Célia Martinie

David Navarre

Philippe Palanque

Exploiting Tasks Descriptions to Assess and Ensure Effectiveness during the programming of interactive Java software

Keywords: Human-centered computing → Human computer interaction (HCI), Interactive Systems and Tools, Interactive applications, task modeling

. The interactive application shall implement all of the needed functionalities and these functionalities shall be accessible when needed (e.g. made available to the user in an order that is compatible with the one needed to reach the goals). Ensuring the effectiveness of an interactive application thus requires an explicit description of users' goals and tasks. Task models is one of the very few means for explicitly and exhaustively describing user tasks at design time. The only mean for ensuring that this need is met requires an explicit description of users' goals and tasks. How to understand these goals and tasks and how to demonstrate that an interactive application supports them all is the main objective of this tutorial.

OVERVIEW OF THE TUTORIAL

Ensuring the effectiveness of an interactive application also requires to verify that each user actions are feasible with the interactive application (e.g. being able to click on a button that is enabled at the appropriate time) and this verification should be possible along the whole development process. On the completion of the tutorial, attendees will be able to:

• understand and navigate through HAMSTERS task models [START_REF] Martinie | Structuring and Composition Mechanisms to Address Scalability Issues in Task Models[END_REF] • annotate parts of Java code to tag them as related to interactive user tasks • identify and set correspondences between interactive tasks (in task models) and lines of code (of the interactive software)

• assess compatibility between a task model and an interactive application (via co-execution of interactive Java applications and HAMSTERS task models) • ensure (via modification of either task models or software code) compatibility between a task model and an interactive application • setup and develop a Java-HAMSTERS project using the TOUCAN environment [START_REF] Martinie | TOUCAN: An IDE Supporting the Development of Effective Interactive Java Applications[END_REF]. This point is critical as tutorial attendees will then be able to use content learnt in the tutorial for other interactive software (than the ones used in the tutorial) • use HAMSTERS tool to modify users' goals and tasks

DESCRIPTION AND CONTENT

Following the presentation of a set of small examples (of increasing level of complexity), the tutorial will provide an interactive hands-on exercise extracted from an industrial case study (Flight Control Unit in an aircraft cockpit, see Fig. 1).

Figure 1. Screenshot of the prototype of the FCU Software

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This case study will be used as a support for presenting and explaining:

• How the effectiveness factor of usability can be checked using task models and interactive software application • How a description of user interactive task in a task model may be connected to lines of codes in an interactive software application • How an Integrated Development environment may support the co-development of an interactive application and its associated task models Fig. 2 presents and excerpt of the pilot tasks (described with HAMSTERS).

AUDIENCE AND PREREQUISITES

This tutorial is open to researchers, practitioners, educators and students with an experience in software programming (Object Oriented and Java preferably). An experience in task analysis and task modeling is preferable but not mandatory.

RESOURCES PROVIDED TO ATTENDEES

The TOUCAN IDE (Integrated Development Environment) enables developers and HCI specialists to edit correspondences between interactive user tasks and event sources and renderers in software applications. This support is achieved through the automatic extraction of interactive input and output tasks in the HAMSTERS task models and the automatic extraction of event sources and renderers from annotated applications using Java technology. TOUCAN editor also presents a view of the correspondence coverage, thus allowing developers to check the completeness of the defined correspondences. Once the correspondence between interactive tasks and event sources and renderers is completed, the TOUCAN tool provides three different means for the co-execution of the interactive application and its associated task models: task model driven co-execution, system driven co-execution and scenario driven coexecution.

Philippe Palanque

is Professor in Computer Science at University of Toulouse 3. He has been teaching HCI and task engineering classes for 20 years and is head of the Interactive Critical Systems group at the Institut de Recherche en Informatique de Toulouse (IRIT) in France. Since the late 80s he has been working on the development and application of formal description techniques for interactive system. He has worked on research projects to improve interactive Ground Segment Systems at the Centre National d'Études Spatiales (CNES) for more than 10 years and is also involved in the development of software architectures and user interface modeling for interactive cockpits in large civil aircraft (funded by Airbus). He is also involved in the research network HALA! (Higher Automation Levels in Aviation) funded by SESAR program which targets at building the future European air traffic management system. The main driver of Philippe's research over the last 20 years has been to address in an even way Usability, Safety and Dependability in order to build trustable safety critical interactive systems. As for conferences he is a member of the program committee of conferences in these domains such as SAFECOMP 2013 (32nd conference on Computer Safety, Reliability and Security), DSN 2014 (44th conference on Dependable

Figure 2 .

 2 Figure 2. Task model of the pilot's task of checking for thunderstorm on the flight path (using the FCU Software application)

Figure 3 .

 3 Figure 3. Screenshot of the TOUCAN environment

Systems and Networks), EICS 2014 (21st annual conference on Engineering Interactive Computing Systems) and was cochair of CHI 2014 (32nd conference on Human Factors in Computing Systems) and research papers co-chair of INTERACT 2015.


Esra Zehra

Vincent Ilhan

Catherine Brochard

Nicolas Philippe

Stéphane Lapaque

Patricia Auvin

Lepage

B Fragilis

P Merdae

B Ovatus

B Uniformis

B Caccae

O Splanchnicus

P Copri

P Vulgatus

P Plebeius

R Gnavus

S Salivarius

R Intestinalis

L Reuteri

S Caprae

E Faecalis

C Scindens

C Ramosum

F Plautii

D Longicatena

A Rectalis

B Luti

V Parvula

M Elsdenii

E Coli

K Aerogenes

S Maltophilia

A Muciniphila

B Breve

B Bifidum

B Longum

C Aerofaciens

B Intestinalis

P Merdaa

A Muciniphila Maltitol

Acesulfame K Sorbitol

Aspartame Maltitol

Bi-directional interactions between anti-seizure medications

come

Antiseizure medications (ASMs) fail to reduce symptoms of epilepsy in 1/3 of the children.

Early life microbiome is vulnerable to xenobiotics.

ASMs disturb early-life gut microbiome dynamics.

Dissect reciprocal interactions between the gut microbiome species and ASMs under in-vitro conditions.

