Khaled Gaaloul
email: khaled.gaaloul@sap.com

François Charoy
email: charoy@loria.fr

Andreas Schaad
email: andreas.schaad@sap.com

Hannah Lee

Collaboration for Human-Centric eGovernment Workflows

Keywords: eGovernment, R4eGov, workflow coordination, workflow collaboration, delegation

The execution of cross-domain eGovernment processes is a challenging topic. In earlier work, we presented an approach based on collaborative workflows to support eGovernment interoperability. However, such collaborative workflows often appear to be lacking transparency and control supporting concepts and mechanisms. These are needed as eGovernment workflows appear to be heavily human-centric. What is in many cases described as collaboration appears to be a mere coordination and synchronization of processes, often ignoring human-centric interactions. One type of transparency and control supporting mechanism in human-centric collaboration is that of task delegation. In this paper we aim to analyse the gap between coordination and collaboration in the context of workflow management for eGovernment. First, we present a real case study to identify the key distinguishing factors regarding collaboration as opposed to coordination. Based on this, we present our approach to support cross-organisational collaboration. In particular, we will focus on the concept of delegation in the context of heavily human-centric collaborative workflows. Finally, we propose a delegation extension and structured set of future requirements regarding a coordination architecture presented in earlier work.

Introduction

Electronic government (eGovernment) is the civil and political conduct of government, including services provision, using information and communication technologies. The concept of eGovernment has been gaining ground from initial isolated to extensive research and applications. The prerequisites for an e-Government enactment strategy are the achievement of a technological interoperability of platforms and a deeper cooperation and security at the organisational level. Those requirements are related with the environment in which the public agencies operate, strictly constrained by norms, regulations, and result-oriented at the same time [START_REF]Electronic Government: Third International Conference, EGOV 2004[END_REF]. Actually, most governmental organisations offer electronic services within a collaborative environment. However, inter-organisational collaboration, especially by means of workflows, is not as widespread.

The R4eGov project consists of inter-organisational collaboration between European administrations [START_REF]Towards e-Administration in the large[END_REF]. An example domain for such collaboration is Europol 4 (European Police Office) and Eurojust5 (European Judicial Cooperation Unit). It describes an interagency collaboration within the areas of law enforcement and justice. One of the objectives is to establish a collaboration, including information exchange between both parties based on legal constraints, such as European laws, to which they have to comply to, but sustain effective degrees of freedom for each department to solve their issues in the way they think is the most efficient and effective [START_REF] Traunmüller | SIXTH FRAMEWORK PROGRAMME, Information Society Technologies, R4eGov[END_REF]. Those objectives can be achieved using collaborative workflows [START_REF] Schulz | Facilitating cross-organisational workflows with a workflow view approach[END_REF][START_REF] Contenti | A Distributed Architecture for Supporting e-Government Cooperative Processes[END_REF]. This is a novel approach supporting interoperability between organisations without the burden of centralized workflow management systems. The perspective is to enable a particular workflow model to be executed collaboratively by different workflow engines located on the private network perimeters of their respective owners. However, recent works [START_REF] Wolter | Collaborative Workflow Management for eGovernment[END_REF][START_REF] Indrakanti | Prototype extended collaborative workflow tool[END_REF] presented new requirements such as control and transparency in collaborative workflows. What is in many cases described as collaboration appears to be coordination and synchronization of processes by ignoring human-centric interactions. Actually, we need to consider all the relevant participating systems and workflows even if they are not directly involved in the current control-flow sequence of the workflow. This paper expands on earlier work we have done in R4eGov to support interorganisational collaboration between European administrations based-workflow. We aim to elicit the collaborative requirements between Europol and Eurojust and the definition of methods and tools to support such an human-centric collaboration. This collaboration requires transparency and control supporting concepts and mechanisms. The concept of delegation had not yet been treated in sufficient detail in the context of heavily human-centric collaborative workflows, and is the subject of this paper to foster transparency and control mechanisms in collaborative workflows according to global policies and European law regulations in R4eGov.

The remainder of this paper is organized as follows. Section 2 presents a workflow example inspired from an R4eGov scenario and shows the difference between workflow coordination and collaboration. We motivate in section 3 the use of collaborative workflow management for eGovernment and present our approach to support cross-organisational collaboration. In particular, we will focus on the concept of delegation. Section 4 extends our approach and presents some future requirements. Section 5 presents some related work regarding delegation. Section 6 concludes and presents some future works. We introduce in this section an R4eGov workflow scenario related to the European administrations collaboration. Europol and Eurojust are two key elements of the European system of international collaboration within the areas of law enforcement and justice. They carry out very specific tasks in the context of dialogues, mutual assistance, joint efforts and cooperation between the police, customs, immigration services and justice departments of the EU member states [START_REF] Traunmüller | SIXTH FRAMEWORK PROGRAMME, Information Society Technologies, R4eGov[END_REF]. During their collaboration, Eurojust and Europol are involved and a number of legal instruments are used. A Specific scenario for this collaboration is the Mutual Legal Assistance (MLA) 6 .

Mutual Legal Assistance (MLA)

Figure 1 depicts a global workflow scenario called Mutual Legal Assistance (MLA) involving two national authorities of different European countries regarding the execution of measures for protection of a witness in a criminal proceeding. This simplified collaborative workflow is inspired by and in parts derived from the case studies delivered in the European research project R4eGov [START_REF] Traunmüller | SIXTH FRAMEWORK PROGRAMME, Information Society Technologies, R4eGov[END_REF].

The workflow shows a member of the Europol National Unit in country A asking for an MLA request. The rules of procedure on the processing and protection of personal data at Eurojust refer to a "Case Management System" (CMS). The measure is to be executed in country B. Europol National Unit makes a written request of assistance (witness protection) to Eurojust National Member (EJNM) A. Then, EJNM A opens a Temporary Work File in the CMS, and contacts EJNM B forwarding the request of assistance. The EJNM B contacts the responsible national authority of country B. Finally, steps will be taken by the responsible national authority to provide the requested assistance. As shown in the example, four different parties are involved in this collaboration. After the Europol National Unit A sends the request to the corresponding contact point it waits until the follow-up is established by the other authorities. What is going on between the two EJNMs A and B is not mediated to the Europol National Unit A. Hence, this appears to be more than coordination and synchronization of processes between governmental organisations. Actually, several of the depicted tasks involve human interactions and are possibly time consuming. For instance, steps taken by the responsible national authority to provide the requested assistance might involve several stakeholders that decide on this subject, hereby considering different aspects like the severeness of the case or concurrent investigations. With respect to transparency, the current process status needs to be communicated to EJNM B. In addition, unexpected events can happen during any task (unexpected in the sense that it cannot be modeled beforehand, e.g. changing bilateral agreements between countries A an B) that need to be propagated to every other participant of the collaborative workflow (especially to the Europol National Unit A, the initiator of the collaborative workflow) even if they do not interact directly with each other in terms of control-flow.

Problem Statement

In the common understanding a collaborative workflow consists of one global workflow model composed of public views of each collaborative partner, with each public view abstracting a concrete private workflow behind it [START_REF] Contenti | A Distributed Architecture for Supporting e-Government Cooperative Processes[END_REF]. Existing choreography and collaboration approaches support the control-flow related message exchange that is part of the workflow model itself. Apparently this message exchange in case of status notification or cancellation is not part of the collaborative workflow model [START_REF] Wolter | Collaborative Workflow Management for eGovernment[END_REF]. What is in many cases described as collaboration or collaborative mechanisms appears to be coordination and synchronization of processes by ignoring human-centric interactions. The requirements for interactions and monitoring can be summarized as transparency and control [START_REF] Jensen | Collaboration, Leadership, Control, and Conflict Negotiation in the NetBeans.org Community[END_REF]. Transparency addresses the revelation of collaborative dependencies. This allows to react accordingly to exceptions and compensations implied by law regulations. Control fosters the behaviour of partners according to the collaborative policies (e.g. European laws).

This scenario depicts that we need to consider all the relevant participating systems and workflows even if they are not directly involved in the current control-flow sequence. Moreover, emergency situations can necessitate delegation of some activities intra and inter-organisations. For instance, EJNM B can delegate its part of the work to the last party. This delegation has to be legal and compliant with the R4eGov laws regulations policies. In the next section, we motivate the concept of delegation as a support for transparency and control within an human-centric collaboration between Europol and Eurojust.

An Extended Analysis of Collaboration for eGovernmental Workflows

As stated before, control and transparency are important for a successful eGovernment collaboration. Hence, we follow a decentralized approach, combining the local workflows to form a collaborative workflow, integrating the existing systems of the involved partners, and adding a decentralized collaborative administration architecture to support human interactions (e.g. during a delegation request).

Workflow Engine Encapsulation

One re-occurring requirement is to enable collaborative workflows across different organisations without changes to the existing IT landscape of each organisation.

A solution that enables collaborative workflows therefore needs to be built on top of existing solutions. Considering that assumption, the purpose of the workflow engine encapsulation is to offer a common interface to collaborative components (that would need to be deployed on each participants system) independent of the underlying workflow engine in place. Such an interface needs to work in both directions: The collaborative components need to access engine and process specific functionalities. The process, during its execution, needs to publish events or performs requests to the collaborative components. Dealing with those requirements, we propose to set up a workflow to workflow collaboration by realizing a layer which we call an Administrative Communication Layer (ACL).

Administrative Communication Layer (ACL)

Wolter and al. [START_REF] Wolter | Collaborative Workflow Management for eGovernment[END_REF] proposed an abstract modular infrastructure for collaborative workflow management and identified key components to leverage an existing workflow system onto decentralized collaboration. As indicated by Figure 2 the proposed architecture is divided into a control-flow layer and an administration layer.

In [START_REF] Indrakanti | Prototype extended collaborative workflow tool[END_REF], we developed a prototype extended collaborative workflow tool to support the collaboration between the MLA partners. The term Administrative Communication Layer refers to the distinction of administrative events (e.g. starting/completion of one task) with control events (e.g. triggering a workflow instance) in each workflow engine. Our collaborative communication is eventbased, on demand, or a combination of both. Therefore, ACL enables administrative information exchange by mediating information to the collaborative event management and process management components of the collaborative

Delegation Scenarios in MLA Request

Delegation is an important factor for secure distributed computing environments.

