Explicit formula for the solution of the Szeg\"o equation on the real line and applications

We consider the cubic Szeg\"o equation i u_t=Pi(|u|^2u) in the Hardy space on the upper half-plane, where Pi is the Szeg\"o projector on positive frequencies. It is a model for totally non-dispersive evolution equations and is completely integrable in the sense that it admits a Lax pair. We find an explicit formula for solutions of the Szeg\"o equation. As an application, we prove soliton resolution in H^s for all s>0, for generic data. As for non-generic data, we construct an example for which soliton resolution holds only in H^s, 0<s<1/2, while the high Sobolev norms grow to infinity over time, i.e. \lim_{t\to\pm\infty}|u(t)|_{H^s}=\infty if s>1/2. As a second application, we construct explicit generalized action-angle coordinates by solving the inverse problem for the Hankel operator H_u appearing in the Lax pair. In particular, we show that the trajectories of the Szeg\"o equation with generic data are spirals around Lagrangian toroidal cylinders T^N \times R^N.

(1. 1) i∂ t u + ∆u = |u| 2 u, (t, x) ∈ R × M, Gérard and Grellier [17] remarked that there is a lack of dispersion when M is a sub-Riemannian manifold (for example, the Heisenberg group). In this situation, many of the classical arguments used in the study of NLS no longer hold. As a consequence, even the problem of global well-posedness of (1.1) on a sub-Riemannian manifold still remains open. In [16,17], Gérard  Endowing L 2 (R) with the usual scalar product (u, v) = R uv, we define the Szegö projector Π : L 2 (R) → L 2 + (R) to be the projector onto the non-negative frequencies, For u ∈ L 2 + (R), we consider the Szëgo equation on the real line: (1.2) i∂ t u = Π(|u| 2 u), (t, x) ∈ R × R.
Endowing L 2 + with the symplectic structure ω(u, v) = 4Im R uv, we have that the Szegö equation is a Hamiltonian evolution associated to the Hamiltonian E(u) = R |u| 4 dx defined on L 4 + (R). From this structure, we obtain the formal conservation law of the energy E(u(t)) = E(u(0)). The invariance under translations and under modulations provides two more conservation laws, the mass Q(u(t)) = Q(u(0)) and the momentum M (u(t)) = M (u(0)), where Q(u) = . First of all we recall some notions and properties concerning the Szegö equation. We refer the readers to [29] for more details. The main property of the Szegö equation is that it is completely integrable in the sense that it possesses a Lax pair structure [29,Proposition 1.4]. We first define two important classes of operators on L 2 + , the Hankel and Toeplitz operators. The Lax pair is given in terms of these operators in Proposition 1.1.
Then, as it was shown in Lemma 3.5 in [29], H u is Hilbert-Schmidt and C-antilinear. Moreover, it satisfies the following identity: As a consequence, H 2 u is a self-adjoint linear operator. A Toeplitz operator T b : L 2 Then, T b is C-linear and bounded. Moreover, T b is self-adjoint if and only if b is real-valued.
Proposition 1.1 (Proposition 1.5 in [29]). Let u ∈ C(R; H s + ) for some s > 1 2 . The cubic Szegö equation (1.2) is equivalent to the following evolution equation: In other words, the pair (H u , B u ) is a Lax pair for the cubic Szegö equation on the real line.
According to the classical theory developed by Lax [21], a direct consequence of the above proposition is the following corollary: Another consequence of the Lax pair structure is the existence of an infinite sequence of conservation laws. More precisely, the following corollary holds. Corollary 1.3. Define J n (u) := (u, H n−2 u u) for all n ≥ 2. Then J 2k (u), k ∈ N * , are conserved quantities for the Szegö equation. In particular, J 2 (u) = Q(u), J 4 (u) = E(u) 2 , and we recover the conservation laws of the mass and energy. Remark 1.4. Using the Mikhlin multiplier theorem, we can prove that J 2k (u) ≤ u 2k L 2k . Then, by the Sobolev embedding we have that J 2k (u) ≤ u 2k  In order to prove this statement, we recall a Kronecker-type theorem. Proposition 1.6 (Theorem 2.1 in [29]). Let u ∈ H . . m k = N , and Im(p j ) < 0 for all j = 1, 2, . . . , k then, we have that (1.8) Ran(H u ) = span C 1 (x − p j ) l j , j = 1, 2, . . . , k and l j = 1, 2, . . . , m j .
Proof of Proposition 1.5. By equation (1.6) and Proposition 1.6, we have that if u 0 ∈ M(N ), then rk(H u(t) ) = rk(H u 0 ) = N . Thus the corresponding solution u(t) ∈ M(N ) for all t ∈ R.
As a corollary of the Kronecker-type theorem [29, Remark 2.2], we also have that if u ∈ M(N ) then u ∈ Ran(H u ), i.e. there exists a unique element g ∈ Ran(H u ) such that This yields Π(u(1 −ḡ)) = 0, which gives: An important property of Hankel operators, that will be a key point in this paper, is their characterization using the shift operatorsT λ : L 2 More precisely, the bounded operator H : L 2 + → L 2 + is a Hankel operator if and only if (1.11)T * λ H = HT λ for all λ > 0, [26, p. 273]. The adjointT * λ : , is very inconvenient to use. Then, for rational functions u, we define the infinitesimal shift operator T : Ran(H u ) → Ran(H u ), and prove that In the general case, when u is not a rational function, u does not always belong to Ran(H u ). Thus, g satisfying (1.9) does not always exist. If such g does not exist, the above definition (1.12) of T does not make sense. We then propose, in Section 3, to extend a definition for T * (see (3.3) below) and pursue our work using T * rather than T .