-Hydrophobic ASMs were more inhibitory towards the growth of gut bacterial strains.

-Artificial sweeteners and parabens, positively and negatively impacted the growth of gut strains, respectively.

-Mixed microbial communities reduced the bioavailability of lamotrigine and carbamazepine (mechanisms under investigation).

-Host cells were impacted by ASMs and this effect could be modulated by gut bacterial supernatants (e.g. for drug resistance gene expression).

Carbamazepine and lamotrigine impaired the growth of Gr (-) and Gr (+) bacteria, respectively.

Propyl-paraben inhibited growth of 26 strains out of 32.

Sweeteners stimulated the growth of majority of the strains.

Feces-derived microbial communities reduced carbamazepine and lamotrigine concentration up to 26% (≈ 100 mM).

Inoculum composition had significant impact on the rate of carbamazepine and lamotrigine dissipation.

  Bi-directional interactions between anti-seizure medications and gut microbial communities mediate host drug response Zehra Esra Ilhan a , Vincent Brochard a , Catherine Philippe a , Nicolas Lapaque a , Stéphane Auvin b,c,d , Patricia Lepage a a Universite Paris-Saclay, INRAE, AgroParisTech, Micalis Institute, Domaine de Vilvert, Jouy-en-Josas, 78350, France; b Universite de Paris, INSERM NeuroDiderot, Paris, France; c APHP, Robert Debré University Hospital, Pediatric Neurology Department, Paris, France; d Institut Universitaire de France, (IUF), Paris, France ASMs based on PCA analysis.


Rabah Labbas
email: rabah.labbas@univ-lehavre.fr

Stéphane Maingot
email: stephane.maingot@univ-lehavre.fr

Alexandre Thorel
email: alexandre.thorel@univ-lehavre.fr

Solvability of a fourth order elliptic problem in a bounded sector, part I

Keywords: Fourth order boundary value problem, conical domain, weighted Sobolev spaces. 2020 Mathematics Subject Classification: 35B65, 35J40, 35J75, 35K35, 46E35

The purpose of this article (composed of two parts) is the study of the generalized dispersal operator of a reaction-diffusion equation in L p -spaces set in the finite conical domain S ω,ρ of angle ω > 0 and radius ρ > 0 in R 2 .

This first part is devoted to the behavior of the solution near the top of the cone which is completely described in the weighted Sobolev space W 4,p 3-1 p (S ω,ρ0 ), ρ 0 ρ, see Theorem 2.2.

Introduction

This work required the use of many calculations and non-trivial checks linked, among other, to the theory of sums of linear operators. This is why we were forced to split this work into two more or less independent parts.

We consider the following bounded conical domain S ω,ρ = {(x, y) = (r cos θ, r sin θ) : 0 < r < ρ and 0 < θ < ω} , and its three edges:

     Γ 0 = (0, ρ) × {0} Γ ω = {(r cos ω, r sin ω) : 0 < r < ρ} Γ ρ = {(ρ cos θ, ρ sin θ) : 0 < θ < ω} ,
where ρ > 0 is given and ω ∈ (0, 2π].

Let k be a positive number, f * be a non-linear reaction function, u 0 be a given function and n be the outwards normal unit vector to ∂S ω,ρ .

We recall that the resolution of the generalized following reaction-diffusion problem

                       ∂u ∂t (t,
x, y) = -∆ 2 (x,y) u(t, x, y) + k∆ (x,y) u(t, x, y) + f * (u(t, x, y)) in R + × S ω,ρ u(0, x, y) = u 0 (x, y), (x, y) ∈ S ω,ρ

u(t, σ) = ∂u ∂n (t, σ) = 0, (t, σ) ∈ R + × Γ 0 ∪ Γ ω u(t, σ) = ∂ 2 u ∂n 2 (t, σ) = 0, (t, σ) ∈ R + × Γ ρ , (1)
needs, in a first step, to solve the following linear stationary problem

               ∆ 2 u -k∆u = f ∈ L p (S ω,ρ ), p ∈ (1, +∞) u = ∂u ∂n = 0 on Γ 0 ∪ Γ ω u = ∂ 2 u ∂n 2 = 0 on Γ ρ .
(

This first study will allow us to determine, by an analogous calculus, the resolvent of the corresponding operator by solving the spectral problem

               ∆ 2 u -k∆u -λu = f ∈ L p (S ω,ρ ) u = ∂u ∂n = 0 on Γ 0 ∪ Γ ω u = ∂ 2 u ∂n 2 = 0 on Γ ρ ,
and then to estimate the resolvent operator in view to obtain the generation of an analytic semigroup as in Labbas, Maingot and Thorel [START_REF] Labbas | Generation of analytic semigroup for some generalized diffusion operators in Lp-spaces[END_REF]. In the last step, the fixed point method can be applied to solve [START_REF] Barton | Higher-order elliptic equations in non-smooth domains: history and recent results[END_REF].

In this first part, we will focus ourselves on the study of problem [START_REF] Costabel | Mellin analysis of weighted Sobolev spaces with nonhomogeneous norms on cones[END_REF]. This work is a natural continuation of the one studied in Labbas, Maingot, Manceau and Thorel [START_REF] Labbas | On the regularity of a generalized diffusion problem arising in population dynamics set in a cylindrical domain[END_REF].

The originality of this work lies in the fact that the open set S ω,ρ is conical whereas in the cited works above, it was cylindrical and in the fact that the spatial operator is composed by a linear combination of a laplacian and a bilaplacian operators.

Such problems, set in conical domains, model many concrete situations related to pollution for instance.

After some variables and functions changes, problem (2) will be written as a abstract sum of unbounded linear operators in a Banach space, see (14). Therefore, we will use the sum theory carried out in part II to solve (14), see Labbas, Maingot and Thorel [START_REF] Labbas | Generalized diffusion problems in a conical domain, part II[END_REF].