It consists of delegating a part of a work to another partner according to laws regulations policies. Delegation can be motivated by many factors (e.g. lack of resources, organisational policies, etc.) [START_REF] Atluri | Supporting conditional delegation in secure workflow management systems[END_REF] and can take place depending on the delegator/delegatee agreement. This agreement is closely related to the delegation criteria. In the following, we identify two different criteria of delegation from the MLA scenario:

Role-based delegation: The basic idea behind a role-based delegation is that users themselves may delegate role authorities to others to carry out some functions authorized to the former. Our interest is in the Eurojust's side, the main actors involved in Eurojust's information workflow are: Eurojust National Members and National Correspondents (NCs). Eurojust National Member of country A can play a role of a senior, an experienced prosecutor, or a judge. As a leader of the Eurojust organisation, EJNM, confronts problems that are particularly perplexing, collaborations are necessary for information sharing with members from the same organisation. Since EJNM believes in delegating responsibility, he would like to delegate certain responsibilities to the NC member where the former define the delegation condition based on the organisation Role Hierarchy (RH) (see figure 3). The delegation request is inspired from RDM2000 proposed by Zhang and al. [START_REF] Zhang | A rule-based framework for role-based delegation and revocation[END_REF]. Since the latter organisation role is JA, we assume that a delegation request can be motivated by the factor of specialisation. Nevertheless, the condition of delegation doesn't depend on the delegation factor (the specialisation of the JA member). Actually, a delegation request can be done if and only if the latter organisation offers the same service (set of tasks) to ensure the well and coherent deployment of the process. Coherency is closely related to the global policies of the MLA request. Here, our concern is the output of the task and not the "how" of the task. Since, the delegatee (JA member) provides tools, platforms and solutions to handle this delegation, there is no need to go deeper in his process attributes and then disclose his privacy. The agreement can be defined according to tasks alignment between organisations. This alignment is defined in the global policies of the process.

Extended Architecture

The scope of our approach is to address user-to-user delegation supporting human-centric collaborative workflows. We propose an abstract modular architecture to extend the architecture presented in [START_REF] Wolter | Collaborative Workflow Management for eGovernment[END_REF]. Once a delegator needs to delegate, ACL communicates a delegation request to the delegatee where the former invites the latter to accept to be in charge of one or more tasks of his local process. A preliminary prototype implementation of the delegation mechanisms has been developed. It is based on mail request where a delegator sends a delegation request and wait for the approval of the receiver (the delegatee). The delegation dialogue between the two actors is depicted in figure 5. The acceptance/decline of the request and the delegating access rights (e.g. credentials, resources, etc.) will depend on the MLA global policies. Actually, "Receive Access Rights" step is more complicated: it depends on the delegation criteria (e.g. role, permission) and the availability of the resources to the delegatee. This step needs more investigations and time and will be a part of our future works.

Delegation offers a suitable solution to support both ad-hoc and process-based interactions in an eGovernment context. In this highly dynamic environment, delegation is closely related to other concepts and mechanisms such as authorizations policies and revocation to allow the compliance and on-the-fly shift of responsibilities with respect to an ongoing collaboration both on a (atomic) task level and on a (global) process level.

Authorization Policy Model from the MLA Scenario

Authorization policy model can be used to specify public roles and their privileges and provides means to specify role mapping to internal and external roles of participating organisations. This specification will be used to identify the delegating access rights requirements afterwards. The Mutual Legal Assistance (MLA) scenario is being used as an example to illustrate the requirements of the authorization policy specification. Some of the related authorization policies rules are listed below:

1. Europol National Unit A: Only national units, liaison officers, and the Director, Deputy Directors or duly empowered Europol officials shall have the right to input data directly into the information system and retrieve it therefrom. 2. Eurojust National Member A: The case management system shall allow National Members to define the specific items of personal and non-personal data to which they wish to give access to other National Member(s), Assistant(s) or authorized staff members that are involved in the handling of the case. 3. Eurojust National Member B: When a National Member gives access to a temporary work file or a part of it to one or more involved National member(s), the case management system shall ensure that the concerned users have access to the relevant parts of the file but that they cannot modify the data introduced by the original author. 4. Responsible National Authority B: The case management system shall mark such data in a way that will remind the person who has introduced the data in the system of the obligation to keep these data for a limited period of time.

The involved participants are highly heterogeneous, and they intend to stay autonomous in terms of controlling their resources and executing tasks responsible for them. Moreover, fairly fixed, globally known roles for the collaboration are already established, and lastly, their privileges are often derived from European laws and regulations that must be followed by all participants. The last two characteristics make the Role Based Access Control (RBAC) model an extremely attractive choice to specify public roles and their privileges [START_REF] Belokosztolszki | Policy Contexts: Controlling Information Flow in Parameterised RBAC[END_REF].

Revocation

Revocation is an important process that must accompany the delegation. It is the subsequent withdrawal of previously delegated objects such as a role or a task. A vast amount of different views on the topic can be found in literature [START_REF] Wainer | DW-RBAC: A formal security model of delegation and revocation in workflow systems[END_REF][START_REF] Hagstrom | Revocations-A Classification[END_REF] where each author having their own assumptions and opinions on how to model revocation. For simplification, our model of revocation is closely related to the delegation model based user-to-user. Actually, the decision of revocation is issued from the delegator in order to take away the delegated privileges, or the desire to go back to the state before privileges were delegated. The privileges consist of the delegating access right provided to the delegatee. Basically, delegating access rights issued from the delegator describes the permission given to access to the task resources such as rogatory letters or legal requests in the MLA scenario.

Related Work

In this section, we present a literature review related to the delegation requirements. Basically, we aim to come up with a delegation classification of models, policies, and technologies that will fit with the motivation criteria for delegation in our future works.

The eXtensible Access Control Markup Language is an XML-based, declarative access control policy language that lets policy editors to specify the rules about who can do what and when. As an OASIS standard, its greatest strength lies in interoperability [START_REF] Moses | eXtensible Access Control Markup Language (XACML) Version 2.0, OASIS[END_REF]. Unlike other application-specific, proprietary accesscontrol mechanisms, this standard can be specified once and deployed beyond the boundaries of organisations and countries. In [START_REF] Rissanen | Administrative Delegation in XACML[END_REF], Rissanen and Firozabadi add new structured data-types to express chains of delegation and constraints on delegation. The main result of their research is an administrative delegation. It is about creating new long-term access control policies by means of delegation in a decentralised organisation. However, this approach does not cover ad-hoc interactions and seems to not support decentralized delegation in the context of MLA.

In [START_REF] Kesselman | X.509 Proxy Certificates for Dynamic Delegation[END_REF][START_REF] Rits | A Secure Public Sector Workflow Management System[END_REF], they tackle new requirements for delegation such as delegating in a dynamic and light-weight manner, performing single sign-on, and reusing existing protocols and software with minimal modifications. Welch and al. define Proxy Certificates allowing an entity holding a standard X.509 public key certificate to delegate some or all of its privileges to another entity. This delegation can be performed dynamically, without the assistance of a third party. However the problem with the X.509 proxy certificates is that commercial tooling for Web Services does not necessarily recognize and properly process these certificates [START_REF] Humphrey | Extending the Security Assertion Markup Language to Support Delegation for Web Services and Grid Services[END_REF]. Wang and Del Vecchio try to leverage and extend existing Web Services standards, without breaking the existing tooling by exploiting the Security Assertion Markup Language (SAML) inherent extensibility to create a delegation framework. They develop a set of verification rules for delegation tokens that rely on WSSecurity X.509 signatures, but do not force any trust relationship between the delegatee and the target service. However, this approach support heavily computing and is time consuming that may slack the MLA deployment during a delegation request.

Role-based access control (RBAC) is recognized as an efficient access control model for large organisations. Most organisations have some business rules related to access control policy. Delegation of authority is among these rules [START_REF] Belokosztolszki | Policy Contexts: Controlling Information Flow in Parameterised RBAC[END_REF]. In [START_REF] Barka | Framework for role-based delegation models[END_REF][START_REF] Zhang | PBDM: a flexible delegation model in RBAC[END_REF], authors extend the RBAC96 model by defining some delegations rules. Barka and Sandhu proposed a role-based delegation model. They deal with user-to-user delegation. The unit of delegation in them is a role. However, users may want to delegate a piece of permission from a role [START_REF] Zhang | PBDM: a flexible delegation model in RBAC[END_REF]. Zhang and al. propose a flexible delegation model named Permission-based Delegation Model (PBDM). PBDM supports user-to-user and role-to-role delegations with features of multi-step delegation and multi-option revocation. It also supports both role and permission level delegation, which provides great flexibility in authority management. However, neither RBAC nor PBDM support the task-based delegation criteria described in the MLA delegation scenario.

Conclusion and Future Directions

In this paper we presented a novel approach to support an human centric collaborative workflow for eGovernment. Our primary concern is to analyse the gap between coordination and collaboration by distinguishing factors regarding collaboration as opposed to coordination. Actually, transparency and control supporting concepts and mechanisms are not taken into account in the context of heavily human-centric collaborative workflows. To satisfy this need we propose an extended architecture supporting task delegation as a mechanism in human-centric collaboration. Further, we discussed future requirements regarding a coordination architecture presented in earlier work.

We consider this paper as a primer for future related work in the areas of collaboration and security. Our concern will be to come up with a secure delegation mechanism supporting privacy and dynamic human interactions by addressing the delegating access rights issue. Moreover, we plan to further investigate the area of compliancy accordingly to the R4eGov laws regulations policies.

Fig. 1 .

 1 Fig. 1. Mutual Legal Assistance scenario

Fig. 2 .

 2 Fig. 2. Administrative Communication Layer

Fig. 3 .

 3 Fig. 3. Role-based delegation

Fig. 4 .

 4 Fig. 4. Extended architecture supporting delegation

Fig. 5 .

 5 Fig. 5. Dialogue states and actions

http://www.europol.eu.int/

http://www.eurojust.europa.eu/

This case study has been performed in joint collaboration between Eurpol, Eurojust and Unisys in the context of the EU FP6 IST Integrated Project R4eGov.


Joseph Thirouin

Classification of traveling waves for a quadratic Szegő equation

Keywords: MSC 2010 37K10, 35C07, 37K45 Hamiltonian systems, Szegő equation, traveling waves, instability

We give a complete classification of the traveling waves of the following quadratic Szegő equation :

and we show that they are given by two families of rational functions, one of which is generated by a stable ground state. We prove that the other branch is orbitally unstable.

