = 0 for 0 ≤ k ≤ n − 2, in other words, P (z) = (z − α) n + β(z − α) n−1 -taking without loss of generality the coefficient of (z − α) n to be 1.
This leaves us with the stationary wave u 1+b 2 z , which we need to normalize to have mass one. Using the identity e −|w| 2 +aw+cw dL(w) = πe ac , we obtain (noticing after the first equality that −|z| 2 This leads to the formula for ψ b = u b u b L 2 ; proceeding similarly, one computes H(ψ b ) and P (ψ b ). By Lemma 5.2, Q(ψ b ) = 0.
Step 4: Q of degree 2, µ = 0. In other words, we now look for solutions of λu = Π(|u| 2 u) of the type P (z)e Az 2 +Bz− 1 2 |z| 2 , where A, B ∈ C and P is a polynomial. We start from the following Gaussian integral. For any complex numbers a, b, c, d such that the integral converges absolutely, .
Notice that the convergence of the integral implies Re(1 − cd) > 0, so that the square root of 1 − cd is defined classically. This identity implies, for a polynomial P of w and w 1 π e −2|w| 2 +aw+bw+cw 2 +dw 2 P (w, w) dL(w) = P (∂ a , ∂ b ) 1 2 √ 1 − cd e da 2 +cb 2 +2ab 4(1−cd) . (6.9) Therefore, For u to be a stationary wave, the coefficients of z 2 and z in da 2 +cb 2 +2ab 4(1−cd) , with a = 2B, b = z + B, c = 2A, d = A, must be A and B respectively. A small computation shows that the coefficients of z 2 agree if A = A 2(1−2|A| 2 ) , which gives A = 0 (in which case we are back to step 2), or |A| = 1 2 . By rotation invariance, we can assume A = 1 2 ; but then the coefficients of z agree if B = is, with s real.
Finally, observe that, if the degree of P is n, the degree of the polynomial Q such that Π|u| 2 u = Q(z)e − 1 2 |z| 2 + 1 2 z 2 +isz , as determined by the formula above, is 3n. Therefore, n = 0.

Moreover, the above estimate is an equality if and only if
for some λ, α ∈ C.
Proof. The proof is inspired from the one of Lemma 1 of [15]. Recall that u = +∞ n=0 c n ϕ n , with ϕ n (z) = 1 √ πn! z n e − |z| 2 2 , so that We then classically write , and observe that We notice that the functions ϕ ℓ ϕ 0 are orthogonal in L 2 (C) and that Consequently, where we used the Cauchy-Schwarz inequality. Furthermore, equality holds if and only if, for every ℓ ≥ 0, there exists γ ℓ such that or √ n!c n = λα n for some α, λ ∈ C. Plugging this information into the formula, we get exactly The proof is complete.
Next, we aim at classifying maximizing sequences of H at M fixed. This will be achieved through the following profile decomposition lemma, in the spirit of [31], [14], [6], [25], [30].
Lemma 7.2. Consider a sequence (u n ) ∈ E with u n L 2 (C) = 1. Then there exist (v j ) ∈ E, a sequence (α j n ) ∈ C with |α j n − α k n | −→ +∞, n −→ +∞, j = k, and such that we have, up to a subsequence, the decomposition and for all J Proof. If u n L ∞ (C) −→ 0, we can take J = 0 and w n = u n . If not, then there exists ǫ 1 > 0 such that, up to a subsequence, and for n large enough ǫ 1 ≤ u n L ∞ (C) ≤ 2ǫ 1 and there exists α 1 n ∈ C such that |u n (α 1 n )| ≥ ǫ 1 . We define v 1 n = R −α 1 n u n ∈ E which satisfies |v 1 n (0)| ≥ ǫ 1 .

(7.2)
Next we write v 1 n (z) = f 1 n (z)e −|z| 2 /2 , where f 1 n is entire. By the Carlen inequality, for all z ∈ C, Therefore, for all K > 0 and n ≥ 1, we get |f 1 n (z)| ≤ C K , |z| ≤ K. By the Montel theorem, there exists an entire function f such that, up to a subsequence f 1 n −→ f 1 , uniformly on any compact of C, and we can set v 1 (z) = f 1 (z)e −|z| 2 /2 ∈ E. Moreover (7.2) implies v 1 L ∞ (C) ≥ ǫ 1 . Next, up to a subsequence v 1 n ⇀ v 1 in L 2 (C). We define w 1 n = R α 1 n (v 1 n − v 1 ), thus u n = R α 1 n v 1 + w 1 n , and u n 2 . Now we repeat the procedure for the sequence (w 1 n ). Either w 1 n L ∞ (C) −→ 0 or there exists ǫ 2 > 0 such that ǫ 2 ≤ w 1 n L ∞ (C) ≤ 2ǫ 2 . Then similarly, n ∈ C and w 2 n ∈ E. Similarly we check the almost orthogonality condition If we had that, for a subsequence α 1 n − α 2 n −→ ℓ ∈ C, this would be in contradiction with the fact that R −α 1 n w 1 n , R −α 2 n w 2 n ⇀ 0 in L 2 (C) and v 2 = 0. As long as the remainder term does not converge to 0 in L ∞ , we construct a sequence (v j ) ∈ E such that with v j L ∞ ≥ ǫ j and with κ J n −→ 0 when n −→ +∞. Then from Carlen and the previous line which implies that ǫ J −→ 0 and therefore w J n L ∞ (C) −→ 0. Here is a classical consequence of this profile decomposition. .
Then, up to extracting a subsequence, there exists β n ∈ C and θ ∈ T such that Proof. Up to extracting a subsequence, we apply the profile decomposition From Hölder's inequality, we infer lim sup and, using we have