This work is inspired by the one done by G. Geymonat and P. Grisvard in [START_REF] Geymonat | Diagonalisation d'opérateurs non autoadjoints et séparation des variables[END_REF] and P. Grisvard in [START_REF] Grisvard | Diagonalisation d'opérateurs non-autoadjoints et séparation des variables[END_REF], where the authors have considered, in a hilbertian framework, the following boundary problem

     ∆ 2 u = 0 in S ω u = ∂u ∂n = 0 on Γ 0 ∪ Γ ω ,
where S ω is the infinite conical domain

S ω = {(x, y) = (r cos θ, r sin θ) : r > 0 and 0 < θ < ω} .
The author has proved that the solution of this problem is written as a "superposition" of particular solutions with separate variables of the form χ 1,j (r)χ 2,j (θ), for j ∈ N. The basic tools used are based on the compact operators belonging to the so-called Carleman class and the Fredholm determinants.

Let us recall some known results concerning the biharmonic equation in a conical domain or in a Lipschitz domain. In Pipher and Verchota [START_REF] Pipher | The Dirichlet problem in L p for the biharmonic equation on Lipschitz domains[END_REF], the authors gave many estimates concerning the solution of the Dirichlet problem in L p for the biharmonic equation in Lipschitz domain. In Costabel, Dauge and Nicaise [START_REF] Costabel | Mellin analysis of weighted Sobolev spaces with nonhomogeneous norms on cones[END_REF], the authors have used Mellin transformation in view to give an optimal characterization of the structure of weighted Sobolev spaces with nonhomogeneous norms on finite cones. In Barton and Mayboroda [START_REF] Barton | Higher-order elliptic equations in non-smooth domains: history and recent results[END_REF], many results are given for general higher-order elliptic equations in non smooth-domains. In Tami [START_REF] Tami | Étude d'un problème pour le bilaplacien dans une famille d'ouverts du plan[END_REF], the author has studied the following problem

∆ 2 u = f in S ω,1 u = ∆u = 0 on ∂S ω,1 ,
where f ∈ L 2 (S ω,1 ). He has proved the two following results

1. If ω < π, the variational solution is written, in the neighbourhood of the corner, as

u ω = u 1,ω + u 2,ω + u 3,ω , with u 1,ω ∈ H 1+ π ω -ε , u 2,ω ∈ H 2+ π ω -ε and u 3,ω ∈ H 4 , for a small ε > 0. 2. If ω = π,
in the neighbourhood of the corner, the variational solution verifies

u π ∈ H 4 .
This article is organized as follows. In Section 2, we state our problem in polar coordinates and our main result in Theorem 2.2. In Section 3 we reformulate our problem as a sum of linear unbounded operators. Then, Section 4 is devoted to the proof of Theorem 2.2.

Statement of the main result

We introduce the following polar variables function

v(r, θ) = u(r cos θ, r sin θ) = u(x, y).
It is known that the laplacian and the bilaplacian, in polar coordinates, are respectively written as ∆u = 1

r 2 r ∂ ∂r 2 + ∂ 2 ∂θ 2 v = ∂ 2 v ∂r 2 + 1 r ∂v ∂r + 1 r 2 ∂ 2 v ∂θ 2 := Λ 1 v, (3)
and

∆ 2 u = ∂ 2 ∂r 2 + 1 r ∂ ∂r + 1 r 2 ∂ 2 ∂θ 2 2 v = ∂ 4 v ∂r 4 + 2 r 2 ∂ 4 v ∂r 2 ∂θ 2 + 1 r 4 ∂ 4 v ∂θ 4 + 2 r ∂ 3 v ∂r 3 - 2 r 3 ∂ 3 v ∂r∂θ 2 - 1 r 2 ∂ 2 v ∂r 2 + 4 r 4 ∂ 2 v ∂θ 2 + 1 r 3 ∂v ∂r := Λ 2 v. ( 4
)
Remark 2.1. We can generalize this work to the dimension n :

∆u = ∂ 2 v ∂r 2 + n -1 r ∂v ∂r + 1 r 2 ∆ ′ v,
where ∆ ′ is the Laplace-Beltrami operator on the sphere.

We set f (x, y) = f (r cos θ, r sin θ) = g(r, θ).

Let γ ∈ R. In the case of the unbounded conical domain S ω = {(x, y) = (r cos θ, r sin θ) : r > 0 and 0 < θ < ω} , we will use the following weighted Sobolev spaces of Kondrat'ev type

L p γ (S ω ) = v measurable on S ω : Sω |v(r, θ)| p r pγ dr dθ < +∞ ,
and

W m,p γ (S ω ) = v ∈ L p γ (S ω ), ∀ (i, j) ∈ N 2 : 0 i + j m, r γ+i+j ∂ i+j v ∂r i ∂θ j ∈ L p (S ω ) , (5)
see, for instance, Definition 2.1 in Costabel, Dauge and Nicaise [START_REF] Costabel | Mellin analysis of weighted Sobolev spaces with nonhomogeneous norms on cones[END_REF].

In the sequel, the definitions above, for the bounded conical domain S ω,ρ , coincide with

L p γ (S ω,ρ ) = v measurable on S ω,ρ : Sω,ρ |v(r, θ)| p r pγ dr dθ < +∞ ,
and

W m,p γ (S ω,ρ ) = v ∈ L p γ (S ω,ρ ), ∀ (i, j) ∈ N 2 : 0 i + j m, r γ ∂ i+j v ∂r i ∂θ j ∈ L p (S ω,ρ ) , ( 6
)
since clearly ( 6) implies (5) in the bounded conical domain S ω,ρ . Note that f ∈ L p (S ω,ρ ) means that g ∈ L p

1 p (S ω,ρ ). In fact, we have

Sω,ρ |f (x, y)| p dxdy = Sω,ρ |g(r, θ)| p rdrdθ, where g(r, θ) = f (r cos θ, r sin θ) , 0 < r < ρ, 0 < θ < ω.
Then, in polar coordinates, problem (2) is written in the following form

             Λ 2 v -kΛ 1 v = g in S ω,ρ v(r, 0) = v(r, ω) = ∂v ∂θ (r, 0) = ∂v ∂θ (r, ω) = 0 v(ρ, θ) = ∂ 2 v ∂r 2 (ρ, θ) = 0, (7)
where Λ 1 and Λ 2 are given by ( 3) and ( 4).