Introduction

This paper is devoted to the continuation of the study of the following Hamiltonian PDE :

i∂ t u = 2JΠ(|u| 2 ) + Ju 2 , (1)
where u : R t × T x → C, the factor J = J(u) := T |u| 2 u ∈ C, and Π is the Szegő projector onto nonnegative frequencies :

Π +∞ k=-∞ u k e ikx = k≥0 u k e ikx .
The framework of such an evolution equation was first explored by Gérard and Grellier in their work on the so-called cubic Szegő equation :

i∂ t u = Π(|u| 2 u). (2)
In a vast series of papers [START_REF] Gérard | The cubic Szegő equation[END_REF][START_REF] Gérard | Invariant tori for the cubic Szegő equation[END_REF][START_REF] Gérard | An explicit formula for the cubic Szegő equation[END_REF][START_REF] Gérard | The cubic Szegő equation and Hankel operators[END_REF], these authors have shown that equation (2) enjoys very rich dynamical properties, which can be summarized in two words : "integrability" and "turbulence". Such a discovery paves the way for the study of a various range of other non-dispersive equations, among which the conformal flow on S 3 [START_REF] Bizoń | Conformal flow on S 3 and weak field integrability in AdS 4[END_REF][START_REF] Bizoń | Ground state of the conformal flow on S 3[END_REF] or the cubic lowest Landau level (LLL) equation [START_REF] Gérard | On the cubic lowest Landau level equation[END_REF].

In this spirit, our aim would be to understand the behavior of solutions of equation [START_REF] Amick | Uniqueness and related analytic properties for the Benjamin-Ono equation-a nonlinear Neumann problem in the plane[END_REF], which shares some features with (2) but is far less understood by now. Let us underline some basic facts about [START_REF] Amick | Uniqueness and related analytic properties for the Benjamin-Ono equation-a nonlinear Neumann problem in the plane[END_REF]. To stick to the notations of [START_REF] Gérard | The cubic Szegő equation and Hankel operators[END_REF], we denote by L 2 + (T) = Π(L 2 (T)) the Hardy space on the disc (or simply L 2 + ), and if G is a subspace of L 2 (T), G + will designate the intersection G ∩ L The usual scalar product on L 2 can be restricted to L 2 + , with the convention that for u, v ∈ L 2 + , we have (u|v) = T uv (with respect to the normalized Lebesgue measure dθ/2π). Through the standard symplectic form on L 2 + given by ω(u, v) := Im(u|v), we can see [START_REF] Amick | Uniqueness and related analytic properties for the Benjamin-Ono equation-a nonlinear Neumann problem in the plane[END_REF] as the Hamiltonian flow associated to the functional

E(u) := 1 2 |J| 2 = 1 2 T |u| 2 u 2 .
Since the energy E is conserved along flow lines, the modulus of J remains constant, so [START_REF] Amick | Uniqueness and related analytic properties for the Benjamin-Ono equation-a nonlinear Neumann problem in the plane[END_REF] can be regarded as a quadratic system, and we call it therefore the quadratic Szegő equation.

The invariance of E under phase rotation (u → e -iθ u) and phase translation (u → u(ze -iα )) yields two other (formal) conservation laws : the mass Q(u) := T |u| 2 and the momentum M (u) := (-i∂ x u|u).

In [START_REF] Thirouin | Optimal bounds for the growth of Sobolev norms of solutions of a quadratic Szegő equation[END_REF], several properties of (1) were enlightened, relying on the crucial fact that (1) admits a Lax pair. Firstly, as in the case of the cubic Szegő equation [START_REF] Gérard | The cubic Szegő flow at low regularity[END_REF], equation (1) admits a well-defined flow on BM O + (T), where BM O(T) stands for the space of functions with bounded mean oscillation introduced by John and Nirenberg. This flow propagates every additional regularity, in the sense that if the initial datum also belongs to H s + (T) for some s > 0, then the corresponding solution will stay in H s + for all time. We thus recover a flow on the natural energy space H 1/2 + where E, Q and M are all well-defined. Secondly, we showed the existence of finite-dimensional submanifolds of L 2 + consisting of rational functions of z which are stable by the flow of [START_REF] Amick | Uniqueness and related analytic properties for the Benjamin-Ono equation-a nonlinear Neumann problem in the plane[END_REF], and where weak turbulence phenomena can occur, in contrast with what happens for the cubic Szegő equation [START_REF] Amick | Uniqueness of Benjamin's solitary-wave solution of the Benjamin-Ono equation[END_REF]. More precisely, we have proved that there is an invariant manifold of homographies starting of which the solutions of (1) in C ∞ + (T) are bounded in the H 1/2 norm, but grow exponentially fast in the H s topology for any s > 1/2.

The purpose of the present work is to investigate another typical feature of integrable systems : we intend to give a complete classification of all traveling waves that are solutions of equation ( 1) in H 1/2 + , and to discuss their stability. Recall that a traveling wave is a solution v(t, z) of the Cauchy problem associated to [START_REF] Amick | Uniqueness and related analytic properties for the Benjamin-Ono equation-a nonlinear Neumann problem in the plane[END_REF] for which there exists ω, c ∈ R such that

v(t, z) = e -iωt v 0 (ze -ict ), ∀t ∈ R, ∀z ∈ D, (3)
where v 0 = v(0, •) is the initial state. We refer to ω as the pulsation, and to c as the velocity.

Hereafter, we only look for such solutions with c = 0 or ω = 0, for the set of steady solutions (ω = c = 0) coincides with the subset {J(v) = 0} of H 1/2 + which seems hard to characterize (see Appendix A for a discussion about that point).

Theorem 1 (Classification of traveling waves). The initial state

v 0 (z) ∈ H 1/2 +
gives rise to a traveling wave solution of (1) with (ω, c) = (0, 0) if and only if it is a constant, or if there exists λ, p ∈ C with 0 < |p| < 1 and an integer N ≥ 1 such that one of the following holds :

(i) v 0 (z) = λ 1 -pz N , in which case ω = |λ| 4 3 -|p| 2 (1 -|p| 2 ) 3 and c = |λ| 4 N 1 (1 -|p| 2 ) 2 ; (ii) v 0 (z) = -λ 1 + |p| 2 1 -|p| 2 + λ 1 -pz N , in which case ω = |λ| 4 |p| 4 (1 + 5|p| 2 )(3 + 5|p| 2 ) (1 -|p| 2 ) 4 and c = - |λ| 4 N |p| 4 3 + 5|p| 2 (1 -|p| 2 ) 3 .
Remark 1. Traveling waves with c = 0 and ω = 0 are called standing waves. In particular, Theorem 1 asserts that there are no standing waves for (1) in

H 1/2
+ apart from constants. However, if we remove the condition on the regularity of v 0 and only ask that v 0 ∈ BM O + , then we will prove that v 0 is a standing wave if and only if

v 0 = λ • Π(½ B ),
where λ ∈ C and B is any Borel subset of T.

Here again, the situation appears to be very different from the one described in [START_REF] Gérard | The cubic Szegő equation[END_REF]Theorem 1.4] for the cubic Szegő equation. Indeed equation (2) admits many stationary waves in H 1/2 + that are given by Blaschke products (see the end of Section 3 for a precise definition). Moreover, it also admits a wider variety of traveling waves, whereas those of Theorem 1 all deduce from one another by an invariance argument (see Proposition 3.5).

The proof of Theorem 1 mainly consists in translating (3) into an algebraic relation between operators thanks to the Lax pair. The key of the proof consists in describing the spectrum of a perturbation of the operator D = 1 i ∂ x by a bounded Toeplitz operator over L 2 + . It is worth noticing that this kind of operators precisely arises in the study of the Lax pair associated to the Benjamin-Ono equation (see e.g. [START_REF] Wu | Simplicity and finiteness of discrete spectrum of the Benjamin-Ono scattering operator[END_REF]). See also the pioneering work of Amick and Toland on the solitary waves for the Benjamin-Ono equation [START_REF] Amick | Uniqueness and related analytic properties for the Benjamin-Ono equation-a nonlinear Neumann problem in the plane[END_REF][START_REF] Amick | Uniqueness of Benjamin's solitary-wave solution of the Benjamin-Ono equation[END_REF], also dealing with a non-local operator such as Π.

In the present paper, we would also like to address the question of the nonlinear stability of the traveling wave solutions. Relatively to some norm • X , we will say that v 0 is X-orbitally stable if for any ε > 0, there exists

η > 0 such that if u 0 -v 0 X ≤ η, then, for all t ∈ R, inf (θ,α)∈T 2 u(t) -e -iθ v 0 (ze -iα ) X ≤ ε,
where t → u(t) refers to the solution of (1) starting from u 0 at time t = 0.

Classically, we have a global Gagliardo-Nirenberg inequality on H

1/2 + : Proposition 1.1. For any u ∈ H 1/2 + , E(u) ≤ 1 2 Q(u) 2 (Q(u) + M (u)). (4)
Equality in (4) holds if and only if u(z) = λ 1-pz for some λ, p ∈ C with |p| < 1. This enables us to prove the following stability result : Corollary 1.2 (Stability of the ground states). For λ, p ∈ C with |p| < 1, the traveling wave

z → λ 1-pz is H 1/2
+ -orbitally stable. Apart from this family of ground states, we are also going to study the stability of the second branch of the traveling waves of Theorem 1. Using the Lax pair, we know (see Corollary 2.2) that the set of functions of the form

z → b + cz 1 -pz , b, c, p ∈ C, c = 0, c -bp = 0, |p| < 1,
is left invariant by the flow of (1), and on this set (which is called V(3)), (1) reduces to a system of coupled ODEs on b, c and p, which have been made explicit in [START_REF] Thirouin | Optimal bounds for the growth of Sobolev norms of solutions of a quadratic Szegő equation[END_REF] and is recalled in Appendix B. Designing an appropriate perturbation of the "translated ground states" inside V(3), and studying the leading order of the new equations of motion, will yield the following result, which also makes use of the invariance argument mentioned above :

Proposition 1.3. For λ, p ∈ C with |p| < 1, for N ∈ N, the traveling waves z → -λ 1+|p| 2 1-|p| 2 + λ 1-pz N are not H 1/2
+ -orbitally stable. The question of the stability of traveling waves of the form λ 1-pz N , with N ≥ 2, is left open. In the case of the cubic Szegő equation, a negative answer is given in [START_REF] Gérard | Invariant tori for the cubic Szegő equation[END_REF] for all traveling waves of degree greater than 2, but the proof heavily relies on the use of action-angle coordinates which are still missing in our case.