Next, we recall the definition and the characterization of soliton solutions for the Szegö equation. See [29] for details. Definition 1.2. A soliton for the Szegö equation on the real line is a solution u with the property that there exist c, ω ∈ R, c = 0 such that In [29,Theorem 2] it was proved that all the solitons for the Szegö equation on R are of the form where φ C,p = C x−p , ω = |C| 2 4(Imp) 2 , c = |C| 2 −2Imp , C, p ∈ C, and Imp < 0. Hence, a soliton of the Szegö equation on R is a simple fraction u(t, x) = Ce −iωt x−ct−p ∈ M(1), where Im(p) < 0. We are now ready to state the main results of this paper. In the first place we find an explicit formula for the solutions of the Szegö equation with rational function data. Theorem 1.7 (Explicit formula in the case of rational function data). Suppose that u 0 ∈ M(N ) and H 2 u 0 has positive eigenvalues λ 2 1 ≤ λ 2 2 ≤ · · · ≤ λ 2 N . We will assume that λ j > 0 for all j = 1, 2, . . . , N . Choose a complex orthonormal basis We define an operator S(t) on Ran(H u 0 ) in the following way. Fix j ∈ {1, . . . , N }, and let λ 2 j be an eigenvalue of multiplicity m j . Moreover, let M j ⊂ N be the set of all indices k such that H u 0 e k = λ j e k . Then, S(t) in the basis {e j } N j=1 is defined by the matrix (1.15) Then, we have the following explicit formula for the solution of the Szegö equation: We extend the explicit formula to more general initial data, that are not necessarily rational functions, in the following corollary. Corollary 1.8 (A first generalization of the explicit formula). Let u 0 ∈ H 1/2 + be a general initial condition. Denote by {λ 2 j } ∞ j=1 the positive eigenvalues of the operator H 2 u 0 . We assume that λ j > 0 for all j ∈ N. Choose a complex orthonormal basis . We define an operator S(t) on Ran(H u 0 ) in the following way. Fix j ∈ N * , and let λ 2 j be an eigenvalue of multiplicity m j . Moreover, let M j ⊂ N * be the set of all indices k such that H u 0 e k = λ j e k . Then, S(t) is defined by Denote byS the closure of the operator S.
If the sequence {β j } j∈N is in ℓ 2 , then there exists g 0 ∈ Ran(H u 0 ) such that u 0 = H u 0 (g 0 ). Moreover, for Imz > 0, the following formula for the solution of the Szegö equation with initial condition u 0 holds: The condition {β j } j∈N ∈ ℓ 2 characterizes all initial data satisfying u 0 ∈ RanH u 0 . In particular, by (1.9), it is satisfied by all rational functions. However, simple non-rational functions, like e iαx x+i with α > 0, do not satisfy it, and hence Corollary 1.8 is not applicable. In the following theorem, we extend the explicit formula to even more general initial data. Theorem 1.9 (Explicit formula for general data). Let u 0 ∈ H s + , s > 1 2 , xu 0 ∈ L ∞ (R). With the notations in Corollary 1.8, we define an operator S * (t) on Ran(H u 0 ) in the following way. Fix j ∈ N * . If λ 2 j is an eigenvalue of multiplicity m j and M j ⊂ N is the set of all indices k such that H u 0 e k = λ j e k , then Let A be the closure of S * . Then, for Imz > 0, the solution of the Szegö equation writes Let S * λ be the semi-group of contractions whose infinitesimal generator is −iA. Then, the above formula is equivalent to a.e. λ ∈ R. Definition 1.3. A function u 0 ∈ M(N ) is called generic if the operator H 2 u 0 has simple eigenvalues 0 < λ 2 1 < λ 2 2 < · · · < λ 2 N and |(u 0 , e j )| = 0, for all j = 1, 2, . . . , N . We denote by M(N ) gen the set of generic rational functions in M(N ).
A function u 0 is called strongly generic if it is generic and, in addition, |(u 0 , e j )| = |(u 0 , e k )| for all k = j. We denote by M(N ) sgen the set of strongly generic rational functions in M(N ).
The sets M(N ) gen and M(N ) sgen are indeed generic, in the sense that they are open, dense subsets of M(N ). As in [16,Theorem 7.1], we have that det J 2(m+n) 1≤m,n≤N = 0 if and only if H 2k u (g), k = 1, 2, . . . , n, are linearly independent. Decomposing g, H 2 g, . . . , H 2(N −1) g in the basis {e j } N j=1 , we obtain that the determinant of the matrix which contains these vectors as columns is: where ν j := 1 λ j |(u, e j )|. Thus, the fact that g, H 2 g, . . . , H 2(N −1) g are linearly independent is equivalent to (u, e j ) = 0, j = 1, 2, . . . , N and λ j are all distinct. Therefore, is an open, dense subset of M(N ). By Theorem 1.14 below, we obtain that χ : M(N ) gen → Ω (see (1.16) blow) is a diffeomorphism. Since M(N ) sgen corresponds, through χ, to an open dense subset of Ω, it results that M(N ) sgen is also generic. Definition 1.4. We say that soliton resolution holds in H s for a solution u(t) of the Szegö equation, if u(t) can be written as the sum of a finite number of solitons and a remainder ε(t, x) with the property that lim t→±∞ ε(t, x) H s = 0.