In this article, we will focus ourselves on the resolution of problem [START_REF] Labbas | Generalized diffusion problems in a conical domain, part II[END_REF] to obtain the behavior of the solution v to problem [START_REF] Labbas | Generalized diffusion problems in a conical domain, part II[END_REF] in L p -weighted spaces, in the neighborhood of the top of the cone. To this v corresponds a solution u 0 to problem (2) by applying the inverse changes of variables and functions.

To solve problem ( 7), we will use results given in Labbas, Maingot and Thorel [START_REF] Labbas | Generalized diffusion problems in a conical domain, part II[END_REF] and to this end, we need to consider τ = min

j 1 |Im(z j )| > 0,
where (z j ) j 1 are the solutions of the following transcendent equation

(sinh(z) + z) (sinh(z) -z) = 0, with Re(z) > 0.
Actually, according to Fädle [START_REF] Fädle | Die Selbstspannumgs-Eigenwertfunktionen der Quadratischen Scheibe[END_REF], we have

τ ≃ 4.21239.
We assume that

ω 3 - 2 p < τ. ( 8
)
Recall that 3 -2 p is exactly the Sobolev exponent of the space W 3,p in two variables.

Let us remark that we have two cases:

1. If 0 < ω τ 3 ≃ 0.45π, then (8) is satisfied for all p ∈ (1, +∞). 2. If τ 3 < ω < τ ≃ 1.34π, then (8) is satisfied for 1 < p < 2ω 3ω -τ .
Our main result is the following:

Theorem 2.2. Let g ∈ L p 1 p
(S ω,ρ ) with p satisfying assumption [START_REF] Labbas | Generation of analytic semigroup for some generalized diffusion operators in Lp-spaces[END_REF]. Then, there exists ρ 0 > 0 such that problem (7) has a unique solution v satisfying

v ∈ W 4,p 3-1 p (S ω,ρ 0 ).
Remark 2.3. Note that, we can also verify that we have in addition: for i = 0, 1, 2 and j = 0, 1, 2, 3, 4 such that 0 i + j 4

∂ i+j v ∂r i ∂θ j ∈ L p (S ω,ρ 0 ).
In fact, we have

∂ 3 v ∂r 3 , ∂ 4 v ∂r 3 ∂θ ∈ L p 2-1 p (S ω,ρ 0 ) and ∂ 4 v ∂r 4 ∈ L p 3-1 p (S ω,ρ 0 ),
see Section 4.2.2; but, these two weighted spaces are not embedded into the space L p (S ω,ρ 0 ).

3 Reformulations of problem (7)

Some preliminary calculus

Let us introduce the auxiliary function v r . Then r ∂ ∂r

2 v r = r ∂ ∂r r ∂ ∂r v r = v r - ∂v ∂r + r ∂ 2 v ∂r 2 = v r -2 ∂ ∂r r. v r + r 1 r ∂v ∂r + ∂ 2 v ∂r 2 = - v r -2r ∂ ∂r v r + r 1 r ∂v ∂r + ∂ 2 v ∂r 2 , so ∆u = 1 r ∂v ∂r + ∂ 2 v ∂r 2 + 1 r 2 ∂ 2 v ∂θ 2 = 1 r r ∂ ∂r 2 v r + v r + 2r ∂ ∂r v r + 1 r ∂ 2 ∂θ 2 v r .
Moreover, we have 4 , and

Π 1 := r ∂ ∂r 2 r ∂ ∂r 2 v r = r ∂ ∂r 2 r ∂ ∂r - v r + ∂v ∂r = r ∂ ∂r 2 v r - ∂v ∂r + r ∂ 2 v ∂r 2 = r ∂ ∂r - v r + ∂v ∂r + r 2 ∂ 3 v ∂r 3 = v r - ∂v ∂r + r ∂ 2 v ∂r 2 + 2r 2 ∂ 3 v ∂r 3 + r 3 ∂ 4 v ∂r
Π 2 := 2 ∂ 2 ∂θ 2 -1 r ∂ ∂r 2 v r + ∂ 2 ∂θ 2 + 1 2 v r = 2 ∂ 2 ∂θ 2 -1 v r - ∂v ∂r + r ∂ 2 v ∂r 2 + 1 r ∂ 2 ∂θ 2 + 1 2 v = 2 ∂v ∂r -2r ∂ 2 v ∂r 2 + 4 r ∂ 2 v ∂θ 2 -2 ∂ 3 v ∂θ 2 ∂r + 2r ∂ 4 v ∂θ 2 ∂r 2 + 1 r ∂ 4 v ∂θ 4 - v r .
Therefore, we obtain that

r 3 ∆ 2 u = Π 1 + Π 2 .
We set w = v r .

Then, in S ω,ρ , problem [START_REF] Labbas | Generalized diffusion problems in a conical domain, part II[END_REF] becomes

                                   1 r 3   r ∂ ∂r 2 r ∂ ∂r 2 w + 2 ∂ 2 ∂θ 2 -1 r ∂ ∂r 2 w + ∂ 2 ∂θ 2 + 1 2 w   - k r r ∂ ∂r 2 w + 2 r ∂ ∂r w + w + ∂ 2 w ∂θ 2 = g w(r, 0) = w(r, ω) = ∂w ∂θ (r, 0) = ∂w ∂θ (r, ω) = 0 w(ρ, θ) = ∂ 2 w ∂θ 2 (ρ, θ) = ∂ 2 w ∂θ 2 (ρ, θ) = 0. ( 9
) Corollary 3.1. Let g ∈ L p 1 p
(S ω,ρ ) with p satisfying assumption [START_REF] Labbas | Generation of analytic semigroup for some generalized diffusion operators in Lp-spaces[END_REF]. Then, there exists ρ 0 > 0 such that problem (9) has a unique solution w satisfying

w ∈ W 4,p 4-1 p (S ω,ρ 0 ).
This result is a direct consequence of Theorem 2.2.