Remark 2 (A remark on the traveling waves on R). As equation (1) can be posed on R as well, the same question of finding traveling waves holds. Adapting rigourously the argument of O. Pocovnicu [START_REF] Pocovnicu | Traveling waves for the cubic Szegő equation on the real line[END_REF] leads to the same result as for the cubic Szegő equation on the line : the only solitons are given by the profiles

v 0 (z) = α z -p , ∀z ∈ {ζ ∈ C | Im ζ > 0},
where α, p ∈ C \ {0} with Im p < 0. Moreover, these solitons are orbitally stable. This remark strengthens an observation that was already made in [START_REF] Thirouin | Optimal bounds for the growth of Sobolev norms of solutions of a quadratic Szegő equation[END_REF] : the cubic and quadratic Szegő equation look very similar on the line, unlike what happens on the torus. That is why it would be also very interesting to study the stability of the above solitons under a perturbation of the quadratic Szegő equation itself, as was done in [START_REF] Pocovnicu | Soliton interaction with small Toeplitz potentials for the Szegö equation on R[END_REF] for a perturbation of the cubic Szegő equation by a Toeplitz potential.

This paper is organized as follows. In Section 2, we give the definition of the operators arising in the study of equation ( 1) and recall some results about them. In Section 3, we give the proof of the classification theorem. Section 4 is devoted to the questions about the stability of the traveling waves that are exhibited in Theorem 1.

The author is most grateful to Pr. Patrick Gérard for suggesting the study of this problem, and for most valuable comments and remarks during this work.

Notations and preliminaries

Hankel and Toeplitz operators

In this paragraph, we recall some notations, the Lax pair result from [START_REF] Thirouin | Optimal bounds for the growth of Sobolev norms of solutions of a quadratic Szegő equation[END_REF], and some of its consequences.

For u ∈ BM O + (T), we denote the Hankel operator of symbol u by H u : h → Π(u h). This defines a bounded C-antilinear operator on

L 2 + . Since ∀h 1 , h 2 ∈ L 2 + , we have (H u (h 1 )|h 2 ) = (H u (h 2 )|h 1 ), we see that H 2
u is C-linear, positive and self-adjoint. Similarly, for b ∈ L ∞ (T), the Toeplitz operator of symbol b is given by T b : h → Π(bh). We also have

T b ∈ L(L 2 + ), but T b is C-linear. The adjoint of T b is (T b ) * = Tb.
A fundamental example of a Toeplitz operator is the shift on the right, which we call S := T e ix . Combining these definitions, we can define the shifted Hankel operator of symbol u ∈ BM O + : for h ∈ L 2 + , we set

K u (h) := S * H u (h) = H u S(h) = H S * u (h).
Now we can state the algebraic identities associated to the Lax pair for (1).

Theorem 2. Let u ∈ H s + for some s > 1/2. Set X(u) := 2Π(|u| 2 ) + u 2 . Then we have

K X(u) = A u K u + K u A u H X(u) = A u H u + H u A u -(u|•)u,
where A u := T u + T ū is a bounded self-adjoint operator on L 2 + . Proof. The proof simply follows from the arguments of [START_REF] Thirouin | Optimal bounds for the growth of Sobolev norms of solutions of a quadratic Szegő equation[END_REF], but we recall it for the convenience of the reader. Taking h ∈ L 2 + , we see that

Π(Π(|u| 2 ) h) = Π(|u| 2 h) = Π(uΠ(uh)) = Π(ūΠ(u h)),
and on the other hand, decomposing

Π(u 2 h) = Π(uΠ(u h)) + Π(u(I -Π)(u h)), we observe that (I -Π)(u h) = Π(ūh) -(u|h), so Π(u 2 h) = T u H u (h) + H u T ū(h) -(u|h)u. Now, writing H X(u) (h) = 2JΠ(Π(|u| 2 ) h) + JΠ(u 2 h) leads to the second identity. Besides, for h ∈ L 2 + , (S * A u -A u S * )h = Π(zΠ((u + ū)h) -(u + ū)Π(zh)) = Π(zuh) -Π(uΠ(zh)) -Π(z(I -Π)(ūh)) = (h|1)S * u, because zh = Π(zh) + z(h|1), and z(I -Π)(ūh) ⊥ L 2 + . Then K X(u) (h) = S * H X(u) (h) = A u K u (h) + K u A u (h) -(u|h)S * u + [S * , A u ]H u (h) = A u K u (h) + K u A u (h), since (H u (h)|1) = (H u (1)|h) = (u|h).
A consequence of this theorem is the existence of finite-dimensional submanifolds of the energy space, of arbitrary complex dimension, which are stable by the flow of ( 1), as it is also true for the cubic Szegő equation (see [START_REF] Gérard | Invariant tori for the cubic Szegő equation[END_REF]).

Definition. Let N ∈ N. We denote by V(2N ) the set of functions u ∈ H 1/2 + (T) such that rk H u = rk K u = N , and by V(2N + 1) the set of those such that rk H u = N + 1 and rk K u = N .
It is easy to see that d≥0 V(d) is simply the set of symbols u such that H u (and K u ) is a finite-rank operator. From a theorem by Kronecker, the V(d)'s can be described explicitely :

Proposition 2.1 ([6, 8]). Let d ∈ N. A function u(z) ∈ H 1/2 + belongs to V(d) if and only if u(z) = A(z) B(z) , ∀z ∈ D,
where A and B are two polynomials such that A ∧ B = 1, B(0) = 1, B has no root in D, and in addition

• deg A ≤ N -1 and deg B = N if d = 2N ; • deg A = N and deg B ≤ N if d = 2N + 1.
In particular, each V(d) is composed of rational, hence smooth functions. Now we can state the announced consequence of Theorem 2 :

Corollary 2.2. For each d ∈ N, V(d) is preserved by the flow of (1).
Proof. From [START_REF] Thirouin | Optimal bounds for the growth of Sobolev norms of solutions of a quadratic Szegő equation[END_REF] we already know that V(2N ) ∪ V(2N + 1) is preserved by the flow of [START_REF] Amick | Uniqueness and related analytic properties for the Benjamin-Ono equation-a nonlinear Neumann problem in the plane[END_REF]. Thus it suffices to show that V(2N ) is stable. Let u 0 ∈ V(2N ) and t → u(t) be the corresponding solution. Since u 0 belongs to C ∞ + (T), so does u(t) for all time t ∈ R, so we can apply Theorem 2 and compute, for a given

h ∈ L 2 + , i d dt H u (h) = H 2JΠ(|u| 2 )+ Ju 2 (h) = A Ju H u (h) + H u A Ju (h) -J(u|h)u,
or in other words, using the C-antilinearity of H u , and the fact that

(u|h) = (H u (h)|1), d dt H u = -iA Ju + i 2 J(•|1)u H u + H u iA Ju - i 2 J(•|u)1 .
We give a name to the operator on the left, say Y u := -iA Ju + i 2 J(•|1)u, and note that since A Ju is self-adjoint, we have in fact

d dt H u = Y u H u + H u Y * u . ( 5
)
This identity is close to a Lax pair, but

Y * u = -Y u .
It is still enough to deduce the stability of V(2N ). Indeed, let V (t) be the (global) solution to the following linear Cauchy problem on

L(L 2 + ) : V ′ (t) = -Y * u(t) V (t), V (0) = I. Since V ∈ GL(L 2 +
) at time t = 0, this remains true for all time. Besides, compute

d dt (V * H u V ) = (-V * Y u )H u V + V * d dt H u V + V * H u (-Y * u V ) = 0 by (5), hence V * (t)H u(t) V (t) = H u 0 for all t ∈ R. In particular, Ran H u(t) = (V * (t)) -1
Ran H u 0 , so both spaces have the same dimension. This proves that ∀t ∈ R, u(t) ∈ V(2N ).

Remark 3. As a weaker version of a Lax pair, an identity such as ( 5) also holds for H 2 u . It can be interpreted saying that H 2 u remains equivalent to H 2 u 0 as a quadratic form, whereas K 2 u remains equivalent to K 2 u 0 as an operator on L 2 + (see [START_REF] Thirouin | Optimal bounds for the growth of Sobolev norms of solutions of a quadratic Szegő equation[END_REF]).

The spectral theory of K u

For the sake of completeness, we recall below some properties of K u which will be useful in the course of the proof of Theorem 1. For a more general picture, we refer to [9, Section 3].

Let u ∈ H 1/2 + . Then H 2 u , K 2 u are positive compact self-adjoint operators [START_REF] Gérard | The cubic Szegő equation[END_REF] which satisfy the relation

H 2 u = K 2 u + (•|u)u.
For ρ, σ ≥ 0, we denote by

E u (ρ) := ker(H 2 u -ρ 2 I), F u (σ) := ker(K 2 u -σ 2 I). Proposition 2.3 ([9]). Let s > 0 such that {0} E u (s) ∪ F u (s).
Then one of the following holds true :

(i) dim E u (s) = 1 + dim F u (s), u ⊥ E u (s) and F u (s) = E u (s) ∩ u ⊥ (s is H-dominant) ; (ii) dim F u (s) = 1 + dim E u (s), u ⊥ F u (s) and E u (s) = F u (s) ∩ u ⊥ (s is K-dominant).
Denote by Σ K u the set of K-dominant eigenvalues of K 2 u (plus 0). For σ ∈ Σ K u , let u σ be the projection of u onto F u (σ). By the above proposition, we have u σ = 0 for σ > 0, and

u = σ∈Σ K u u σ . (6)
We will also need a simple lemma :

Lemma 2.4. Let σ ∈ Σ K u \ {0}. Then E u (σ) ⊆ F u (σ) ∩ 1 ⊥ . Proof. Suppose h ∈ E u (σ). Then H u (h) ∈ E u (σ), so H u (h) ⊥ u by the preceding proposition. Thus 0 = (H u (h)|u) = (1|H 2 u (h)) = σ 2 (h|1).

Proof of the main theorem

As usual, we derive from (3) an equation on v 0 : indeed, if v satisfies both ( 1) and ( 3), then

i∂ t v = ωv + cDv,
and J(v) = e -iωt J 0 , where we set

J 0 := J(v 0 ) = (H 3 v 0 (1)|1).
As usual, D := -i∂ x = z∂ z . Note that J 0 = 0, since we assume that u is not a steady solution. Thus, we find an equation for the initial data v 0 :

ωv 0 + cDv 0 = 2J 0 Π(|v 0 | 2 ) + J 0 v 2 0 . (7)
Our goal is thus to solve this differential equation. We begin with the necessary conditions and assume that there exists v 0 ∈ H

1/2
+ such that (7) holds.

The case of standing waves

Let us first examine the case when c = 0. To simplify our analysis, we write v 0 = ωJ 0 |J 0 | 2 u so that equation [START_REF] Gérard | Invariant tori for the cubic Szegő equation[END_REF]

translates into u = 2Π(|u| 2 ) + u 2 . (8)
In order to use Theorem 2, we first need to show that the operator A u can be properly defined.