Using the above explicit formula for the solution, we prove the following result: Theorem 1.10 (Solition resolution for strongly generic data). Let u 0 ∈ M(N ) sgen be a strongly generic initial data for the Szegö equation. Then, the corresponding solution satisfies the property of soliton resolution in H s for all s ≥ 0. More precisely, with the notations in Theorem 1.7, we have , and lim t→±∞ ε(t, x) H s + = 0 for all s ≥ 0.
Studying the case of non-generic initial data u 0 ∈ M(2), such that H 2 u 0 has a double eigenvalue λ 2 1 = λ 2 2 , we can prove that the soliton resolution holds in H s only for 0 ≤ s < 1/2. It turns out that the H s -norms with s > 1/2 of such non-generic solutions grow to ∞ as t → ±∞. Theorem 1.11 (Partial solition resolution for non-generic data). Let u 0 ∈ M(2) be such that H 2 u 0 has a double eigenvalue λ 2 > 0. Then the corresponding solution satisfies the property of soliton resolution in H s for 0 ≤ s < 1/2. More precisely, where the first term is a soliton with |C| = , and ε(t, x) → 0 in the all the H s -norms with 0 ≤ s < 1/2. However, ε(t, x) stays away from zero and is bounded in the L ∞ -norm and H 1/2 -norm.
As a consequence, we obtain the following result: Corollary 1.12 (Growth of high Sobolev norms). The Szegö equation admits solutions u(t) whose high Sobolev norms H s , for s > 1/2, grow to infinity: More precisely, there exists a solution u of the Szegö equation and a constant C > 0 such that u(t) H s ≥ C|t| 2s−1 for sufficiently large |t|.
Remark 1.13. Corollary 1.12 presents an example of solutions whose high Sobolev norms grow to infinity. We could observe this phenomenon by considering non-generic initial data u 0 such that the operator H 2 u 0 has a double eigenvalue. We believe that the nondispersive character of the Szegö equation plays an important role in the occurrence of this phenomenon. For example, consider the dispersionless NLS, iu t = |u| 2 u. Then, However, the situation is more subtle for the Szegö equation, due to the conservation of the H 1/2 -norm. In particular, this explains why, for the Szegö equation, only the H s -norms with s > 1/2 grow to infinity. Corollary 1.12 shows that the energy is supported on higher frequencies while the mass is supported on lower frequencies. This phenomenon is called "forward cascade" and is consistent with some predictions in the weak turbulence theory.
Previously, Bourgain constructed, in [4,5,6], solutions with Sobolev norms growing to infinity. He considered, however, Hamiltonian PDEs involving a spectrally defined Laplacian. For general (dispersive) Hamiltonian PDEs, such a phenomenon is not known, but there are several partial results in this direction. In [16,Corollary 5], Gérard and Grellier noticed the growth of Sobolev norms for the Szegö equation on T. However, their construction of a sequence of solutions u ε (t ε ) whose Sobolev norms become larger depends on the small parameter ε. In [9], Colliander, Keel, Staffilani, Takaoka, and Tao constructed solutions for the defocusing cubic NLS on T 2 whose high Sobolev norms become greater than any fixed constant at some time. Kuksin considered in [19] the case of small dispersion NLS, −i∂ t u + δ∆u = |u| 2 u, 1 with odd periodic boundary condition, where δ is a small parameter. He proved that Sobolev norms of solutions with relatively generic data of unit mass, grow larger than a negative power of δ. However, these constructions do not give an example of solution such that sup t u(t) H s = ∞.
In the following theorem we introduce generalized action-angle coordinates for the Szegö equation in the case of generic rational functions. Theorem 1.14 (Generalized action-angle coordinates). For u ∈ M(N ) gen denote by 0 < λ 2 1 < λ 2 2 < . . . λ 2 N the simple positive eigenvalues of H 2 u and by {e j } N j=1 an orthonormal basis of Ran(H u ) such that H u e j = λ j e j . Denote ν j = |(g, e j )|, φ j = arg(g, e j ), and γ j = Re (T e j , e j ).
As a corollary, we obtain that in the generic case, the trajectories of the Szegö equation spiral around toroidal-cylinders T N × R N , N ∈ N * . Then, u(t) ∈ T C(u 0 ) for all t ∈ R, and the set T C(u 0 ) is diffeomorphic to a toroidal cylinder T N × R N parameterized by the coordinates (2φ j , γ j ) N j=1 , where γ j ∈ R, 2φ j ∈ T. It seems difficult to extend Theorem 1.14 and Corollary 1.15 to arbitrary generic functions, which are not necessarily rational, as we did in Theorem 1.9. The main reasons are the lack of compactness and the fact that we are unable to characterize the conditions u 0 ∈ H s + , s > 1/2 and xu 0 (x) ∈ L ∞ (R) in terms of the spectral data. The present paper was inspired by [15], where Gérard and Grellier introduced actionangle coordinates for the Szegö equation on T. However, [15] does not treat the question of soliton resolution and growth of high Sobolev norms. Different difficulties are to be overcome in the two settings. In the case of R, these difficulties are mostly related to the infinitesimal shift operator T in (1.12), which does not appear in the case of T.
