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On a damped Szego equation (with an appendix in collaboration with Christian Klein)

We investigate how damping the lowest Fourier mode modifies the dynamics of the cubic Szeg{\"o} equation. We show that there is a nonempty open subset of initial data generating trajec-tories with high Sobolev norms tending to infinity. In addition, we give a complete picture of this phenomenon on a reduced phase space of dimension 6. An appendix is devoted to numerical simulations supporting the generalisation of this picture to more general initial data.


Introduction
In the last decade, a number of papers have tried to display growth of Sobolev norms of high regularity for solutions of globally wellposed nonlinear Hamiltonian partial differential equations. This question, raised by Bourgain in [1], [2] for the defocusing nonlinear Schrödinger equation on the torus, led to several contributions constructing solutions with a small initial Sobolev norm of high regularity and a big Sobolev norm at some later time, see [4], [5], [8], [13], [16], [19], [15], [14], [12]. The actual existence of unbounded trajectories was proved in [21], [17], [18], [7], [23], [24], [3], [22], [9]. In this paper, we intend to study a case where a weak damping can promote unbounded trajectories in Sobolev spaces with high regularity. This unexpected phenomenon will be displayed in the particular case of a weak damping applied to the cubic Szegő equation. It would be interesting to investigate how weak damping can perturb other Hamiltonian dynamics in the same way.
The cubic Szegő equation was introduced in [5] as a toy model of degenerate nondispersive Hamiltonian dynamics. A natural phase space is the intersection of the Sobolev space H where Π is the orthogonal projector from L 2 (T) onto L 2 + (T). An important property of equation (1.1) is the existence of a Lax pair structure, leading to action-angle variables [6]. On the other hand, the conservation laws do not control high Sobolev regularity of the solutions, allowing for long term infinitely many transitions between low and high frequencies, as proved in [7] -see also the lecture notes [9]. However, such solutions are very unstable. The goal of this paper is to investigate how these properties are changed when adding a damping term, which breaks both the Hamiltonian and the integrable structures. Such a problem for integrable systems seems very difficult. In order to make it more amenable, we choose a specific damping term which keeps part of the above structure. This leads to the following equation.
where α > 0 is a given parameter, which could be made equal to 1 after a scaling transformation, and (u|1) is the Fourier coefficientû(0) of u. Note that the momentum M(u) := (Du|u) = k≥1 k|û(k)| 2 is preserved by the flow. An easy modification of the arguments in [5] shows that (1.2) is globally wellposed on H s + := L 2 + ∩ H s for every s ≥ 1 2 . Our goal is to study the behaviour of solutions of (1.2) as t → +∞, in particular the growth of Sobolev norms H s for s > 1 2 .
Furthermore, this subset has an empty interior, since it does not contain any trigonometric polynomial. It turns out that the introduction of the damping term drastically modifies this asymptotic behaviour. Indeed, our main result is the following.
In fact, we obtain an explicit sufficient condition on initial data which drives to an exploding orbit in H s + (see Theorem 4 below). Theorem 1 calls for a number of natural questions.
(1) Is the open set Ω ∩ H s + dense in H s + ? (2) What is the rate of the growth of u(t) H s ? At this stage, we do not have a complete answer to these questions. Nevertheless, the evolution of (1.2) admits an invariant finite dimensional submanifold on which a complete description of the dynamics is available, providing a precise answer to questions (1), (2) in this particular setting. We denote by W the subset of functions u on T of the form where b, c, p ∈ C, c = 0, |p| < 1. Note that W is a closed submanifold of dimension 6 in H 1 2 + . One can prove that, if u is a solution of (1.2) with u(0) ∈ W, then u(t) ∈ W for every t ∈ R (see section 2 below). Given M > 0, we define the following hypersurface of W We also denote by C M the circle made of functions of the form which is a closed orbit for (1.2).