Remark 3.2. Note that, we can also verify that we have in addition: for i = 0, 1, 2 and j = 0, 1, 2, 3, 4 such that 0 i + j 4

∂ i+j w ∂r i ∂θ j ∈ L p (S ω,ρ 0 ).
In fact, we have ∂ 3 w ∂r 3 , ∂ 4 w ∂r 3 ∂θ

∈ L p 3-1 p (S ω,ρ 0 ) and ∂ 4 w ∂r 4 ∈ L p 4-1 p (S ω,ρ 0 ),
see Section 4.2.2; but, these two weighted spaces are not embedded into the space L p (S ω,ρ 0 ).

Vector formulation of problem (9)

Now, let us consider the vector variable Ψ(r, θ) :

Ψ =    w r ∂ ∂r 2 w    ,
and the following matrix

A =     0 1 - ∂ 2 ∂θ 2 + 1 2 -2 ∂ 2 ∂θ 2 -1     .
We have

r ∂ ∂r 2 Ψ -AΨ =      r ∂ ∂r 2 w r ∂ ∂r 2 r ∂ ∂r 2 w      +       -r ∂ ∂r 2 w ∂ 2 ∂θ 2 + 1 2 w + 2 ∂ 2 ∂θ 2 -1 r ∂ ∂r 2 w       =     0 r ∂ ∂r 2 r ∂ ∂r 2 w + 2 ∂ 2 ∂θ 2 -1 r ∂ ∂r 2 w + ∂ 2 ∂θ 2 + 1 2 w     = 0 Π 1 + Π 2 = 0 r 3 ∆ 2 u .
We set

A 0 =   0 0 ∂ 2 ∂θ 2 + 1 1   and B 0 =   0 0 2 r ∂ ∂r 0   .
We will precise the domain of all these operators in section 3.4. It is clear that the action of these operators are independent. Then

(A 0 + B 0 ) Ψ =    0 0 ∂ 2 ∂θ 2 + 1 1       w r ∂ ∂r 2 w    +    0 0 2 r ∂ ∂r 0       w r ∂ ∂r 2 w    , hence (A 0 + B 0 ) Ψ =    0 ∂ 2 ∂θ 2 w + w + r ∂ ∂r 2 w    +    0 2 r ∂ ∂r w    =    0 2r ∂ ∂r w + ∂ 2 ∂θ 2 w + w + r ∂ ∂r 2 w    = 0 r∆u = r 0 ∆u .
The generalized diffusion equation becomes 1

r 3 r ∂ ∂r 2 Ψ -AΨ - k r (A 0 + B 0 ) Ψ = 0 g .
Finally, we obtain the following complete equation

r ∂ ∂r 2 Ψ -AΨ -kr 2 A 0 Ψ -kr 2 B 0 Ψ = 0 r 3 g . ( 10
)
Note that linear operators A and A 0 act with respect to variable θ whereas operator B 0 acts with respect to variable r ∂ ∂r .

New formulation in a finite cone

We apply the following variables and functions change r = ρe -t , φ(t, θ) = w(ρe -t , θ) and g(ρe -t , θ) = G(t, θ), then, it is easy to verify that r ∂ ∂r w = -∂φ ∂t and r ∂ ∂r

2 w = r ∂ ∂r + r 2 ∂ 2 ∂r 2 w = ∂ 2 φ ∂t 2 .
We set Φ(t, θ) = Ψ(ρe -t , θ); then

Φ =    φ ∂ 2 φ ∂t 2    .
Therefore, equation ( 10) is now set on the strip Σ = (0, +∞) × (0, ω) in the following form

r ∂ ∂r 2 Ψ -AΨ -kr 2 A 0 Ψ -kr 2 B 0 [Ψ] = ∂ 2 Φ ∂t 2 -AΦ -kρ 2 e -2t A 0 Φ + kρ 2 e -2t B 1 [Φ] = 0 ρ 3 e -3t G ,
where

B 1 =   0 0 -2 ∂ ∂t 0   .
The boundary conditions, in problem (9), on w become

φ(r, 0) = φ(r, ω) = ∂φ ∂θ (r, 0) = ∂φ ∂θ (r, ω) = 0.
As usual, we will use the vector valued notation:

Φ(t)(θ) := Φ(t, θ) =    φ(t, θ) ∂ 2 φ ∂t 2 (t, θ)    =    φ(t, .) ∂ 2 φ ∂t 2 (t, .)    (θ) :=    φ(t) ∂ 2 φ ∂t 2 (t)    (θ).
Hence, we deduce the following abstract vector valued equation

Φ ′′ (t) -AΦ(t) -kρ 2 e -2t A 0 Φ(t) + kρ 2 e -2t [B 1 Φ] (t) = 0 ρ 3 e -3t G(t)
,

where

[B 1 Φ] (t) =    0 0 -2 ∂ ∂t 0       φ ∂ 2 φ ∂t 2    (t) =    0 -2 ∂φ ∂t (t)    .
Note that A and A 0 act on Φ(t) with respect to θ, while B 1 acts on Φ with respect to t. Now, we have to solve on (0, +∞) the following problem

     Φ ′′ (t) -AΦ(t) -kρ 2 e -2t A 0 Φ(t) + kρ 2 e -2t [B 1 Φ] (t) = 0 ρ 3 e -3t G(t) Φ(0) = 0. ( 11
)
Remark 3.3. Note that the boundary condition at t = +∞, will be included in the vector valued space containing the solution Φ.

Sums of linear operators

In this section, we are going to write problem [START_REF] Pipher | The Dirichlet problem in L p for the biharmonic equation on Lipschitz domains[END_REF] as a sum of linear operators, firstly in the following Banach space X = W 2,p 0 (0, ω) × L p (0, ω), see (13) below and secondly, in L p (0, +∞; X), see (14) below.