1.3. Structure of the paper. We conclude this introduction by discussing the structure of the paper with some details. In Section 2, we prove Theorem 1.7, i.e. find an explicit formula for the solution of the Szegö equation with rational function initial condition. In the case of other completely integrable equations like KdV and one dimensional cubic NLS, an explicit formula for solutions was determined by the inverse scattering method [2,10,14]. Since in our case the operator H u is compact, we will not apply the inverse scattering method. We find a direct approach to solve the inverse spectral problem for the Hankel operator H u , using the Lax pair structure and the commutation relation (1.13) between the operator H u and the infinitesimal shift T .
The inverse spectral problem for Hankel operators was considered in several papers, among which we cite [1,25]. Our results are more precise than the previous ones and allow us to have a formula for the symbol u of the Hankel operator H u only in terms of the spectral data.
Let us describe our strategy in Section 2. First we notice thatû(λ) = (u, e iλx g), λ > 0. Then, we introduce the operators S λ (t) = P u 0 U * (t)T λ (t)U (t), S(t) = U * (t)T (t)U (t) acting on Ran(H u 0 ). Exploiting the Lax pair structure, we obtain that Since S is defined using U (t) and since the definition of U (t) (1.5) depends on u(t) itself, the above formula is a vicious circle. To break it, we determine S without using U (t). The explicit expression for S is obtained by computing the commutator [H 2 u 0 , S] and the derivative d dt S(t). In Section 3, we prove Corollary 1.8 and Theorem 1.9. The proof of Theorem 1.9 uses an approximation argument, based on the remark that u ∈ Ran(H u ) for all u ∈ H 1/2 + . The crucial step is to define the "adjoint of the infinitesimal shift operator", T * , for functions which are not necessarily rational functions (it seems more delicate to define the operator T directly).
Notice that in Theorem 1.7, S is a matrix whose eigenvalues are not real and thus the inverse (S−xI) −1 can be explicitly computed. The result obtained in Theorem 1.9 is weaker. The operator S * acts between infinite dimensional spaces. Explicitly computing (A − zI) −1 or the semi-group S * λ comes down to solving an infinite system of linear differential equations. Therefore, Theorem 1.9 actually states that we can transform our nonlinear infinite dimensional dynamical system into a linear one.
In Section 4, we prove Theorem 1.10. The soliton resolution conjecture is believed to be true for many dispersive equations for which the non-linearity is not strong enough to create finite-time blow-up. However, this was proved only for few equations like KdV [11] and one dimensional cubic NLS [27,23], for which an explicit formula for the solution is available. For KdV, soliton resolution was proved in L ∞ and it was noticed that it is unlikely to hold in H 1 (R) (the remainder may carry a part of the initial energy). For NLS, soliton resolution was proved in L 2 . In this case, in addition to solitons, the solution contains a radiation term, which is a solution of the linear Schrodinger equation. For both KdV and NLS, the conjecture holds only for "generic" data. In Theorem 1.10 we prove that for strongly generic, rational functions solutions of the Szegö equation, soliton resolution holds in all H s , s ≥ 0.
In Section 5, we prove Theorem 1.11 and Corollary 1.12. We show that soliton resolution still holds, even for non-generic solutions, but only in H s , 0 ≤ s < 1/2. This is probably due to the fact that H 1/2 is the space of critical regularity.
The starting point in proving Theorems 1.10 and 1.11 is the explicit formula found in section 2, which we are able to write as a sum of simple fractions The key remark is that the complex conjugates of the poles of u(t), E j (t), are the eigenvalues of the operator T (t) acting on Ran(H u(t) ). In the strongly generic case, the eigenvalues of and thus, one of the poles of the solution approaches the real line as |t| → ∞. This causes u(t) H s to grow to ∞ if s > 1/2.
In Section 6, we prove Theorem 1.14 and Corollary 1.15. The Szegö equation is an infinite dimensional, completely integrable system. The Lax pair structure yields the existence of an infinite sequence of prime integrals J 2n = (u, H 2n−2 u u), n ∈ N. Since the finite dimensional manifolds M(N ) are invariant under the flow, by restricting the Szegö equation to M(N ), we obtain a 4N -dimensional, completely integrable system. The common level sets of the prime integrals J 2n are not compact. Then, a generalization of the Liouville-Arnold theorem [13,12] to the case of a 4N -dimensional, completely integrable system, with non-compact level sets, states the existence of generalized action-angle coordinates, if certain conditions are satisfied. In these coordinates (2N invariant action coordinates, k angle coordinates belonging to T, and 2N − k generalized angle coordinates belonging to R) the equation can be easily integrated. In Theorem 1.14, we explicitly introduce generalized action-angle coordinates in terms of the spectral data.
Our strategy is to use the Szegö hierarchy, i.e. the the infinite family of completely integrable systems corresponding to the Hamiltonian vector fields of J 2n . The difficulty consists in proving that γ j = Re (T e j , e j ) are the generalized angles.

Explicit formula for the solution in the case of rational function initial data
In this section we find the explicit formula for the solution in the case of rational functions data.
By the residue theorem we have that Using the Plancherel formula, we obtain that and thus (D l j −1 f )(p j ) = 0, for all j = 1, 2, . . . , k and l j = 1, 2, . . . , m j . Then, the classical property [26,Corollary 3.7 Moreover, equation (1.8) yields that 1 − b u ∈ Ran(H u ) and by (1.9) we have that Since H u is one to one on its range, we conclude that 1 − b u = g.