Let us say a few words about the ingredients of the proofs of Theorems 1 and 2. The first important feature of the damped Szegő equation Using LaSalle's invariance principle associated to this identity, one infers that limit points of u(t) as t → +∞ in the weak H 1 2 topology are initial data of solutions v of (1.1) satisfying (v(t)|1) = 0 for every t ∈ R. The second important argument relies on the Lax pair structure for the Szegő equation (1.1), which is given by where H u , K u are Hankel operators associated to u, and B u , C u are antiselfadjoint operators. It turns out that, though the identity for H u does not hold anymore for solutions of the damped Szegő equation (1.2), identity for K u remains valid for (1.2). As a consequence, the spectrum of the positive trace class operator K 2 u is conserved by the dynamics of (1.2). The connection between the two above arguments is made thanks to a characterization of initial data of solutions v of (1.1) satisfying (v(t)|1) = 0 for every t ∈ R, in terms of the spectral theory of H v and K v . If u is a solution whose H s norm does not tend to +∞, this allows to calculate the limit of the L 2 norm of u(t) in terms of the spectrum of K 2 u , leading to Theorem 1. As for Theorem 2, the Lax pair identity for K u implies that the dynamics of (1.2) preserves functions u such that operator K u has a given finite rank. Manifold W precisely corresponds to operators K u of rank 1. A careful study of the ODE system defined by (1.2) on W then leads to Theorem 2. This paper is organised as follows. Section 2 is devoted to recalling important facts about the Szegő equation and Hankel operators, and to establishing general properties of solutions of the damped Szegő equation. Theorem 1 is proved in Section 3, and Theorem 2 is proved in Section 4.
Let us mention that some introductory material to the Szegő equation can be found in [9], and in [10], where the results of this paper were announced.

Generalities on the damped and undamped Szegő equations
In the following, we denote by t → S(t)u 0 (respectively by t → S α (t)u 0 ) the solution of the Szegő equation (1.1) (respectively of the damped Szegő equation (1.2)) with initial datum u 0 .
2.1. The Lyapunov functional. As emphasized in the introduction, an important tool in the study of the damped Szegő equation (1.2) is the existence of a Lyapunov functional. Precisely, the following lemma holds.
Hence, t → u(t) 2 L 2 admits a limit at infinity and since On the other hand, we claim that From both observations, we conclude that |(u(t)|1)| tends to zero as t goes to infinity.
From Lemma 1 and the conservation of the momentum, the H 1/2 norm of S α (t)u 0 remains bounded as t → +∞, hence one can consider limit points u ∞ of S α (t)u 0 for the weak topology of H 1/2 as t → +∞. Another general lemma describes more precisely these limit points, according to LaSalle's invariance principle.
Proof. Denote by Q the limit of the decreasing non-negative function t → S α (t)u 0 2 L 2 . By the weak continuity of the flow in H 1/2 weakly in H 1/2 as n → ∞. Hence, thanks to the Rellich theorem, as n tends to infinity. On the other hand, by Lemma 1, Recall that, from (2.1), It forces (S α (t)u ∞ |1) = 0 for all t. Hence S α (t)u ∞ = S(t)u ∞ is a solution to the Szegő equation without damping.
In order to characterize u ∞ , we need to recall some results about the cubic Szegő equation.

2.2.
Hankel operators and the Lax pair structure. In this paragraph, we recall some basic facts about Hankel operators and the special structure of the cubic Szegő equation (1.1). We keep the notation of [7] and we refer to it for details. For u ∈ H 1 2 + , we denote by H u the Hankel operator of symbol u namely It is well known that, for u in H One can also consider the shifted Hankel operator K u corresponding to H S * u where S * denotes the adjoint of the shift operator Sf (x) := e ix f (x). This shifted Hankel operator is Hilbert-Schmidt as well, with Observe in particular that u 2 L 2 = Tr(H 2 u ) − Tr(K 2 u ). A crucial property of the cubic Szegő equation is its Lax pair structure. Namely, if u is a smooth enough solution to (1.1), then there exists two antiselfadjoint operators B u , C u such that Classically, these equalities imply that H u(t) and K u(t) are isometrically equivalent to H u(0) and K u(0) (see [7] for instance). In particular, both spectra of H u and K u are preserved by the cubic Szegő flow. It motivated the study of the spectral properties of both Hankel operators that we recall here. For u ∈ H 1 2 + , let (s 2 j ) j≥1 be the strictly decreasing sequence of positive eigenvalues of H 2 u and K 2 u . Following the terminology of [7], σ 2 is an . From the min-max formula and the fact that K 2 u = H 2 u − (·|u)u, it is possible to prove that the s 2 2j−1 correspond to H-dominant eigenvalues of H 2 u while the s 2 2j correspond to K-dominant eigenvalues of K 2 u . Furthermore, the eigenvalues of H 2 u and of K 2 u interlace and, as a consequence, if m j := dim ker(H 2 u − s 2 j I) andm j := dim ker(K 2 u − s 2 j I), then m 2j−1 =m 2j−1 + 1 asm 2j = m 2j + 1.