Here X is equipped, for instance, with the following norm

ψ 1 ψ 2 X = ψ 1 W 2,p 0 (0,ω) + ψ 2 L p (0,ω) , where W 2,p 0 (0, ω) = ϕ ∈ W 2,p (0, ω) : ϕ(0) = ϕ(ω) = ϕ ′ (0) = ϕ ′ (ω) = 0 .
Then, we define the linear operator A by

             D(A) = W 4,p (0, ω) ∩ W 2,p 0 (0, ω) × W 2,p 0 (0, ω) ⊂ X A ψ 1 ψ 2 =    ψ 2 - ∂ 2 ∂θ 2 + 1 2 ψ 1 -2 ∂ 2 ∂θ 2 -1 ψ 2    , ψ 1 ψ 2 ∈ D(A).
In the same way, we define operator A 0 by

           D(A 0 ) = W 2,p 0 (0, ω) × L p (0, ω) = X A 0 ψ 1 ψ 2 =    0 ∂ 2 ∂θ 2 + 1 ψ 1 + ψ 2    , ψ 1 ψ 2 ∈ D(A 0 ).
It is clear that D(A) ⊂ D(A 0 ). Note that operator A 0 is continuous from X into itself since

A 0 ψ 1 ψ 2 X =    0 ∂ 2 ∂θ 2 + 1 ψ 1 + ψ 2    X = ∂ 2 ∂θ 2 + 1 ψ 1 + ψ 2 L p (0,ω) ψ 1 W 2,p 0 (0,ω) + ψ 2 L p (0,ω) = ψ 1 ψ 2 X .
Equation ( 11) is set in the Banach space X.

Recall that the second member in problem (2) satisfies

f ∈ L p (S ω,ρ ) , for p ∈ (1, +∞). Set t → e -3t G(t)(.) = e -3t G(t, .) = H(t, .) = H(t)(.).
Therefore, we have

Sω,ρ |f (x, y)| p dxdy = Sω,ρ |g(r, θ)| p rdrdθ = ρ 2 Σ |G(t, θ)| p e -2t dtdθ = ρ 2 Σ e 3-2 p t H(t, θ) p dtdθ = ρ 2 +∞ 0 e 3-2 p t p ω 0 |H(t)(θ)| p dθ 1/p p dt = ρ 2 +∞ 0 e (3p-2)t H(t) p L p (0,ω) dt.
It follows that the second member H is in the weighted Sobolev space as recalled in ( 6)

H : t → e 3-2 p t H ∈ L p (Σ) = L p ν (0, +∞; L p (0, ω)),
where

ν = 3 - 2 p ∈ (1, 3),
which is exactly the Sobolev exponent of the space W 3,p (Σ) in dimension 2. Then, since it would not be easy to work in weighted Sobolev spaces, we will use the following new vector valued function :

V (t) = e νt Φ(t) = e νt φ(t) e νt φ ′′ (t) = V 1 (t) V 2 (t) . ( 12
)
Since we have

Φ(t) = e -νt V 1 (t) e -νt V 2 (t) , we deduce that Φ ′ (t) = -νe -νt V (t) + e -νt V ′ (t) = e -νt (∂ t -νI) V (t), and Φ ′′ (t) = ν 2 e -νt V (t) -2νe -νt V ′ (t) + e -νt V ′′ (t) = e -νt (∂ t -νI) 2 V (t).
Moreover, we obtain

[B 1 Φ] (t) = 0 0 -2∂ t 0 e -νt V 1 (t) e -νt V 2 (t) = e -νt 0 0 -2(∂ t -νI) 0 V 1 (t) V 2 (t) = e -νt [B 2,ν V ] (t),
where

B 2,ν = 0 0 -2(∂ t -νI) 0 .
Hence, problem [START_REF] Pipher | The Dirichlet problem in L p for the biharmonic equation on Lipschitz domains[END_REF] becomes

         e -νt (∂ t -νI) 2 V (t) -e -νt AV (t) -kρ 2 e -νt e -2t A 0 V (t) -kρ 2 e -νt e -2t [(B 2,ν V )] (t) = 0 ρ 3 H(t) V (0) = 0, then      (∂ t -νI) 2 V (t) -AV (t) -kρ 2 e -2t A 0 V (t) -kρ 2 e -2t [(B 2,ν V )] (t) = 0 ρ 3 e νt H(t) V (0) = 0. (13) We have t → ρ 3 e νt H(t) ∈ L p (Σ) = L p (0, +∞; L p (0, ω)),
and

t → 0 ρ 3 e νt H(t) ∈ L p 0, +∞; W 2,p 0 (0, ω) × L p (0, ω) = L p (0, +∞; X).
Finally, let us introduce the following abstract linear operators:

   D(L 1,ν ) = V ∈ W 2,p (0, +∞; X) : V (0) = V (+∞) = 0 [L 1,ν (V )] (t) = (∂ t -νI) 2 V (t) = V ′′ (t) -2νV ′ (t) + ν 2 V (t), with ν = 3 - 2 p ∈ (1, 3), D(L 2 ) = {V ∈ L p (0, +∞; X) : for a.e. t ∈ (0, +∞), V (t) ∈ D(A)} [L 2 (V )] (t) = -AV (t), D(P 1 ) = {V ∈ L p (0, +∞; X) : for a.e. t ∈ (0, +∞), V (t) ∈ D(A 0 )} [P 1 (V )] (t) = -e -2t A 0 V (t),
and

   D(P 2,ν ) = W 1,p (0, +∞; X) [P 2,ν (V )] (t) = -e -2t (B 2,ν V ) (t).
Then, problem (13) can be written as the following abstract equation

(L 1,ν + L 2 ) V + kρ 2 (P 1 + P 2,ν ) V = F ν , (14)
set in L p (0, +∞; X), with p ∈ (1, +∞), where, for almost every t ∈ (0, +∞)

F ν (t) = 0 ρ 3 e νt H(t) .
4 Proof of Theorem 2.2

Resolution of equation (14)

Equation ( 14) will be completely studied in the second part of this work by using the sum theory of linear operators, where the main result described by Theorem 1.1 in Labbas, Maingot and Thorel [START_REF] Labbas | Generalized diffusion problems in a conical domain, part II[END_REF] states that there exists ρ 0 > 0 such that for all ρ ∈ (0, ρ 0 ], there exists a unique solution

V ∈ D(L 1,ν + L 2 ) to equation (14) that is V ∈ W 2,p (0, +∞; X) ∩ L p (0, +∞; D(A)).
Thus, we know that there exists a continuous extension from W 2,p (0, +∞; X) into W 2,p (R; X) and also from L p (0, +∞; D(A)) into L p (R; D(A)); it suffices, for instance to use the wellknown Babich techniques. Set V , the extension of V ; it is then written as

V = V 1 V 2 . So V 1 ∈ W 2,p R; W 2,p 0 (0, ω) ∩ L p R; W 4,p (0, ω) ∩ W 2,p 0 (0, ω) ,
and

V 2 ∈ W 2,p (R; L p (0, ω)) ∩ L p R; W 2,p 0 (0, ω) .
From this two properties, we obtain

V 1 , V 2 ∈ W 2,p (R × (0, ω)) ,
and

V 1 ∈ W 2,p R; W 2,p 0 (0, ω) ∩ L p R; W 4,p (0, ω) ,
by using the following lemma.