Proof. Denoting by F the Fourier transform, we have that By (1.10) we have thatū(1 − g) ∈ L 2 + . Thus, the first term is the Fourier transform at −λ < 0 of a function in L 2 + , and hence it is zero. Lemma 2.3. If u(t) is the solution of the Szegö equation corresponding to the initial condition u 0 ∈ M(N ) at time t and g(t) ∈ Ran(H u(t) ) is such that u(t) = H u(t) (g(t)), then we have: Proof. Differentiating with respect to t and using equations (1.4), (1.2), and (1.6), we have d dt Since U * (0) = U (0) = I, this yields the first equality. By equation (1.6) and using the fact that the operator H u is skew-symmetric, we can rewrite (2.4) as and since H u 0 is one to one on Ran(H u 0 ), the second equality follows.
In the following we denote the unitary operator e i t 2 H 2 u 0 by W (t). The skew-symmetry of the Hankel operator H u 0 yields We also set With these notations we have, by equation (1.6), that Definition 2.1. Let us denote by P u the orthogonal projection on Ran(H u ). We also denote by T λ , λ > 0, the compressed shift operators acting on Ran(H u ) by If u(t) is the solution of the Szegö equation with initial condition u 0 and T λ (t) acts on Ran(H u(t) ), then we define the operators S λ (t), λ > 0, t ∈ R on Ran(H u 0 ) by Notice that using (1.7), we have we define The infinitesimal shift operator is the linear operator T defined on Ran(H u ) by: is the solution of the Szegö equation with initial condition u 0 and T (t) is the operator T acting on Ran(H u(t) ), we introduce the family of operators S(t) acting on Ran(H u 0 ), by Lemma 2.4. The eigenvalues of T and S are the complex conjugates of the poles of u. In particular, the eigenvalues of T and S have strictly positive imaginary part.
Proof. Since T and S are conjugated, they have the same eigenvalues. If T f = λf , then we have that (x − λ)f = Λ(f )b u . Taking x = λ, we obtain that b u (λ) = 0. Then, Lemma 2.1 yields that λ =p j .
Remark 2.5. Notice that we can extend the definition of Λ to We then use formula (2.9) to extend the definition of T to T |u| 2 Ran(H u ) .
Lemma 2.6. The operator iS is the infinitesimal generator of the semi-group S λ , i.e. S λ = e iλS for all λ > 0.
Proof. Because of the definitions of S and S λ in terms of T and T λ , it is enough to prove that where T and T λ act on Ran(H u ). Define the linear operator L : where p 1 , p 2 , . . . , p k are the poles of u and m j is the multiplicity of the pole p j . Then, In particular, L |Ran(Hu) : is the solution of the Szegö equation corresponding to the initial data u 0 ∈ M(N ), then the following formula holds: Proof. Using the Cauchy integral formula, Plancherel's identity, equation ( The above formula also holds for x ∈ R since, by Lemma 2.4, the eigenvalues of S are not real numbers. Notice that in this formula for u(t), the operator S(t) is defined using U (t) whose definition depends on u(t). Our goal is to characterize S(t) without using U (t). In order to do that, we need to determine the derivative in time of S(t)h, for any h ∈ Ran(H u 0 ). This derivative is expressed in terms of commutators of T with Hankel and Toeplitz operators, that we compute in the following.
Therefore we obtain To conclude, we only need to rewrite the two parenthesis so that they do not depend on U . By equation (1.6), the definitions of g,ẽ, and equation (1.3), we have: In order to express S without using U (t), we also need to determine the adjoint S * of the operator S and prove the commutation relation S * H u 0 = H u 0 S. We first determine T * .
Lemma 2.11. The adjoint of the operator T on Ran(H u ) is the operator T * defined by Proof. By equations (2.11), (2.1), and (2.12), for all f 1 , f 2 ∈ L 2 + we have that: Proof. By projecting equation (1.11) on Ran(H u ), we obtain T * λ H u = H u T λ . Then, by Lemma 2.6 it follows that T * H u = H u T . This and equation (1.6) yield for all h ∈ Ran(H u 0 ) that Notice that (2.13) yields that Then, (2.20) follows immediately by conjugating the above relation with U (t).
Proof of Theorem 1.7. By conjugating equation (2.17) by U (t), we obtain: for all h ∈ Ran(H u 0 ). Applying this to h = e j we have Suppose that λ j is an eigenvalue of multiplicity m j and that M j is the set of all indices k such that H u 0 e k = λ j e k . Pluggingẽ = N k=1 (ẽ, e k )e k in the above formula we have: Since (ẽ, e j ) = (e i t 2 H 2 we have Identifying the coefficients of (H 2 (e j ,ẽ)(ẽ, e k ) + (ẽ, e j )(e k ,ẽ) Therefore, for k ∈ M j we have where c k j (0) = (S(0)e j , e k ) = (T e j , e k ).