To complete the spectral analysis of these Hankel operators, we need to recall the notion of Blaschke product. A function b is a Blaschke product of degree m if for some p j ∈ C with |p j | < 1, j = 1 to m. As proved in [7] -see also [11] for a generalisation to non compact Hankel operators-, for any H-dominant eigenvalue s 2 2j−1 , there exists a Blaschke product Ψ 2j−1 of degree m 2j−1 − 1 such that, if u j denotes the orthogonal projection of u on the eigenspace ker( Analogously, for any K-dominant eigenvalue s 2 2k , there exists a Blaschke product Ψ 2k of degreem 2k such that, ifũ k denotes the orthogonal projection of u on ker( We proved in [7] that the sequence ((s 2 j ), (Ψ j )) characterizes u, and that it provides a system of action-angle variables for the Hamiltonian evolution (1.1). Namely, if u(0) has spectral coordinates ((s 2 j ), (Ψ j )) then u(t) has spectral coordinates We are now in position to characterize the asymptotics of the damped Szegő equation. We first remark that the equation inherits one Lax pair, the one related to the shifted Hankel operator K u . It comes easily from the fact the shifted Hankel operator associated to a constant symbol is identically 0 and so, if u(t) := S α (t)u 0 , then where C u is the antiselfadjoint operator given by (2.2). As a usual consequence, K Sα(t)u 0 is unitarily equivalent to K u 0 and for instance, the class of symbol u with K u of fixed finite rank is preserved by the damped Szegő flow. In particular is invariant by the flow since it corresponds to the set of symbol whose shifted Hankel operators are of rank 1. Another consequence is the following result. k } k denotes the strictly decreasing sequence of the eigenval- By the spectral analysis of the Hankel operator recalled above, there exists a Blaschke product . The evolution of the Blaschke product is given by It implies that (Ψ 2j−1 (0)|1) = 0 so that the degree of Ψ 2j−1 is at least 1 and hence, the multiplicity of s 2 2j−1 is at least 2. From the interlacement property, this eigenvalue is also an eigenvalue for K 2 u . Let {σ 2 k } k denote the strictly decreasing sequence of the eigenvalues of K 2 u . We denote by m k the multiplicity of σ 2 k as an eigenvalue of H 2 u and bym k its multiplicity as a eigenvalue of K 2 u . From the interlacement property, if k is odd, m k =m k + 1 and if k is even, m k =m k − 1. We now compute the L 2 norm of u(t):

Exploding trajectories
In this section, we consider trajectories of (1.2) in H s , s > 1 2 , along which the H s norm of u(t) tends to infinity as t → +∞. Let us define the functional where (σ 2 k ) k is the strictly decreasing sequence of positive eigenvalues of K 2 u . We prove the following result.
then the H s -norm of the solution of the damped Szegő equation Proof. Let us proceed by contradiction and assume that there exists a sequence t n → +∞ such that u(t n ) := S α (t n )u 0 is bounded in H s . We may assume that u(t n ) is weakly convergent to some u ∞ in H s + . By the Rellich theorem, the convergence is strong in H 1 2 where Σ(u 0 ) denotes the set of eigenvalues of K 2 u 0 and m(σ) the multiplicity of σ 2 ∈ Σ(u 0 ). By the Lax pair structure, the eigenvalues of K 2 u(tn) are the same as the eigenvalues of K 2 u 0 , with the same multiplicities, hence every eigenvalue σ 2 of K 2 u∞ must belong to Σ(u 0 ), with a multiplicity not bigger than m(σ). In view of identity (3.1), we infer that Σ(u ∞ ) = Σ(u 0 ) , with the same multiplicities. On the other hand, from Proposition 1, we know that u ∞ generates a solution of the cubic Szegő equation which is orthogonal to 1 at every time. Consequently, Theorem 3 gives remains constant and necessarily, by the Lyapunov functional identity (2.1), (S α (t)u 0 |1) = 0 so that in particular, (u 0 |1) = 0. Hence, the case u 0 2 L 2 = F (u 0 ) and (u 0 |1) = 0 drives to an exploding orbit in H s as well as the case u 0 L 2 < F (u 0 ). It ends the proof of Theorem 4.