Now apply the Carlen inequality to each profile
We obtain This implies that all the inequalities above are equalities, in particular there is only one j -say j = 1-such that v j = 0, and w J n = w n → 0 in L 2 . In particular, v 1 is a minimizer of the L 4 − L 2 Carlen inequality with mass π, so there exists α ∈ C such that v 1 = e iθ R α e − |z| 2 2 , which yields the result by setting β n = −α 1 n − α. By using the phase rotation we can assume that Imċ 0 (0) = 0, which givesċ 0 (0) = 0. An immediate computation shows that (everything being evaluated at s = 0) d ds Making use of (7.3), this reduces to d ds Therefore ϕ 0 is a strict local minimizer iff 8 2 n + 2µn − 4 > 0 for any n ∈ N; but this is equivalent to µ > 1 2 .
(iii) This will be a direct implication of (iv) and (v), combined with Theorem 6.1.
To show that ϕ n cannot be a local minimizer for n ≥ 3, observe that, if 0 < ǫ < 1, Since 1 2 n−2 − (2n)! 2 2n−1 (n!) 2 < 0 for n ≥ 3, ϕ n cannot be a local minimizer. (v) A direct computation shows that Then for µ = 1/2, a variation of b may decrease this quantity, excepted in the case b = 0, but then ψ 0 = ϕ 1 which is treated in point (ii).
(ii) For µ ≥ √ 3 − 1, ϕ 0 is the unique global minimizer of G µ over {u ∈ E, M (u) = 1}. (iii) For µ ∈ (0, 5 32 ), the global minimizer of G µ has an infinity of zeros. Proof. (i) Consider a minimizing sequence (u n ) in {u ∈ E, M (u) = 1} of G µ . Then P (u n ) and M (u n ) are uniformly bounded. On the one hand, by (3.1), u n is uniformly bounded in B(0, R) for any R, and, by Cauchy's integral formula, so are all its derivatives; on the other hand, the L 2 mass of u n on B(0, R) ∁ is 1 R 2 . Therefore, (u n ) is precompact in L 2 , and a subsequence converges to u ∈ E such that M (u) = 1. By lower semi-continuity of G µ , we obtain that u is a minimizer.
(7.6) Notice that R µ ≥ 0 if µ ≥ 2 3 . Coming back to (7.5), we therefore observe that, for µ ≥ 2 3 , if the remaining real quadratic form in c 0 c 1 , c 1 c 2 is positive, which holds as soon as , this completes the proof of the inequality. If the equality holds for such µ, then R µ = 0, which means c k = 0 for k ≥ 2, and c 2 1 + √ 2c 0 c 2 = 0, so c 1 = 0. Hence u must be proportional to ϕ 0 .
(iii) is an immediate consequence of Proposition 7.4.
Remark 7.6. If one is interested about minimizing G µ among even functions in E, the situation is simpler: • If µ > 1 2 , ϕ 0 is the unique global minimizer. • If µ < 1 2 , the global minimizer has an infinity of zeros. The first claim follows from (7.5) and (7.6) by setting c 2n+1 = 0 for all n ≥ 0. We turn to the second claim. Let µ < 1/2, then by Theorem 6.1, the only possible minimizers with a finite number of zeros are the ϕ 2n , with n ≥ 1. But the proof of Proposition 7.4 (iv) shows that none of them is a local minimizer, among even functions.
As in [4], denote by λ a Lagrange multiplier associated to the problem ( Given M 0 , H 0 > 0, we study min Recall that, by Proposition 7.1, for all u ∈ E, u = 0, one has 8π H(u) M (u) 2 ≤ 1.
(ii) Let α ∈ R and recall the definition (2.1) of R α . Then H(R α u) = H(u) and M (R α u) = M (u), and we can check that Thus, if u realises the minimum in (7.7), we get C (z + z)|u(z)| 2 dL(z) = 0, and the same argument with R iα then implies C z|u(z)| 2 dL(z) = 0 -see also Lemma 5.2.
(iii) For this part, we rely on the classification in Theorem 6.1 of the M P -stationary waves which have a finite number of zeros. By the symmetries of the problem, we can assume that A = 1 and ϕ = 0.
Therefore, by (6.2), for all 2(1+b 2 ) 2 , and thus Hence if we choose M 0 , H 0 as in the proposition, the stationary solution we find has an infinite number of zeros, by Theorem 6.1.
Let us now study the sign of (7.15).
The quadratic form Q 2 + Q 3 is positive definite : Set c j = x j + iy j , then and the claim follows.
Under the constraints (7.10) and (7.12), the quadratic form Q 1 is non-negative : Set c j = x j + iy j . Then (7.10) and (7.12) imply that x 1 = bx 0 and x 2 = − √ 2x 0 . Therefore The matrix of this quadratic form has two positive eigenvalues (b 2 + 1) 2 and (b 2 + 3) , and the eigenvalue 0 has multiplicity 2, which corresponds to the symmetries T γ and L ϕ .
7.4. Stability of stationary waves with finite mass and a finite number of zeros.
Finally, the proof of (v) is mostly contained in [19,Section 8.2], but we include it here for the sake of completeness. Up to the symmetries of the equation, it suffices to consider the stationary wave ϕ n e −iωnt , with ω n = (2n)! π(n!) 2 2 2n+1 .
Our second example provides stationary waves having zeros located on γZ ∪ iπ kγ Z for some γ = 0 and for some integer k = 0.
The corresponding solution to (LLL) is v k e −iλt with λ = 1 8π .