Extension of the formula to general initial data
Proof of Corollary 1.8. The proof of Theorem 1.7 can be adapted to the case of a general initial data, as long as u 0 ∈ Ran(H u 0 ), i.e. there exists g 0 ∈ Ran(H u 0 ) such that u 0 = H u 0 (g 0 ). Writing g 0 = ∞ j=1 (g 0 , e j )e j in the basis {e j } ∞ j=1 , the fact that g 0 ∈ L 2 (R) is equivalent to ∞ j=1 |(g 0 , e j )| 2 < ∞. Since u 0 = H u 0 (g 0 ) yields (u 0 , e j ) = λ j (e j , g) for all j ∈ N * , it follows that {β j } ∞ j=1 = { 1 λ j (u 0 , e j )} ∞ j=1 ∈ ℓ 2 . The main difference with the case of rational functions data is that S is no longer a matrix, but an operator acting between infinite dimensional spaces. Then, the infinitesimal generator of the semi-group S λ is not iS, but its closure iS (like in Proposition 3.4). This explains the operatorS appearing in the explicit formula. Proof. The local well-posedness follows using a fixed point argument in the space (L ∞ t , X), where By equation (2.11), the Hölder inequality, and Sobolev embedding, we have: x . The global well-posedness is a consequence of the Brezis-Gälouet estimate and of Gronwall's inequality. Proof. For h ∈ L 2 + , we have that Taking h ∈ Ker(H u ), it follows that (u, h) = 0 and u ∈ (Ker(H u )) ⊥ = Ran(H u ).
By (1.3), the above equation also yields that for all h ∈ L 2 + , we have that Then, H u 1 1−iεx converges weakly to u in L 2 + . We now intend to prove that, if u ∈ H s (R) and xu(x) ∈ L ∞ (R), then H u This yields that the convergence is strong in L 2 + . Computing the Fourier transform with the residue theorem, we have that By the Sobolev embedding H s (R) ⊂ L ∞ (R) for s > 1 2 , we have that there exists C 0 > 0 such that |u(x)| ≤ C 0 for all x ∈ R. Since u is a holomorphic function in C + , we can write using the Poisson integral for all z ∈ C + . Thus, u is bounded in C + ∪ R. Similarly, since xu(x) ∈ L ∞ (R), we have that zu(z) is bounded in C + ∪ R by a constant C 1 . In particular, we have that i ε u( i ε ) ≤ C 1 and thus lim ε→0 u( i ε ) = 0. Then, by (3.2), we have that H u 1 1−iεx converges pointwise to u(x). Furthermore, Then, the functions H u 1 1−iεx 2 are bounded by an integrable function. By the dominated convergence theorem, it follows that H u The key point in extending the explicit formula for the solution to the case of general initial data is the below definition of the operator T * : Ran(H u ) → L 2 + , If xu ∈ L ∞ (R), by (2.11) we have that Proof. For all f ∈ Ran(H u ) and h ∈ Ker(H u ), we have that Then, T * f ∈ (Ker(H u )) ⊥ = Ran(H u ).
For λ > 0, we introduce the operators T * λ : L 2 Let us now conjugate T * and T * λ using the operators U (t). We obtain S * (t) and S * λ (t): S * (t) = U * (t)T * U (t), S * λ (t) = U * (t)T * λ U (t). Proposition 3.4. The closure of the operator −iS * is the infinitesimal generator of the semi-group S * λ . Moreover, Ran(H u 0 ) is a core for the infinitesimal generator of the semigroup S * λ .
Conjugating with U (t), we obtain that the restriction of the infinitesimal generator of S * λ to Ran(H u 0 ) is −iS * . Moreover, by conjugating T * λ H u = H u T λ with U (t), we obtain S * λ H u = H u S λ . This yields S * λ (Ran(H u 0 )) ⊂ Ran(H u 0 ). By Theorem X.49, vol. II in [30], we have that Ran(H u 0 ) is a core of the infinitesimal generator of S * λ . Then, the infinitesimal generator of S * λ is the closure −iA of −iS * .
Proof of Theorem 1.9. According to Proposition 3.1, we have that u(t) ∈ H s and xu(t, x) ∈ L ∞ (R) for all t ∈ R. Then, by Lemma 3.2, we obtain that By Plancherel's identity, this is equivalent to Since, we obtain that The rest of the proof follows the same lines as the proof of Theorem 1.7, but uses T * and S * instead of T and S. Special attention should be given to the fact that the infinitesimal generator of the semi-group S * λ is not −iS * , but its closure −iA.

Soliton resolution in the case of strongly generic, rational function data
We prove that all the solutions with strongly generic, rational function initial data u 0 ∈ M(N ) sgen resolve into N solitons and a remainder which tends to zero in all the H s -norms for s ≥ 0, when t → ±∞.
Proof of Theorem 1.10. The strategy is to write all the vectors in Ran(H u 0 ) in the basis {e j } N j=1 and make formula (2.10) more explicit. According to Theorem 1.7, we have Since a ji (t) are linear combinations of e ±i t 2 (λ 2 j −λ 2 i ) with constant coefficients, there exists M > 0 such that for all j = i and all t ∈ R. Denoting A j = λ 2 j ν 2 j 2π t + (S(0)e j , e j ), the operator S in the basis {e j } N j=1 can be written as the following matrix: Then, we notice that Therefore, and in H s , s ≥ 0, as t → ±∞, since it is equal to a linear combination of . We notice that, using the definition of the determinant, the terms 1 x−A j do not appear in the above linear combination. Then, where C jj is the cofactor of A j − x, equal to the sum of ( and a linear combination of terms containing at most N − 2 factors (A j − x), and C ij , i = j is the cofactor of a ij , equal to a linear combination of terms containing at most N − 2 factors (A j − x). Then, we have Therefore, Since u 0 = N j=1 (u 0 , e j )e j and by (1.3), (4.3) (u 0 , e j ) = (H u 0 g 0 , e j ) = (H u 0 e j , g 0 ) = λ j (g 0 , e j ) = λ j β j , we have x−Ā j is a soliton of speed c = λ 2 j ν 2 j 2π and frequency ω = λ 2 j . Let us notice that the result in Theorem 1.10 can also be restated in terms of N -solitons.