3.1. The case of Blaschke products. As a particular case of initial datum satisfying u 0 2 L 2 = F (u 0 ) and (u 0 |1) = 0, we consider initial datum given by a Blaschke product.
From the spectral analysis done in [7], and in particular from Lemma 3.5.2, one obtains the explicit description of the eigenspace corresponding to 1. If Ψ is of degree N, this eigenspace is of dimension N + 1 hence 1 is H-dominant, of multiplicity N + 1 and the representation of Ψ through the non-linear Fourier transform is Ψ itself. In particular, the rank of H Ψ is N + 1. On the other hand, from the interlacing property, 1 is a singular value of K Ψ of multiplicity N and the rank of K Ψ is N. Hence K Ψ has only 1 as possible non zero singular value and F (Ψ) = 1. As Ψ L 2 = 1 we get the norm explosion as a corollary of Theorem 4.

3.
2. An open condition. As a second corollary of Theorem 4, we get the following result, which implies Theorem 1.
as t tends to +∞.
Proof. By elementary perturbation theory, it is easy to prove that function F is continuous at those u of H 1/2 + (T) such that K 2 u has simple non zero spectrum. Furthermore, in the particular case it is easy to check that K 2 u has rank one with 1 (1−|p| 2 ) 2 as simple eigenvalue. As u 2 L 2 = 1 1−|p| 2 , this function belongs to Ω, and moreover it belongs to every H s . In view of Theorem 4, this completes the proof.
We end this section by giving a simple example of functions in Ω: Example 1. The set of functions u 0 whose nonzero eigenvalues of H 2 u 0 and K 2 u 0 are all simple, and form the decreasing square summable list is a subset of Ω.

A special case
In this section, we restrict ourselves to the set introduced in (2.4). As recalled in section 2.2, W corresponds to rational functions u with shifted Hankel operator K u of rank 1 and hence it is preserved by the damped Szegő flow. It is straightforward to check that the system on variables b, c, p reads In this section, we provide a panorama of the dynamics of the damped Szegő equation on W.
In particular, we prove that the set of functions u 0 such that, for some t, condition In the next statement, we normalise the H s norm as follows. • If u 0 ∈ Σ M,α , then S α (t)u 0 tends to C M as t → +∞, and with λ(α, M) > 0 .
Indeed, let us consider the latter case. As lim t→+∞ |p(t)| 2 < 1, the set {u(t) , t ≥ 0} is relatively compact in any H s (T). Denote by u ∞ any H 1/2 limit of (S α (t)u 0 ). As W is closed in H Proposition 2. If u is a solution of (1.2) on W, either u(t) is bounded in any H s , s > 1/2, as t → +∞, and u 0 2 L 2 ≥ M, or the trajectory is exploding in the sense that u(t) H s tends to infinity for any s > 1 2 . So we can rephrase the statement of Theorem 5 in view of these observations. Theorem 5 claims that those data of momentum M for which p(t) → 0 as t → +∞ form the disjoint union of the circle C M and of a three dimensional submanifold Σ M,α , which is a union of trajectories converging exponentially to C M as t → +∞. Furthermore, outside of this set, 1 − |p(t)| 2 → 0 as t → +∞, with a universal rate where ρ(α, M) = α 2 +M 2 2αM 2 > 0. We split the proof of Theorem 5 into two parts. The first one is a careful analysis of the case |p(t)| → 1 through the differential equations corresponding to (1.2) on some reduced variables. The second part consists in reducing the system to a scattering problem.
Our task is to prove, under the additional assumption γ(t) → 0 as t → +∞, that As a first step, let us establish that ∞ 0 γ(t) 2 dt < +∞ .
We write the equation on ζ in system (4.4) as Notice that r ∈ L 1 (R + ) and that, as t → +∞, Integrating from 0 to T the imaginary part of both sides of (4.5), we obtain, usingγ = −2Imζ, The second step consists in coming back to (4.5) and integrating the imaginary part of both sides from t to +∞. Using |ζ| = O(γ √ β) and r = O(βγ) , we infer Using again that β ∈ L 1 (R + ), this yields Using the monotonicity of and, coming back to (4.6), we finally obtain (1)) .