Observe that since |u| 2 takes real values, we have 2 , we infer that Re u = r ± almost everywhere on T, where r ± stands for the roots of the polynomial

|u| 2 = Π(|u| 2 ) + Π(|u| 2 ) -Q = Re(u(1 -u)) -Q. Writing |u| 2 + Re(u 2 ) = |u| 2 + (Re u) 2 -(Im u) 2 = 2(Re u)
2X 2 -X + Q. In particular, Re u ∈ L ∞ (T), which proves that A u = T u + T ū = 2T Re u defines a self-adjoint bounded operator on L 2 + .
The Lax pair of Theorem 2 then applies, and yields that

K u = A u K u + K u A u .
This shows that A u and K 2 u commute. Now suppose that K 2 u is not zero. As K 2 u is a compact operator (it is even trace class), there exists V ⊆ L 2 + an eigenspace of K 2 u of finite dimension d ≥ 1. A u stabilizes V , and A u | V is self-adjoint. By the theory of Hermitian operators in finite dimension, A u | V then admits a non-zero eigenvector, i.e. there exists

ϕ ∈ L 2 + \ {0} and λ ∈ R such that A u ϕ = λϕ. Hence 2Π(ϕ Re u) = λϕ, which means that (2 Re u -λ)ϕ ⊥ L 2 + .
Multiplying by φ yields that (2 Re u -λ)|ϕ| 2 only has negative Fourier modes, but as this last function takes real values, we must have (2 Re u-λ)|ϕ| 2 ≡ 0, or equivalently 2 Re u ≡ λ on {ϕ = 0}. However, a classical result on the Hardy space L 2 + (see [START_REF] Rudin | Real and complex analysis[END_REF]Theorem 17.18]) ensures that since ϕ is not identically zero, {ϕ = 0} has zero measure in T. So 2 Re u ≡ λ almost everywhere. Let us write u = λ 2 + iψ with ψ a real function. As

ψ = 1 i (u-λ 2 ) ∈ L 2 +
, which means that ψ is constant, and so is u. This contradicts the assumption that K 2 u = 0. Hence K 2 u = 0, and K u = 0 (because ker K u = ker K 2 u ), and finally S * u = 0, which means that u (therefore v 0 ) is constant.

Conversely, the set of constant functions corresponds to the manifold

V(1) = {u ∈ H 1/2 + | K u = 0}
which is preserved by the flow. On this manifold, (1) becomes the following ODE : if ′ = 3|f | 4 f . Its solutions satisfy |f | 2 = cst, and so turn out to be standing waves.

Remark 4. However, it appears from the previous analysis that equation [START_REF] Gérard | An explicit formula for the cubic Szegő equation[END_REF] admits many nontrivial solutions on BM O + (T), and we can also classify them. Indeed, pick some real number 0 < r -< 1 6 , and set r + := 1 2 -r -. Define With these definitions, we notice that f ∈ L 2 (T), and thanks to the Poisson kernel, we see that Re u equals r + (resp. r -) on B + (resp. B -). It also appears that u is holomorphic on D, and expanding (1 -ze -ix ) -1 as a power series, it can be checked that u

θ := r - r + -r - = 2r - 1 -4r - ∈ (
= 2Π(f ) -(f |1), so u ∈ Π(L ∞ ) = BM O + (T).
As above, we have 2(Re u) 2 -Re u + 2r -r + = 0, which implies that

|u| 2 = Re(u(1 -u)) -2r -r + .
Applying the Szegő projector, we get

Π(|u| 2 ) = 1 2 u(1-u)+ 1 2 (1|u)(1-(1|u))-2r -r + . Furthermore, (u|1) = (f |1) = r + θ + r -(1 -θ)r -= 2r -, so we compute 1 2 (1|u)(1 -(1|u)) -2r -r + = r -(1 -2r --2r + ) = 0.
This proves that u solves [START_REF] Gérard | An explicit formula for the cubic Szegő equation[END_REF]. Note that we have

u = 1 1 + 2θ • Π(½ B + ).

The general case

Traveling waves are rational functions. From now on, we assume that c = 0. Observe that from [START_REF] Gérard | Invariant tori for the cubic Szegő equation[END_REF], it implies that v 0 ∈ C ∞ + (T). If we make the ansatz v 0 = cJ 0 |J 0 | 2 u, then equation ( 7) reduces to the following equation on the profile u :

̟u + Du = 2Π(|u| 2 ) + u 2 , (9)
where we have set ̟ := ω/c.

Writing K Du = DK u + K u D + K u and H Du = DH u + H u D, the Lax pair identities from Theorem 2 read (̟ + 1)K u = (A u -D)K u + K u (A u -D), (10)
̟H u = (A u -D)H u + H u (A u -D) -(u|•)u. (11)
Now, since D is a selfadjoint operator with compact resolvent and A u is a selfadjoint bounded operator, D-A u is also a selfadjoint operator with compact resolvent, so we can find an eigenbasis {ε j } j∈N of L 2 + and real eigenvalues λ j going to +∞, such that (D -A u )ε j = λ j ε j , ∀j ∈ N. Plugging this into [START_REF] Gérard | The cubic Szegő flow at low regularity[END_REF], we find

(-̟ -1 -λ j )K u (ε j ) = (D -A u )K u (ε j ), so -̟-1-λ j is also an eigenvalue of D-A u , unless K u (ε j ) = 0. Since ((D-A u )h|h) ≥ -M h 2
for some M > 0, we deduce that K u (ε j ) = 0 for j large enough, hence K u has finite rank. This proves that u is a rational function by Proposition 2.1.

We are now going to deduce further informations on u thanks to equation ( 9) itself. Indeed expand u(z) as a linear combination of

z k , 0 ≤ k ≤ m 0 , and 1 (1 -p ℓ z) k , 1 ≤ k ≤ m ℓ ,
where P = {p ℓ | 1 ≤ ℓ ≤ N } is some finite set of points of D \ {0}, and the m ℓ 's are nonnegative integers. Now Π(|u| 2 ) = H u (u) ∈ Im H u , so it is a combination of the same fractions (see [6, appendix 4] for a complete description of Im H v when v is a rational function). As D = z∂ z , Du is a linear combination of z k , 0 ≤ k ≤ m 0 , and of (1 -p ℓ z) -k , 1 ≤ k ≤ m ℓ + 1, p ℓ ∈ P. However, the u 2 term in (9) generates terms like z 2m 0 and (1 -p ℓ z) -2m ℓ , and they cannot be compensated by other terms unless m 0 = 0, and m ℓ = 1 for all ℓ. In other words, u must have simple poles, and we write

u = β + N ℓ=1 α ℓ 1 -p ℓ z ,
for some β, α 1 , . . . , α N ∈ C.

Let us compute

Π(|u| 2 ) = N ℓ=1 β + N κ=1 α ℓ α κ 1 -p ℓ p κ 1 1 -p ℓ z + |β| 2 + N ℓ=1 α ℓ β, u 2 = N ℓ=1 α 2 ℓ (1 -p ℓ z) 2 + N ℓ=1   2βα ℓ + κ =ℓ 2p ℓ α ℓ α κ p ℓ -p κ   1 1 -p ℓ z + β 2 , Du = N ℓ=1 α ℓ (1 -p ℓ z) 2 - N ℓ=1 α ℓ 1 -p ℓ z .
Considering the multiples of 1 (1-p ℓ z) 2 which only appear when computing Du and u 2 , we get that α ℓ = 1, for all 1 ≤ ℓ ≤ N (assuming without loss of generality that α ℓ = 0).

Reduction to the case u ∈ V(2N ). From the above formulae on u, Π(|u| 2 ) and u 2 , we also get an equation on β which reads

̟β = 2|β| 2 + 2N β + β 2 .
This equation only has two solutions : either β = 0 or, dividing by β, 2 β + β = ̟ -2N . This shows that β = ̟ -2N -2 Re(β) ∈ R, so we must have

β = 1 3 (̟ -2N ). (12)
Now, introduce ũ = u -β. Then

2Π(|ũ| 2 ) + ũ2 = 2Π(|u| 2 ) + 2|β| 2 -2βΠ(ū) -2 βu + u 2 -2uβ + β 2 = 2Π(|u| 2 ) + u 2 -4βu + 3β 2 -2β(1|u) = (̟ -4β)u + Du + β 2 -2βN = (̟ -4β)ũ + Dũ.
Consequently, up to a modification of ̟, it suffices to treat the case β = 0, so from now on, we consider that u belongs to V(2N ) and is of the form

u = N ℓ=1 1 1 -p ℓ z . (13)
In particular, we have

(u|1) = N. (14)
To conclude, it suffices to describe the possible choices of P = {p ℓ }. Note that the p ℓ 's solve the following system of equations :

̟ -1 2 = N κ=1 1 1 -p ℓ p κ + N κ=1 κ =ℓ p ℓ p ℓ -p κ , ∀ℓ = 1, . . . , N. (15)
We thus have to compute the commutator To see that S * e ∈ F \ {0}, it suffices to notice that σ 2 S * e = S * H 2 u e = K u H u e, and to conclude thanks to the injectivity of H u and K u as in the second step.

[A u -D, S * ]. Since for h ∈ L 2 + , [D, S](h) = z∂ z (zh) -z 2 ∂ z h = zh = S(h),
Fourth step. The eigenvalue λ of A u -D| F is simple.

Otherwise, there would be an eigenvector e ∈ F associated to λ such that e ⊥ u σ . This would mean that e ∈ E, so by the previous point, S * e would be an eigenvector of A u -D| F associated to λ + 1, hence orthogonal to u σ . Therefore, for all j ≥ 0, (S * ) j e would be a (non-zero) eigenvector of A u -D| F associated to λ + j, and this contradicts the fact that dim F < ∞.

Similarly, A u -D| F has no eigenvalue µ > λ, otherwise, iterating the third step, µ + j would be an eigenvalue of A u -D| F for all j ∈ N.

Now we are able to finish the proof. Let µ be the smallest eigenvalue of A u -D on F and ẽ a corresponding eigenvector. Then ν := λ -µ should be a nonnegative integer, or then the (S * ) j ẽ, j ≥ 0, would give rise to an infinite sequence of orthogonal eigenvectors. We also have (S * ) ν ẽ ∈ Cu σ , thanks to the previous remark. Moreover, since S * is injective on E (recall that E ⊆ 1 ⊥ ), all the eigenvalues µ, µ + 1, . . . , µ + ν -1 are simple as well on F . Thus dim F = ν + 1.