If, moreover, there exist δ j ∈ R, j = 1, 2, . . . , N such that we say that the N -soliton is pure or that the collision of the N solitons C j (t) x−p j (t) is elastic, in the sense that there is no loss of energy in the collision. Theorem 1.10 states for s = 1/2 that if u 0 ∈ M(N ) sgen , then the corresponding solution is a pure N -soliton. Moreover, there is no shift in the trajectories of the N solitons, i.e. δ j = 0 for all j = 1, 2, . . . , N . This situation is characteristic to completely integrable equations. For the one dimensional cubic NLS, KdV and mKdV, which are all completely integrable, it is known that N -solitons exist and are pure [32,18]. For the gKdV equation with fourth order nonlinearity, which is not completely integrable, it was proved in [24] that the collision of solitons fails to be elastic by loss of a small quantity of energy.

Asymptotic behavior of the solution in the case of non-generic, rational function data
We show that when u 0 ∈ M(2) is such that H 2 u 0 has a double eigenvalue, then the solution u behaves as the sum of a soliton and a remainder, which tends to zero in the H s -norms, 0 ≤ s < 1/2. However, u(t) H s → ∞ if s > 1/2. An example of such an initial condition is u 0 = 2 x+i − 4 x+2i . The operator H 2 u 0 has the double eigenvalue ( 1 3 ) 2 in this case. Let us consider an orthonormal basis {ẽ 1 ,ẽ 2 } of Ran(H u 0 ) such that H u 0ẽ j = λẽ j . Denotingβ j = (g 0 ,ẽ j ) andν j = |β j | we have g 0 =β 1ẽ1 +β 2ẽ2 and g 0 2 L 2 =ν 2 1 +ν 2 2 . By (2.22), we have thatβ 1β2 ∈ R. We assume thatβ 1β2 =ν 1ν2 , and thusβ j = e iθν j for j = 1, 2.
Therefore, in the basis {e 1 , e 2 }, the operator S is the matrix and its characteristic equation is , we obtain that the discriminant of this equation writes where A, B, C are defined in (5.2). The eigenvalues of S will be written in terms of √ ∆, where we use the principal determination of the square root. In order to do so, we have to make sure that ∆ is not negative. We will show that when |t| is large enough, ∆ cannot be a real number. In what follows we suppose that t > 0. The case t < 0 can be treated similarly.
Proof of Corollary 1.12. Notice that the Sobolev norms of solitons are constant in time.
Then, the solution in Theorem 1.11, having a non-generic initial data u 0 ∈ M(2) such that H u 0 has a double eigenvalue, provides an example of a solution whose H s -norms, with s > 1/2 grow u(t) H s ≥ C|t| 2s−1 if s > 1/2 and |t| is big enough.
This does not contradict the complete integrability of the Szegö equation, since the conservation laws J 2n = (u, H 2n−2 u (u)) can all be controlled by the H 1/2 + -norm, as it was noticed in Remark 1.4.

Generalized action-angle coordinates
On L 2 + (R) we introduce the symplectic form for all u, h ∈ L 2 + (R). If the functions F, G : L 2 + (R) → R admit the Hamiltonian vector fields X F , X G , then we define the Poisson bracket of F and G by: A consequence of the Lax pair is the existence of an infinite sequence of conservation laws as we noticed in Corollary 1.3.
We now introduce the Szegö hierarchy, i.e. the evolution equations associated to the Hamiltonian vector fields of J 2n for all n ∈ N * , and prove that each of these equations possesses a Lax pair. We will need the following lemma: As a consequence, the following identity holds: Proof. The first equation is equivalent to f = Π(f ) + Π(f ) and it follows by passing into the Fourier space. Then Proposition 6.2. Let u ∈ H s + , s > 1 2 . The Hamiltonian vector field associated to J 2n (u) is Moreover, Proof. The proof follows using the above lemma and similar computations as in the proof of Theorem 8.1 in [16]. Denote x n J 2n+2 (u).
A computation shows that Identifying the coefficients of x n , we obtain the desired formula for X J 2n (u). For the second part of the proposition, we use . Identifying once more the coefficients of x n and using the fact that H u is a skew-symmetric operator, we obtain the formula for H X J 2n (u).
As in [16], the following result holds: In what follows we compute g ′ (t) and the commutator [T, B u,n ] and use this result to determine the evolution of the angles and generalized angles along the flow of X J 2n . Lemma 6.4. Let u ∈ C(R, H s + ), s > 1 be a solution of (6.6) and for all t ∈ R let g(t) ∈ Ran(H u(t) ) be such that H u(t) g(t) = u(t). Then Proof. In order to compute g ′ (t), we differentiate with respect to time the equality H u (g) = u: [B u,n , H u ]g + H u (g ′ ) = X J 2n (u). Thus Using the fact that H u is a skew-symmetric operator and is onto on its range, we obtain (6.9).
Since the product of two rational functions has Λ equal to zero, we notice that A similar computation yields the second equation in the statement.