Eventually, one gets which, in view of the expression (4.7) of κ, is the expected result.

4.2.
The stable manifold of the periodic orbit. We come to the second part of the theorem, characterising the trajectories of (1.2) in W which converge to C M as t → +∞. Since C M is a trajectory itself, we may focus on trajectories {u(t)} of momentum M which are disjoint of C M , and which satisfy ∀t ≥ 0 , u(t) 2 L 2 ≥ M. At this stage we are not sure that such trajectories exist. However we are going to establish a necessary condition on the asymptotic behaviour of (b(t), c(t), p(t)) as t → +∞.
As a first step, we will use a linearisation procedure that we now illustrate on a"baby example".
The solutions of the characteristic equation Notice that Re(λ + ) > 0 while Re(λ − ) < 0. This leads to In particular, t 0 |q 0 (s)| 2 ds tends to +∞ with t. Let us come back to the expression of the L 2 norm of our solution, Fix T > 0 such that Let us complete the proof of the second part of Theorem 5. First we extend to this general context the notation introduced in the proof of Proposition 3, Notice that Lemma 2. If {u(t)} is a trajectory of (1.2) in W, of momentum M, disjoint of C M and such that ∀t ≥ 0 , u(t) 2 L 2 ≥ M, then b(t) = 0 for every t, and there exists θ ∈ T such that, as t → +∞, Proof. Since u(t) 2 L 2 = |b(t)| 2 + M(1 − |p(t)| 2 ), the condition on u reads (4.12) ∀t, |b(t)| 2 ≥ M|p(t)| 2 .
Since the trajectory is disjoint from C M , b and p cannot cancel at the same time, and therefore b(t) = 0 for every t. From the ODE on c in (4.1), we have Recall that |b| 2 ∈ L 1 (R + ). Therefore (4.12) implies that and consequently In order to establish the other condition, we fix T ≫ 1 and we set ε = ε(T ) := |b(T )| .
Integrating from T to +∞ the identity we obtain (4.14) Consequently, where the last estimate comes from integrating (4.13). In other words, in any H s space, At this stage, we are going to describe u(t+T ) = S α (t)u(T ) for t ≥ 0 by means of the linearised equation, exactly as in the proof of Proposition 3. We obtain where v is the solution of the linearised problem and, for every R > 0, there exists C s,R independent of T such that This leads to In other words, the condition u(t + T ) 2 ≥ M reads as follows : for every R > 0, there exists c R > 0 such that, for every t ∈ [0, R], for every T > 0, Let us compute v(t). From the linearized problem, we find Let us focus on q 0 . Deriving the first equation, we have We are left with the following second order ODE, Again, the solutions of the characteristic equation are given by Again, Re(λ + ) > 0 > Re(λ − ) and this leads to Rephrasing (4.15), we must have, for every R > 0, for every t ∈ [0, R], for every T > 0, Computing the integral of |q 0 | 2 , we infer, for some uniform constant Taking the upper limit of both sides as T → ∞, we infer 2α a − α lim sup Coming back to the expression of A + (T ) and of c ∞ above, this yields This completes the proof of Lemma 2.
Remark 1. For further reference, it is useful to observe that the result of Lemma 2 can be made uniform. Indeed, assume that we have a sequence of solutions {u n } of fixed momentum M satisfying u n (t) 2 L 2 ≥ M for every t, n and such that (u n (T )|1) −→ T →+∞ 0 uniformly with respect to n. Then we claim that the two convergences established by Lemma 2 are also uniform with respect to n. Indeed, this is straightforward for c n (t), since, as we already noticed, As for the second quantity, we just need to reproduce the above linearisation proof by observing that the linearisation is made near a compact sequence of solutions {c n,∞ e −itM } with a family {ε n (T )} of uniformly small parameters, so that, for every R > 0, the bound is uniform with respect to n. The result then follows from the estimate 2α As a next step, we show that the corresponding solutions of the reduced system on β := |b| 2 , δ := M|p| 2 = M − γ, ζ := Mcpb, satisfy a nonlinear scattering problem.