It is now straightforward to see that ν = n -1, with n = 2λ -̟ as above. On the one hand, K u (u σ ) is an eigenvector of A u -D| F associated to ̟ + 1 -λ, which means that ν = λ -µ ≥ λ-(̟ + 1-λ) = n -1 by minimality of µ. On the other hand, K u (ẽ) is an eigenvector associated to ̟ + 1 -µ, so ν = λ -µ ≥ (̟ + 1 -µ) -µ = ̟ + 1 -2(λ -ν) by maximality of λ, and hence n -1 ≥ ν. This establishes the yielded formula :

(A u -D)u σ = ̟ + dim F 2 u σ ,
in addition to the fact that ̟+1-λ = 1 2 (̟+2-dim F ) is the smallest eigenvalue of A u -D| F . We point out a by-product of the last part of the proof : Corollary 3.2. There exists a complex number ζ σ ∈ C \ {0} such that we have

K u (u σ ) = ζ σ z nσ-1 u σ , ( 16
)
where n σ = dim F as above.

Proof. Indeed, with the terminology of the proof above, K u (u σ ) is non-zero and colinear to ẽ.

So (S * ) nσ-1 K u (u σ ) = ζ σ u σ for some ζ σ = 0. Now if j < n σ -1, then (S * ) j K u (u σ ) ∈ E,
and on E, we have

SS * = I -(•|1)1 = I. Therefore, S nσ-1 (S * ) nσ-1 K u (u σ ) = K u (u σ )
, and ( 16) is proved.

We summarize the results of this paragraph on Figure 1. We first prove a simple but crucial lemma.

. . . µ µ + 1 µ + 2 λ -2 λ -1 λ Sp((A u -D)| F ) ẽ u σ K u H u S *
Lemma 3.3. We have, for each 1 ≤ m ≤ |Σ|,

(̟ + n m -2N ) • (u m |1) = 2 u m 2 . ( 17
)
In particular, (u m |1) is a positive real number.

Proof. We know that u m is a non-zero eigenvector of the operator A u -D. Let λ be the associated eigenvalue. Since (A u -D)1 = Π(u + ū), and by [START_REF] Thirouin | Optimal bounds for the growth of Sobolev norms of solutions of a quadratic Szegő equation[END_REF], Π(ū) = (1|u) = N , we can compute

λ(u m |1) = ((A u -D)u m |1) = (u m |(A u -D)1) = u m 2 + N (u m |1), so that (u m |1)(λ -N ) = u m 2 .
But by Proposition 3.1, we have λ = 1 2 (̟ + n m ), which gives formula (17).

As u m 2 > 0, it remains to show that ̟ + n m -2N is a positive number. Taking the scalar product of equation ( 9) with 1 leads to the following identity :

̟N = 2Q + N 2 . ( 18
)
However, the Cauchy-Schwarz inequality yields

N 2 = |(u|1)| 2 ≤ Q, so ̟N ≥ 3N 2 , which means that ̟ ≥ 3N . A fortiori ̟ +n m -2N ≥ ̟ -2N > 0,
and the proof of the lemma is complete.

We are now able to prove our claim :

Lemma 3.4. Necessarily, Σ is a singleton {σ}, and dim F u (σ) = N .

Proof. We are going to sum identity (17) over m. This gives, using [START_REF] Gérard | The cubic Szegő equation[END_REF] and the fact that

u m ⊥ u m ′ for m = m ′ , (̟ -2N ) • (u|1) + |Σ| m=1 n m (u m |1) = 2 u 2 .
As we know from (18

) that 2Q = ̟N -N 2 = (̟ -N ) • (u|1), we get |Σ| m=1 n m (u m |1) = N • (u|1) = N |Σ| m=1 (u m |1),
or equivalently

|Σ| m=1 (N -n m ) • (u m |1) = 0. ( 19
) But m n m ≤ rk K 2 u = N , so n m ≤ N .
Together with the preceding lemma, this shows that the sum in (19) only involves nonnegative terms. Each of them must then be zero, i.e. we must have

n m = N for all 1 ≤ m ≤ |Σ|. Hence N ≥ m n m = N |Σ|, so finally |Σ| = 1.
Determination of u. Because of ( 6), Lemma 3.4 implies that u is an eigenvector of K 2 u , i.e. K 2 u (u) = σ 2 u for some σ > 0. Hence H 2 u (u) = (σ 2 + Q)u, so u is also an eigenvector of H 2 u . Now, coming back to Proposition 3.1, this means that u σ = u, and dim F = N , hence

(A u -D)u = ̟ + N 2 u.
However,

(A u -D)u = Π(|u| 2 ) + u 2 -Du = ̟u -Π(|u| 2 )
, by equation [START_REF] Gérard | The cubic Szegő equation and Hankel operators[END_REF]. Therefore,

H u (u) = ̟u -(A u -D)u = ̟ -N 2 u.
On the other hand, by Corollary 3.2, we also have, for some ζ = 0,

K u (u) = ζz N -1 u.
Combining the two informations, we can write ζz

N -1 u = K u (u) = S * H u (u) = ̟-N 2 S * u.
Applying S again, we get the existence of some α = 0 such that SS * u = αz N u. Besides,

SS * u = u -(u|1) = u -N . This finishes to prove that u(z) = N 1 -αz N .
In other words, going back to formula [START_REF] Rudin | Real and complex analysis[END_REF], this shows that P in (15) has to be the set of the N th roots of some complex number α such that |α| < 1.

Remark 5. In terms of the inverse spectral transform of [9, Section 5], we have proved that H u has one dominant singular value of multiplicity 1, and that K u has one dominant singular value σ of multiplicity N . Corollary 3.2 says that the angle associated to σ is e iψ z N -1 for some ψ ∈ T.

The reverse statement. Now that we have found what form a traveling wave should have a priori, we need to show that functions of this form are traveling waves for (1) indeed. From a straightforward computation based on [START_REF] Wu | Simplicity and finiteness of discrete spectrum of the Benjamin-Ono scattering operator[END_REF], we can easily find the traveling waves of V(2) and V(3) : the function u(z) = 1 1-pz , for |p| < 1, satisfies

̟u + Du = 2Π(|u| 2 ) + u 2 , with ̟ ∈ R,
and no other function of V(2) does. Besides, ̟ = 3-|p| 2 1-|p| 2 because of [START_REF] Wu | Simplicity and finiteness of discrete spectrum of the Benjamin-Ono scattering operator[END_REF]. Then, if λ ∈ C, set v := λu. We have

J(v) = |λ| 2 λJ(u) = |λ| 2 λ (1 -|p| 2 ) 2 ,
which leads to

2J(v)Π(|v| 2 ) + J(v)v 2 = |λ| 4 λ (1 -|p| 2 ) 2 (2Π(|u| 2 ) + u 2 ) = |λ| 4 (3 -|p| 2 ) (1 -|p| 2 ) 3 v + |λ| 4 (1 -|p| 2 ) 2 Dv.
Thus v gives rise to a traveling wave solution of (1) with

ω = |λ| 4 3 -|p| 2 (1 -|p| 2 ) 3 , c = |λ| 4 1 (1 -|p| 2 ) 2 .
Similarly, in V(3), the only solution of ̟u + Du = 2Π(|u| 2 ) + u 2 has the form u = β + 1 1-pz , and by ( 12) and ( 15), we must have

     ̟ -4β = 3 -|p| 2 1 -|p| 2 , ̟ -3β = 2, or equivalently β = -1+|p| 2 1-|p| 2 and ̟ = -1+5|p| 2 1-|p| 2 . Then v := λu for λ ∈ C is such that J(v) = |λ| 2 λJ(u) = - |λ| 2 λ • |p| 4 (3 + 5|p| 2 ) (1 -|p| 2 ) 3 ,
as show the formulae of Appendix B for instance. As above, we finally get that v is the initial state of a traveling wave with

ω = |λ| 4 |p| 4 (3 + 5|p| 2 )(1 + 5|p| 2 ) (1 -|p| 2 ) 4 , c = -|λ| 4 |p| 4 (3 + 5|p| 2 ) (1 -|p| 2 ) 3 .
The general case of V(2N ) and V(2N + 1) relies on an invariance argument discovered in [START_REF] Xu | The cubic Szegő equation with a linear perturbation[END_REF] in the case of the cubic Szegő equation, and which also applies here.

Definition. An inner function is a function f ∈ L 2 + such that for almost every x ∈ T,

|f (e ix )| = 1.

Among such functions, some are rational functions : they are called Blaschke products, and are of the form

z → L ℓ=1 z -p ℓ 1 -p ℓ z , p ℓ ∈ D, ∀1 ≤ ℓ ≤ L.
The invariance result is the following : Proposition 3.5 (Invariance by composition with an inner function). Let f ∈ L 2 + be an inner function, and t → u(t, z) a solution of (1) starting from u(0, z) = u 0 (z)

∈ BM O + (T). Then t → u(t, zf (z))
is the solution of (1) in BM O + starting from initial data u 0 (zf (z)).

Let us first show how this proposition allows us to conclude. Assume v(t, z) is a traveling wave with v(0, z) = v 0 (z). Equation ( 3) is equivalent to

v(t)(k) = v 0 (k)e -i(ω+c•k)t , ∀k ∈ Z,
where • is the Fourier transform. Now, if N ≥ 1 and f (z) = z N -1 , the invariance proposition states that v(t, z N ) is the solution starting from v 0 (z N ). Writing w(t, z) := v(t, z N ), we get

w(t)(k) = v 0 (k)e -i(ω+ c N •k)t , ∀k ∈ Z s.t. N |k.
Thus, the set of traveling waves for equation ( 1) is stable by composition with z N . Moreover, the pulsation does not change and the velocity is divided by N . This ends the proof of Theorem 1.

To prove Proposition 3.5, we introduce an operator on L 2 (T). If f is an inner function, we set

T f : L 2 (T) -→ L 2 (T) +∞ k=-∞ u k e ikx -→ +∞ k=-∞ u k [e ix f (e ix )]
k . The operator T f satisfies some useful algebraic properties that we sum up in the next lemma. Lemma 3.6. (i) T f is an isometry of L 2 (T).

(ii) T f • Π = Π • T f . (iii) T f induces a * -endomorphism of the C * -algebra L ∞ (T).
Proof. For k ∈ Z, let us introduce e f k (x) = e ikx f (e ix ) k for x ∈ T. To see that T f is an isometry, it suffices to notice that {e f k } k∈Z is a orthonormal family of L 2 . If p ≥ q, we have

(e f p |e f q ) = 1 2π π -π e f p (x)e f q (x)dx = 1 2π π -π
e i(p-q)x f (e ix ) p-q dx = δ pq , for f has nonnegative frequencies only. Hence, for any

u = +∞ k=-∞ u k e ikx ∈ L 2 (T), T f u 2 L 2 = +∞ k=-∞ |u k | 2 = u 2 L 2 .
As T f is linear, this proves (i). Property (ii) comes from the fact that Π(e f k ) = e f k if k ≥ 0, and 0 otherwise. Finally, if v ∈ L ∞ , then v can be identified to a function ṽ such that v = ṽ almost everywhere and |ṽ(x)| ≤ v L ∞ for all x ∈ T. Now, if x ∈ T, T f v(e ix ) = T f ṽ(e ix ) = ṽ(e iy ), where y ∈ T is such that e iy = e ix f (e ix ). This proves that |T f ṽ(e ix )| ≤ v L ∞ , so the restiction of T f to L ∞ gives rise to a continuous endomorphism of L ∞ .