W 2,p (R; L p (I)) ∩ L p (R; W 2,p (I)) ⊂ W 2,p (R × I).

Regularities of v(r, θ) and w(r, θ) = v(r, θ)/r

In the sequel, all the regularities of w are deduced easily from those of v.

Recall that, from (12), we have

V (t) = e νt Φ(t) and Φ(t) = e -νt V (t).
Moreover, since r = ρe -t , we obtain

V 1 (t, θ) = v(ρe -t , θ) (ρe -t ) ν+1 , where ν = 3 - 2 p ∈ (1, 3) and V 2 (t, θ) = r ∂ ∂r 2 v r (ρe -t , θ) = (ρe -t ) ∂ 2 v ∂r 2 (ρe -t , θ) - ∂v ∂r (ρe -t , θ) + v(ρe -t , θ) ρe -t .

Regularity of V 1

Here, we explicit the fact that

V 1 ∈ W 2,p ((0, +∞) × (0, ω)) . ( 15
)
We have Then, we have

+∞ 0 ω 0 |V 1 (t, θ)| p dθ dt = +∞ 0 ω 0 v(ρe -t , θ) ρ ν+1 e -t(
v ∈ L p γ 0 (S ω,ρ ) and w = v r ∈ L p γ 0 +1 (S ω,ρ ), (16)
where

γ 0 = -4 + 1 p . Moreover ∂V 1 ∂t (t, θ) = 1 ρ ν+1 ∂ ∂t v(ρe -t , θ)e (ν+1)t = 1 ρ ν+1 (ν + 1)e (ν+1)t v(ρe -t , θ) -ρe νt ∂v ∂r (ρe -t , θ) , hence 1 (ρe -t ) ν ∂v ∂r (ρe -t , θ) = (ν + 1) (ρe -t ) ν+1 v(ρe -t , θ) - ∂V 1 (t, θ) ∂t .
Thus, in virtue of ( 15) and ( 16), it follows that So, we obtain ∂v ∂r ∈ L p γ 1 (S ω,ρ ) and

(t, θ) -→ 1 (ρe -t ) ν
∂w ∂r ∈ L p γ 1 +1 (S ω,ρ ), (17)
where γ 1 = -3 + 1 p . Furthermore, we have

∂ 2 V 1 ∂t 2 (t, θ) = ν + 1 ρ ν+1 ∂ ∂t v(ρe -t , θ)e (ν+1)
+ 1 ρ ν-1 e (ν-1)t ∂ 2 v ∂r 2 (ρe -t , θ) = (ν + 1) 2 (ρe -t ) ν+1 v(ρe -t , θ) - 2ν + 1 (ρe -t ) ν ∂v ∂r (ρe -t , θ) + 1 (ρe -t ) ν-1 ∂ 2 v ∂r 2 (ρe -t , θ), hence 1 (ρe -t ) ν-1 ∂ 2 v ∂r 2 (ρe -t , θ) = ∂ 2 V 1 ∂t 2 (t, θ) + 2ν + 1 (ρe -t ) ν ∂v ∂r (ρe -t , θ) - (ν + 1) 2 (ρe -t ) ν+1 v(ρe -t , θ).
Thus, in virtue of (15), ( 16) and (17), it follows that (t, θ) -→ 1 (ρe -t ) ν-1 ∂ 2 v ∂r 2 (ρe -t , θ) ∈ L p ((0, +∞) × (0, ω)), and ). In the same way, we deduce that

+∞ 0 ω 0 1 (ρe -t ) ν-1 ∂ 2 v ∂r 2 (ρe -t , θ) p dθ dt = ρ 0 ω 0 1 r ν-1+ 1 p ∂ 2 v ∂r 2 (r, θ) p dθ dr = ρ 0 ω 0 r -2p+1 ∂ 2 v ∂r 2 (
∂ 2 v ∂θ 2 ∈ L p γ 0 (S ω,ρ ) and ∂ 2 v ∂r∂θ ∈ L p γ 1 (S ω,ρ ), hence ∂ 2 w ∂θ 2 ∈ L p γ 0 +1 (S ω,ρ ) and ∂ 2 w ∂r∂θ ∈ L p γ 1 +1 (S ω,ρ
). Now, we explicit the fact that

∂ 3 V 1 ∂t∂θ 2 , ∂ 3 V 1 ∂t 2 ∂θ , ∂ 4 V 1 ∂t 2 ∂θ 2 ∈ L p ((0, +∞) × (0, ω)) . We have ∂ 3 V 1 ∂t 2 ∂θ (t, θ) = (ν + 1) 2 (ρe -t ) ν+1 ∂v ∂θ (ρe -t , θ) - 2ν + 1 (ρe -t ) ν ∂ 2 v ∂r∂θ (ρe -t , θ) + 1 (ρe -t ) ν-1 ∂ 3 v ∂r 2 ∂θ (ρe -t , θ), hence 1 (ρe -t ) ν-1 ∂ 3 v ∂r 2 ∂θ (ρe -t , θ) = ∂ 3 V 1 ∂t 2 ∂θ (t, θ) - (ν + 1) 2 (ρe -t ) ν+1 ∂v ∂θ (ρe -t , θ) + 2ν + 1 (ρe -t ) ν ∂ 2 v ∂r∂θ (ρe -t , θ).
Thus, we obtain ∂ 3 v ∂r 2 ∂θ ∈ L p γ 2 (S ω,ρ ) and