From the first two equations, we infer On the other hand, from Lemma 2, we have as an elementary calculation using (4.11) shows. Combining the above two informations, we obtain (1)) .
In particular, for every ε > 0, there exists C ε such that The same estimate holds for δ(t), and, in view of |ζ| = (M − δ) √ βδ, for |ζ(t)|. Writing we observe that An elementary calculation -involving for instance A − αI-shows that the four eigenvalues of A are Now recall that we have proved hence, in view of the spectrum of A, since Q(X) is a quadratic-cubic expression of X, we infer Consequently there exists X ∞ ∈ R 4 such that e tA X(t) → X ∞ as t → +∞. Integrating from t to +∞, we infer In view of the spectrum of A, we have, for s ≥ t, This implies in particular e −tA X ∞ = O e −(a+α)t for every ε > 0, which, in view of the spectrum of A, imposes that X ∞ is an eigenvector of A for the eigenvalue λ = α + a, hence there exists β ∞ ∈ R such that Since β(t) > 0, this imposes in particular β ∞ > 0, so that, improving the remainder estimates by coming back to equation (4.18), we conclude, using again (4.11), Conversely, given any β ∞ > 0, one can easily solve equation ( Then the extension to the whole real line is ensured by, say, the identities which, combined with the first equation, lead to Finally, let us prove the last statement. From the fixed point argument mentioned above, it is easy to check that there exists K > 0 such that, if |X ∞ | is small enough, then By contradiction, this proves that, if β(0) ≥ C −1 , then β ∞ ≥ (C ′ ) −1 . The proof of the other inequality is slightly more delicate. Assume β(0) ≤ C. As a first step, we are going to prove that β(t) → 0 as t → +∞ uniformly. Indeed, since On the other hand, since β(t) + M − δ(t) is a decreasing function of t, we have so that β(t) ≤ C + M . Coming back to the equation, we infer that there exists B = B(α, M, C) such that Consequently, for every s ∈ t, t + β(t) B , we have and therefore In view of the estimate on ∞ t β, we conclude β(t) ≤ Ke −αt with K = K(α, M, C), which implies the uniform convergence of β(t) to 0. Then, using Remark 1 on uniformity in Lemma 2, we infer that, for every ε > 0, there exists T = T (α, M, C, ε) such that Coming back to identity (4.19), we infer, for every t ≥ T , Since we finally obtain . We have the same estimate for |X(t)|, and, coming back to the identity we conclude, by choosing ε small enough, that Notice that Lemma 3 combined with the estimate Finally, in order to describe the geometric structure of Σ M,α , we give a complete description of the asymptotic properties of u(t) as t → +∞.
Proof. The asymptotic property has already been established for c(t). Let us prove it for b(t) and p(t). Recall from system (4.1) that T > 0 big enough so that M > δ(T ) > 0, ζ(T ) = 0. Consider the solution (b 1 , c 1 , p 1 ) of the system in (b, c, p) with Then, by uniqueness of the Cauchy problem for the reduced system, we have Applying the first part of Lemma 4, there exists (θ 1 , ϕ 1 ) ∈ T × T such that as t → +∞, satisfies the system in b, c, p with the required asymptotic properties. Finally, let us prove the uniqueness of such a solution. Ifb,c,ζ is a solution of the same system with the same asymptotic properties, we first observe that, in view of the uniqueness in Lemma 3, Then we come back to the equations on b/ √ β and √ Mp/ √ δ that we derived in the beginning of this proof. We obtain This implies that ξ 1 : and consequently that ξ(t) ≡ 0 for t large enough. Hence b =b, p =p, and finally c =c from the definition of ζ.
In order to complete the proof of Theorem 5, we consider the mapping where u is the unique solution of (1.2) provided by the second statement of Lemma 4. In view of Lemma 4, the range of J is precisely the set Σ M,α . In order to prove that this set is a submanifold of dimension 3 of E M , it is enough to establish that J is a one to one proper immersion. The injectivity of J is trivial. Its smoothness with respect to β ∞ is a consequence of the fixed point argument in Lemma 3 ; the dependence with respect to (θ, ϕ) is much more elementary, since it reflects the gauge and translation invariances, hence it is smooth as well.