As we obviously have If u is the solution starting from u 0 , then we get

T f 1 = 1, it remains to show that T f (uv) = (T f u)(T f v) and T f ū = T f u whenever u, v ∈ L ∞ . But
i∂ t (T f u) = T f (i∂ t u) = 2J(u)T f Π(|u| 2 ) + J(u)T f (u 2 ) = 2J(u)Π(|T f u| 2 ) + J(u)(T f u) 2 . Furthermore, J(u) = (u 2 |u) = (T f (u 2 )|T f u) = J(T f u),

Stability and instability of the traveling waves

Hereafter we consider the problem of the stability of the solutions that Theorem 1 classifies.

The ground state

We begin with the "ground state", and show its stability via the functional inequality (4) as in [START_REF] Bizoń | Ground state of the conformal flow on S 3[END_REF][START_REF] Gérard | The cubic Szegő equation[END_REF].

E(u) = 1 2 |J(u)| 2 = 1 2 k,ℓ≥0 u k u ℓ u k+ℓ 2 ≤ 1 2   k≥0 |u k | 2     k≥0 ℓ≥0 u ℓ u k+ℓ 2   ≤ 1 2 Q(u) •   ℓ≥0 |u ℓ | 2     k,ℓ≥0 |u k+ℓ | 2   = 1 2 Q(u) 2   m≥0 (1 + m)|u m | 2   = 1 2 Q(u) 2 (Q(u) + M (u)).
If in addition 2E = Q 2 (Q + M ), then the case of equality in Cauchy-Schwarz allows us to say that necessarily, ∀k ≥ 0, there exists γ k ∈ C such that

u ℓ = γ k u ℓ+k , ∀ℓ ≥ 0.
In particular, u ℓ = γ 1 u ℓ+1 , so {u ℓ } must be a geometric sequence. There exists λ, p ∈ C such that u ℓ = λp ℓ , ∀ℓ ≥ 0.

As ℓ |u ℓ | 2 < ∞, we further have |p| < 1. Hence u(z) = λ 1-pz . Conversely, if {u ℓ } is a geometric sequence, both inequalities in the previous computation become equalities. Now it is standard to prove the H 1/2 -stability of the traveling waves z → λ 1-pz (which are even stationary waves in the case when p = 0).

Proof of Corollary

1.2. Let λ, p ∈ C with |p| < 1, and v 0 (z) = λ 1-pz . Simple computations show that Q(v 0 ) = |λ| 2 1 -|p| 2 , M (v 0 ) = |λ| 2 |p| 2 (1 -|p| 2 ) 2 , E(v 0 ) = 1 2 |λ| 6 (1 -|p| 2 ) 4 .
Before starting the proof of this lemma, let us recall how equation (1) looks like on V(3). The exact formulae can be found in Appendix B. Let u be a solution of (1) of the form

u(z) = b + cz 1 -pz , ∀z ∈ D, with b, c, p ∈ C, c = 0, c -bp = 0, |p| < 1.
Then Q, M and J can be computed explicitely in terms of b, c, p :

Q = |b| 2 + |c| 2 1 -|p| 2 , M = |c| 2 (1 -|p| 2 ) 2 , J = |b| 2 b + 2b|c| 2 1 -|p| 2 + |c| 2 cp (1 -|p| 2 ) 2 . Since |c| = √ M • (1 -|p| 2 ) and |b| 2 = Q -|c| √ M
, we can rewrite J in a simpler way as

J = (Q + |c| √ M )b + M cp.
Introduce the doubled energy E := 2E = |J| 2 . From the expression of J, we find

E = (Q + |c| √ M ) 2 (Q -|c| √ M ) + M |c| 2 (M -|c| √ M ) + 2M (Q + |c| √ M ) Re(bcp).
Writing for simplicity x := |c| √ M , and defining ψ by ψ = arg(bcp) whenever bp = 0, and ψ = 0 when b = 0 or p = 0, we get

E = (Q + x) 2 (Q -x) + x 2 (M -x) + 2x(Q + x) (Q -x)(M -x) • cos ψ. (21)
As far as the evolution of x is concerned, recall that

i dc dt = 2bc J + 2 bcJ + 2Jp|c| 2 1 -|p| 2 = 4 Re( bJ) + 2M |p| 2 |c| 2 1 -|p| 2 c + 2|c| 2 (Q + |c| √ M ) bp 1 -|p| 2 , hence d|c| dt = 1 |c| Re c dc dt = 2|c|(Q + |c| √ M ) Im(bcp) 1 -|p| 2 , or equivalently, dx dt = 2x(Q + x) (Q -x)(M -x) • sin ψ.
Using (21), and writing (sin ψ) 2 = 1 -(cos ψ) 2 , we can finally express the evolution of x in terms of conservation laws only :

dx dt 2 = 4x 2 (Q + x) 2 (Q -x)(M -x) -(Q + x) 2 (Q -x) + x 2 (M -x) -E 2 . (22)
Here, the important point is that all the O do not depend on γ nor on δE. After a tedious calculation, we get

dy dt 2 = δE • 16r 4 (1 + r) (1 -r) 5 + O r (y) -δE + - 64r 7 (1 + r) 2 (1 -r) 9 + O r (y) • y. (24)
Now, assume that v r is H 1/2 -orbitally stable. Take ε > 0 to be adjusted, and the corresponding η > 0. Choose γ * > 0 small so that ∀γ ∈ (0, γ * ), we have u γ 0 -v r H 1/2 ≤ η, and consider u the corresponding solution. With the above notations for u, observe that for any (θ, α) ∈ T 2 , and t ∈ R,

u(t, z) -e -iθ v r (ze -iα ) H 1/2 ≥ u(t, z) -e -iθ v r (ze -iα ) z = |c(t) -e -i(θ+α) √ r| ≥ |c(t)| - √ r = 1 √ M r |y(t)| so ∀t ∈ R, |y(t)| ≤ M r • inf (θ,α)∈T 2 u(t) -e -iθ v r (ze -iα ) H 1/2 ≤ M r • ε
because of the stability. Going back to (24), taking ε small enough (in a way that only depends on r), and shrinking γ * if needed to lessen δE, we get

16r 4 (1 + r) (1 -r) 5 + O r (y) -δE ≥ a r , - 64r 7 (1 + r) 2 (1 -r) 9 + O r (y) ≤ -b r ,
for all times t ∈ R, where a r , b r > 0 are some constants only depending on r. At time t = 0, we have y = 0, so

dy dt | t=0 2 ≥ a r δE > 0.
We suppose that dy/dt < 0 at time t = 0 (it suffices to argue on negative times if the converse is true). Then y < 0 in some maximal time interval (0, T ). Thus (24) yields dy dt This yields T = +∞ and ∀t ∈ R + , y(t) ≤ -t √ a r δE, which cannot happen. This ends the proof of the instability of v r .

Let us now prove the instability of the traveling waves of higher degree.

Proof of Proposition 1.3. In Lemma 4.1, we have proved the existence of some ε 0 > 0, and constructed initial data u γ 0 arbitrary close to v r in H 1/2 + , such that the corresponding solution u satisfies u(T, z) -e -iθ v r (ze -iα ) H 1/2 ≥ ε 0 > 0 for some T ∈ R, and all (θ, α) ∈ T 2 . If now N ≥ 2, Proposition 3.5 precisely states that the solution of (1) starting from u γ 0 (z N ) is u(t, z N ). On the one hand, we have

u γ 0 (z N ) -v r (z N ) H 1/2 = u γ 0 (z) -v r (z) H 1/2 ≪ 1,
since u γ 0 only differs from v r by its constant term. On the other hand, for all (θ, α) ∈ T 2 , u(T, z N ) -e -iθ v r (ze -iα ) N H 1/2 ≥ u(T, z) -e -iθ v r (ze -iN α ) H 1/2 ≥ ε 0 ,

where we used the elementary fact that for any w ∈ H

1/2 + , w(z N ) 2 H 1/2 = k≥0 (1 + kN )| ŵ(k)| 2 ≥ k≥0 (1 + k)| ŵ(k)| 2 = w 2 H 1/2 .
This concludes our proof.

Appendices A About equilibrium points

The purpose of this section is to discuss the case of the steady solutions of (1). Notice first that if for some u ∈ H 1/2 + , 2JΠ(|u| 2 ) + Ju 2 = 0, then taking the scalar product with u, we find 3|J| 2 = 0, hence E = 0. Hence all steady solutions are issued from null-energy functions. Describing the set of steady solutions, say, in H 1/2 + amounts to giving a characterization of the set {J = 0}.

As an illustration of how tough this may be, we are now going to give an explicit description of the subset of the steady solutions that are also in V(3) thanks to the inverse transform of Gérard and Grellier [START_REF] Gérard | The cubic Szegő equation and Hankel operators[END_REF]. (x, y) = 2 √ 3 (cos(θ + π 6 ), cos(θ -π 6 )), for some θ ∈ R. However, we must have y > x and x, y > 0, which imposes θ ∈ (0, π 3 ). The matrix involved in the inverse spectral formula reads

C (z) :=     ρ 1 -σe iψ z ρ 2 1 -σ 2 1 ρ 1 ρ 2 + σe iψ z σ 2 -ρ 2 2 - 1 ρ 2     , z ∈ D.
A translation in z enables to reduce ourselves to the case when the angle associated to σ 2 /2, called ψ, is zero. Then we can reconstruct We have

u(z) = C (z) -1 1 1 , 1 1 C 2 = 1-ρ 1 ρ 2 ρ 1 -ρ 2 - z σ(ρ 1 -ρ 2 ) 2 1-ρ 1 ρ 2 + (1 + ρ 1 ρ 2 )(ρ 1 -ρ 2 )z
Q = |b| 2 + |c| 2 1 -|p| 2 , M = |c| 2 (1 -|p| 2 ) 2 , J = |b| 2 b + 2b|c| 2 1 -|p| 2 + |c| 2 cp (1 -|p| 2 ) 2 .
As far as the evolution of u is concerned, [START_REF] Amick | Uniqueness and related analytic properties for the Benjamin-Ono equation-a nonlinear Neumann problem in the plane[END_REF]

translates into              i ṗ = c J , i ċ = 2bc J + 2 bcJ + 2Jp|c| 2 1 -|p| 2 , i ḃ = b 2 J + 2|b| 2 J + 2J|c| 2 1 -|p| 2 .