Proposition 6.5. If u 0 ∈ H s + , s > 1 and u 0 ∈ M(N ) gen , then the solution u(t) of the equation (6.6) is contained in the toroidal cylinder T C(u 0 ) defined by (1.17), for all t ∈ R. Moreover, the angles φ j and the generalized angles γ j evolve along the flow of this equation as follows: Proof. Since the evolution equation (6.6) admits the Lax pair (6.7), the classical theory yields that if u(t) is a solution of (6.6), then where U n (t) is a unitary operator on L 2 + satisfying d dt U n (t) = B u,n U n , U (0) = I.
Therefore, the eigenvalues λ 2 j , j = 1, 2, . . . , N of H 2 u(t) are conserved in time. Moreover, if we denote by {e j (0)} N j=0 an orthonormal basis of Ran(H u 0 ) such that H u 0 e j (0) = λ j e j (0), then e j (t) = U n (t)e j (0) form a basis of Ran(H u(t) ) such that H u(t) e j (t) = λ j e j (t). Then, by (6.9) and using the fact that B u,n is a skew-symmetric operator, we have d dt e j (t), g(t) = B u,n e j (t), g(t) − i 4 e j (t), H 2n−2 u (g(t)) + e j (t), B u,n (g(t)) .
Proof. The proof follows the same lines as the proof of Theorem 1.7. The only difference is that we work with the orthonormal basisẽ j = e iφ j e j . Since H u is anti-linear, the orhonormal basis {e j } N j=1 satisfying H u e j = λ j e j is determined only modulo the sign of e j . Therefore, φ j = arg(e j , u 0 ) is determined modulo π. We intend to introduce generalized action-angle coordinates, and the angles should be defined modulo 2π. Considering the basisẽ j , the formulas we obtain only depend on 2φ j , which are therefore good candidates for the angles.
Proof of Theorem 1.14. Let us first notice that, if we prove that χ is a symplectic diffeomorphism, then the coordinates (2λ 2 j ν 2 j , 4πλ 2 j , 2φ j , γ j ) are canonical. Denote I j = 2λ 2 j ν 2 j . By equation E = 2J 4 = 2 N j=1 λ 4 j ν 2 j = N j=1 λ 2 j I j and using Proposition 6.5, we obtain that for the flow of the Szegö equation we have: Thus, the Szegö equation can be indeed rewritten as . The first step in proving that χ is a symplectic diffeomorphism is to compute the Poisson brackets between actions and (generalized) angles. This will lead to χ being a local diffeomorphism.
Since a bijective local diffeomorphism is a diffeomorphism, and we have by Proposition 6.6 that χ is one to one, we only need to show that χ is onto. A proper local diffeomorphism taking values in a connected manifold is onto. Thus, it is enough to show that χ is proper. 6.3. χ is a proper mapping. Let K ⊂ Ω be a compact set. Set Since I (p) → I andĨ (p) →Ĩ as p → ∞, this yields that u p H 1/2 is bounded. Consequently, there exists u ∈ H 1/2 + such that u p ⇀ u in H 1/2 + . It follows in particular that u p → u in L 2 loc . We denote by λ j (u), ν j (u), φ j (u), and γ j (u) the spectral data for u. By Proposition 6.6, we have that for all j ∈ {1, 2, . . . , N }. By equation (6.19), there exists R > 0 such that T (p) ≤ R 2 for all p ∈ N. Using the Neumann series, we have that if |x| ≥ R, then there exists A > 0 such that for all p ∈ N. This yields that lim R→∞ sup p |x|>R |u p (x)| 2 dx ≤ lim R→∞ |x|>R A 2 |x| 2 dx = 0.
Since u p → u in L 2 loc , this triggers u p → u in L 2 + (R).
Let us now prove that H up (h) → H u (h) in L 2 + , for all h ∈ L 2 + . First, notice that there exists C > 0 such that In particular, it follows that it suffices to prove that H up (h) → H u (h) for h in a dense subset of L 2 + , for example h ∈ L ∞ ∩ L 2 + . For such h, we have that As a consequence, we have that J 2n (u p ) → J 2n (u) as p → ∞. Indeed, we write For the first term we notice that For the other terms, in the case when j is even, we use the self-adjointness of the operator H 2 u . We then obtain: and the first factor tends to zero since H up−u (h) → 0 in L 2 + for all h ∈ L 2 + . For the case when j is odd, we use equation (1.3) and then proceed similarly.
We still need to show that e j (u p ) → ±e j (u). Using λ (p) j → λ j = λ j (u), ν (p) j → ν j = ν j (u), we have that Since u p ⇀ u in H 1/2 + and u → u in L 2 + , it follows that u p → u in H 1/2 + . This yields that H up → H u in the sense of the norm. As a consequence, setting P (p) j h := (h, e j (u p ))e j (u p ).
to be the orthogonal projection onto the eigenspace of H 2 up , corresponding to the eigenvalue λ (p) j 2 and similarly, P j (u)h := (h, e j (u))e j (u) to be the orthogonal projection onto the eigenspace of H 2 u , corresponding to the eigenvalue λ 2 j (u), we have by Theorem VIII.23 in [30], that P (p) j → P j (u). Therefore, (h, e j (u p ))e j (u p ) → (h, e j (u))e j (u) as p → ∞, for all h ∈ L 2 + . Taking h = e j (u), we have that (e j (u), e j (u p ))e j (u p ) → e j (u). Since e j (u p ) and e j (u) are unitary