To prove the immersion property, we just have to check that, for every (β ∞ , θ, ϕ), the three vectors are independent. We claim that the subspace spanned by these three vectors is also spanned by ∂ t u(0), −i∂ x u(0), iu(0). Indeed, in view of Lemma 4 and of the invariances of equation (1.2), one easily checks the following identity, If these three vectors were dependent, this would mean that u is a traveling wave of equation (1.2). This would impose that (u|1) ≡ 0, hence u ∈ C M , which is impossible since Σ M,α is disjoint from C M . Finally, J is proper because of the last statement of Lemma 3.
The last statement to be proved is that Σ M,α ∪ C M is closed. Since C M is compact, it is enough to prove that the closure of Σ M,α is contained into Σ M,α ∪C M . Let (u n ) be a sequence of points of Σ M,α which tends to u ∈ W. Set (β n , θ n , ϕ n ) := J −1 (u n ). Since J is a homeomorphism onto its range Σ M,α , the only cases to be studied are β n → 0 and β n → +∞. Now we appeal to the last statement of Lemma 3. In the first case, we obtain that (u|1) = 0, and more generally that (S α (t)(u)|1) = 0, so that u ∈ C M . In the second case, we infer |(u n |1)| → ∞, which contradicts the fact that u n is convergent.
The proof of Theorem 5 is complete.
Appendix A. Numerical simulations in collaboration with C. Klein (Université de Bourgogne) As mentioned in the introduction, the complete study made in section 4 suggests that the open subset Ω of initial data which give rise to exploding orbits is a dense subset. Furthermore, it is natural to ask about the rate of the Sobolev norms for such trajectories. For instance, is it true that the square of the H 1 norm grows linearly for generic initial data, as in the case of exploding trajectories in W? The numerical simulations below suggest that these two questions have a positive answer.
To numerically study the damped Szegö equation (1.2), we approximate u by a trigonometric polynomial, i.e., we consider the discrete Fourier transform (DFT) of a vector u (in an abuse of notation, we use the same symbol for the function u and its discrete approximation) with components u n = u(x n ), where x n = −π + 2πn/N, n = 1, 2, . . . , N. The DFT can be computed efficiently with a Fast Fourier transform. Note that we work in the Hardy space, thus all coefficientsû k corresponding to negative wave numbers vanish. But for computing the DFT, the negative wave numbers are nonetheless important. The action of the projector Π is simply to put the coefficients of the negative wave numbers equal to zero. The approximation of the function u via a DFT implies that equation (1.2) is approximated via a finite dimensional system of ODEs. The latter is integrated with the standard explicit fourth order Runge-Kutta method. Derivatives with respect to x are computed in standard way by multiplying the coefficientsû k by ik. We apply a Krasny filter, i.e., we put Fourier coefficients with a modulus smaller than 10 −12 equal to 0 to address that we work with finite precision, and to take care of unavoidable rounding errors.
To test the code, we first study an element of W, namely u 0 (x) = e ix 1 − pe ix with p = 0.5 and α = 1, for which we validate the result of Theorem 5, namely, as t → +∞, We use N t = 10 5 time steps for t ∈ [0, 20] and N = 2 12 . During the computation we observe relative conservation of the momentum to the order of 10 −15 , i.e., essentially machine precision, and the Fourier coefficients decrease to the order of the Krasny filter. This means the solution is well resolved both in space and in time. As can be seen in Fig. A.1, the agreement between theoretical prediction and numerics is excellent, the asymptotic regime is reached for comparatively small values of t. The second example studies the case of an initial datum with two poles, u 0 (x) = e ix 1 − p 1 e ix + e ix 1 − p 2 e ix , which corresponds to a more complicated phase space than W, but still finite dimensional, since the rank of K u 0 is 2. We put p 1 = 0.7, p 2 = 0.8 and use the same numerical parameters as before. The relative conservation of the momentum is of the order of 10 −7 . The norm ||S α (t)u 0 || 2 H 1 in dependence of time for this example can be seen in Fig. A.2. The norm appears to grow linearly in time.
The third example illustrates the case of an arbitrary initial datum with a Gaussian profile, u 0 = Π exp(−10x 2 ). We use the same numerical parameters, this time for t ∈ [0, 1000]. The momentum is conserved to the order of 10 −13 . The norm ||S α (t)u 0 || 2 H 1 can be seen for this case in Fig. A.3. Once more the square of the H 1 norm grows linearly in time.