2 +.

 2 It is well-known that elements of L 2 + can either be considered as Fourier series with only nonnegative modes, or as holomorphic functions u(z) on the open unit disc D = {z ∈ C | |z| < 1} satisfying sup iθ )| 2 dθ < +∞.

e

  0, 1), and let B + ⊆ T be any Borel set of measure θ. Let B -:= T \ B + . Lastly, set f : T → R, with f = r + ½ B + + r -½ B -, and introduce u(z) ix + z e ix -z f (e ix )dx, ∀z ∈ D.

  we get [S * , D] = ([D, S]) * = S * . Besides, we know from the proof of Theorem 2 that S * A u -A u S * = (•|1)S * u. But E ⊆ 1 ⊥ by Lemma 2.4. This proves that A u and S * commute on E. Finally, taking e ∈ E as above, we get (A u -D)S * e = S * (A u -D)e + [A u -D, S * ]e = µS * e + [S * , D]e = (µ + 1)S * e.

Figure 1 -

 1 Figure 1 -The action of H u , K u and S * on F .

Remark 6 .

 6 this is immediate once we have interpreted T f as a composition operator. The converse of Lemma 3.6 is true : any operator G :L 2 (T) → L 2 (T) that satisfies properties (i)-(iii) is of the form T f , where f ∈ L 2 + isan inner function. It suffices to set f (e ix ) := e -ix G(e ix ) for x ∈ T. Proof of Proposition 3.5. Firstly, we note that since u 0 = Π(b) for some b ∈ L ∞ (T), then T f u 0 = T f Π(b) = Π(T f b) ∈ BM O + . Indeed, T f b ∈ L ∞ by the previous lemma.

Remark 7 .

 7 because of property (i) of Lemma 3.6. So T f u solves equation[START_REF] Amick | Uniqueness and related analytic properties for the Benjamin-Ono equation-a nonlinear Neumann problem in the plane[END_REF] in BM O + , and T f (u)(t = 0) = T f u 0 . The mapping T f also preserves the class of the standing waves in BM O + , for if Re u only takes two different values on T, then so does Re(T f u).

Proof of Proposition 1 . 1 . 2 +

 112 Let u ∈ H 1/, and write u(z) = k≥0 u k z k . Then by Parserval's theorem and twice the Cauchy-Schwarz inequality,

2 ≥

 2 a r δE + b r |y| ≥ a r δE on (0, T ),and since y is C ∞ , we have dy dt <a r δE on (0, T ).

Proposition A. 1 . 3 √

 13 The subset of V(3) consisting in steady solutions of (1), called V 0 (3), is a submanifold of real codimension 2 given byV 0 9+2 cos 2θ-2 cos 4θ ze ib   λ ∈ R, a, b ∈ T, θ ∈ [0, π 3 )

, and σ = ρ 2 1 + ρ 2 2 - 1 ,

 221 which leads to the claim, by means of a few trigonometric identities. Example. The function u(z) =to an equilibrium point of (1).B Formulae for solutions in V(3)We summarize below some useful formulae from [14, Section 4] for solutions of (1) of the formu(z) = b + cz 1 -pz , ∀z ∈ D,where b, c, p ∈ C, and |p| < 1.

Spectral analysis of A u -D. To solve the above system, we are going to perform a spectral analysis of the operator A u -D, taking advantage from its relation with the self-adjoint finiterank operator K 2 u , in the spirit of [START_REF] Gérard | The cubic Szegő equation[END_REF]Section 9]. Denote by Σ the (finite) set of the K-dominant eigenvalues of K 2 u . In this whole section, we fix σ ∈ Σ. To simplify the notations, we set F := F u (σ) and E := E u (σ) (see paragraph 2.2). Since 0 / ∈ Σ now that u ∈ V(2N ), both F and E are finite-dimensional subspaces of Ran H u = Ran K u . We also define u σ to be the (non-zero) orthogonal projection of u onto F . Our goal is to prove the following statement : Proposition 3.1. We have

where n σ := dim F .

Proof. The proof of this proposition decomposes into several steps.

First step. u σ is an eigenvector of A u -D. From (10), we see that

If h ∈ E, then H u (h) ∈ E, so we apply this identity to H u (h). We get

Assume by contradiction that it is not true. Then by [START_REF] Gérard | The cubic Szegő flow at low regularity[END_REF], K u u σ satisfies (A u -D)K u u σ = (̟ +1-λ)K u u σ . As ̟ +1-λ = λ by assumption, we find that K u u σ ⊥ u σ , and since K u u σ ∈ F , K u u σ ⊥ u, hence K u u σ ∈ E. Applying identity [START_REF] Pocovnicu | Traveling waves for the cubic Szegő equation on the real line[END_REF], we get

Note that H u K u u σ = 0, for K u (resp. H u ) is one-to-one on F (resp. E).

As H u K u u σ ∈ E and also in F , we can restart this argument, and prove by induction that

This of course cannot happen, otherwise A u -D would have infinitely many distinct eigenvalues on F which is a finite dimensional subspace of L 2 + . From now on, we write λ = 1 2 (̟ + n). It remains to show that n = dim F . Third step. The action of S * on E.

Our purpose now is to prove that if e ∈ E \ {0} satisfies (A u -D)e = µe, then S * e ∈ F \ {0} and satisfies (A u -D)S * e = (µ + 1)S * e.

Then by Proposition 1.1, it appears that

In the sequel, we denote by T this two-(or maybe one-) dimensional torus, and we show that it is stable in the energy space.

Suppose that it is not. Then there exists a sequence {u n 0 } of initial data in H

, and a sequence of times {t n } such that the solution u n (t) of ( 1) starting from u n 0 at time t = 0 satisfies

for some ε 0 > 0.

Because of the conservation laws, we know that

+ . Up to extracting a subsequence, there is an element f of the energy space such that u n (t n ) ⇀ f . By Rellich's theorem, we know that u n (t n ) → f strongly in any L p , p < ∞, so

By the weak convergence, we also have

so all the inequalities are in fact equalities. Thus M (f ) = M (v 0 ), so u n (t n ) → f strongly in H 1/2 and f ∈ T . This is a contradiction with (20).

The second branch

Now it remains to consider the case of the second branch of the traveling waves in Theorem 1. We first focus on the simplest one, of degree 1 in z, namely

where λ, p ∈ C \ {0}, and |p| < 1. Observe that thanks to time rescaling, and to the invariance of equation ( 1) under phase rotation and phase translation, it is enough to study the stability of the following traveling waves :

+ -orbitally stable.

Proof of Lemma 4.1. Thanks to the above formulae, we can easily compute

Besides, since v r gives rise to traveling wave solution, |c| (hence x) will be constant, so ∀t ∈ R,

For the same reason, ψ(t) = ψ r := π. Equation ( 22) then gives

Now we are going to perturb equation ( 22) around x r and E r without changing Q r nor M r . Let γ > 0 be a small enough parameter (in a sense that we will further make clear), and set

We have

, where the notation O r means that the constant in the O only depends on r. In addition, 21), we see that only the angle ψ is modified at time 0, and changes from π to π + γ. If γ is sufficiently small, we have cos(π + γ) > -1, and thus we can write E(u γ 0 ) = E r + δE, where δE > 0 and δE = O r (γ).

Denote by u (with an implicit dependence on γ) the solution of (1) starting from u γ 0 . Write x = |c| √ M as above, and decompose x = x r + y, where y is initially zero, and is meant to remain small. Equation ( 22) yields

We must develop this expression. The leading order term vanishes because of (23). Handling the δE-terms with care, we can write

where

Proof. Let us fix u a function of V(3) such that J(u) = 0. We suppose that u is not identically zero, and rescale it so that Q(u) = 1. Denote by σ 2 the unique non-zero eigenvalue of K 2 u . We have σ 2 = Tr K 2 u = M . Moreover, it is an elementary fact (see [START_REF] Gérard | Invariant tori for the cubic Szegő equation[END_REF]) that eigenvalues of H 2 u and K 2 u are interlaced. As rk H 2 u = 2, we denote by ρ 2 1 , ρ 2 2 its eigenvalues, in such a way that

In addition, we have

The case when ρ 1 = ρ 2 can be treated first, because it corresponds to a Blaschke product of degree 1. In that case, u should be of the form

for some p ∈ D and a ∈ T. However, for such a function, J(u) = T |u| 2 u = (u|1) = -e ia p. Therefore, p = 0 and u(z) = e ia z. From now on, we assume that ρ 1 > ρ 2 in (25).

Let us now briefly recall some notations. We denote by E u (ρ j ) the eigenspace of H 2 u associated to the eigenvalue ρ 2 j , j ∈ {1, 2}, and we define u j to be the orthogonal projection of u onto E u (ρ j ).

We can also characterize the action of H u on u j , noting that dim E u (ρ 2 j ) = 1 : there exists ϕ 1 , ϕ 2 ∈ T such that

Last of all, it is possible to compute the L 2 norm of the u j 's in terms of ρ 2 1 , σ 2 , ρ 2 2 (see e.g. [START_REF] Xu | The cubic Szegő equation with a linear perturbation[END_REF]) :

Using the decomposition u = u 1 + u 2 , we get 0 = J(u) = (u|H u (u)) = ρ 1 e -iϕ 1 u 1 2 + ρ 2 e -iϕ 2 u 2 2 .

Up to a rotation of u, we can assume that ϕ 1 = 0. As u 2 2 = 0, e iϕ 2 must then be real, so necessarily, ϕ 2 = π. Now using the expression of u 1 2 , u 2 2 , and multiplying by ρ 2 1 -ρ 2 2 , we get ρ 3 1 (ρ 2 1 -σ 2 ) -ρ 3 2 (σ 2 -ρ 2 2 ) = 0. By (26), ρ 2 1 -σ 2 = 1 -ρ 2 2 and σ 2 -ρ 2 2 = ρ 2 1 -1. Hence ρ 3 1 + ρ 3 2 = ρ 2 1 ρ 2 2 (ρ 1 + ρ 2 ), or rather

This equation means that x := 1 ρ 1 and y := 1 ρ 2 belong to the ellipse of equation x 2 -xy + y 2 = 1. A standard reduction procedure indicates that all the points of this ellipse can be written as