∂ 3 w ∂r 2 ∂θ ∈ L p γ 2 +1 (S ω,ρ
), and in the same way, we also have

∂ 3 v ∂r∂θ 2 ∈ L p γ 1 (S ω,ρ ) and ∂ 4 v ∂r 2 ∂θ 2 ∈ L p γ 2 (S ω,ρ ), hence ∂ 3 w ∂r∂θ 2 ∈ L p γ 1 +1 (S ω,ρ ) and ∂ 4 w ∂r 2 ∂θ 2 ∈ L p γ 2 +1 (S ω,ρ ). Now, we explicit the fact that V 1 ∈ L p (0, +∞); W 4,p (0, ω) , that is, for all i = 1, 2, 3, 4 +∞ 0 ∂ i V 1 ∂θ i (t, θ) p L p (0,ω) dt < +∞. Then, we have +∞ 0 ω 0 ∂ i V 1 ∂θ i (t, θ) p dθ dt = +∞ 0 ω 0 1 (ρe -t ) ν+1 ∂ i v ∂θ i (ρe -t , θ) p dθ dt = ρ 0 ω 0 r -4p+1 ∂ i v ∂θ i (r, θ) p dθ dr, which gives
∂ i v ∂θ i ∈ L p γ 0 (S ω,ρ ) and ∂ i w ∂θ i ∈ L p γ 0 +1 (S ω,ρ ), for i = 1, 2, 3, 4.

Regularity of V 2

In the same way, we explicit the fact that In the same way, we have

V 2 ∈ W
∂ 4 v ∂r∂θ 3 ∈ L p γ 1 (S ω,ρ
) and ∂ 4 v ∂r 3 ∂θ ∈ L p γ 3 (S ω,ρ ), hence ∂ 4 w ∂r∂θ 3 ∈ L p γ 1 +1 (S ω,ρ ) and ∂ 4 w ∂r 3 ∂θ ∈ L p γ 3 +1 (S ω,ρ ). Furthermore, we have

∂ 2 V 2 ∂t 2 (t, θ) = ρe -t 3 ∂ 4 v
∂r 4 (ρe -t , θ) + 2 ρe -t 2 ∂ 3 v ∂r 3 (ρe -t , θ) + ρe -t ∂ 2 v ∂r 2 (ρe -t , θ) -∂v ∂r (ρe -t , θ) -v(ρe -t , θ) ρe -t .

Since ρe -t 3 ∂ 4 v ∂r 4 (ρe -t , θ) = -2 ρe -t 2 ∂ 3 v ∂r 3 (ρe -t , θ)ρe -t ∂ 2 v ∂r 2 (ρe -t , θ) Therefore ∂ 4 v ∂r 4 ∈ L p γ 4 (S ω,ρ ) and ∂ 4 w ∂r 4 ∈ L p γ 4 +1 (S ω,ρ ),

+
where

γ 4 = 3 - 1 p .
It is clear now that the appropriated weight function, cited in [START_REF] Labbas | On the regularity of a generalized diffusion problem arising in population dynamics set in a cylindrical domain[END_REF], is r -→ r γ := r 3-1 p .

Lemma 4 . 1 .

 41 Let I = ]a, b[, a < b be an open bounded subset of R. Then

r

  ν+1) p dθ dt < +∞.Setting r = ρe -t , we obtain -4p+1 |v (r, θ)| p dθ dr.

  ∂v ∂r(ρe -t , θ) ∈ L p ((0, +∞) × (0, ω)),

r

  2p-1 ∂ 3 v ∂r3 (r, θ) p dθ dr < +∞, which means that ∂ 3 v ∂r3 ∈ L p γ 3 (S ω,ρ ), and∂ 3 w ∂r 3 ∈ L p γ 3 +1 (S ω,ρ ),where γ 3 = 2 -1 p .

r

  3 ∂ 4 v ∂r4 (ρe -t , θ) 3p-1 ∂ 4 v ∂r4 (r, θ) p dθ dr < +∞.

  2,p ((0, +∞) × (0, ω)) , =ρe -t 2 ∂ 3 v ∂r3 (ρe -t , θ) -

	where
		V 2 (t, θ) = r	∂ ∂r	2 v r	(ρe -t , θ) = ρe -t ∂ 2 v ∂r 2 (ρe -t , θ) -	∂v ∂r	(ρe -t , θ) +	v(ρe -t , θ) ρe -t .
	It is clear, from Subsection 4.2.1, that
					r	∂ 2 v ∂r 2 ,	∂v ∂r	,	v r	∈ L p γ 1 (S ω,ρ ).
	Moreover, we have
		∂V 2 ∂t	(t, θ) ∂v ∂r	(ρe -t , θ) +	v(ρe -t , θ) ρe -t ,
	hence	ρe -t 2 ∂ 3 v ∂r 3 (ρe -t , θ) = -	∂v ∂r	(ρe -t , θ) +	v(ρe -t , θ) ρe -t -	∂V 2 ∂t	(t, θ).

Acknowledgments

We would like to thank the referee for its valuable comments and corrections which have helped us to improve this paper.

Proof. Let χ ∈ W m,p (a, b). Then χ ∈ C m-1 ([a, b]). Consider the Taylor polynomial

We define the following bump function ϕ ∈ C ∞ ([b, +∞)) such that

Then x -→ T m-1 (x)ϕ(x) belongs to W m,p (b, +∞). Now, we define an extension function on (a, +∞) by

Clearly, χ belongs to W m,p (a, +∞). Moreover, all the derivatives on the left and on the right, up to order m -1, coincide. In the same way, we build an extension to the left of point a. Consequently, there exists an extension operator P which maps continuously

we have

This last space coincides with W 2,p (R 2 ), by Mihlin's theorem (see Mihlin [START_REF] Mihlin | On the multipliers of Fourier integrals[END_REF]). Consequently, χ, the restriction of P χ to R × I, belongs to W 2,p (R × I).

We take this opportunity to indicate a similar result in the case of a bounded open set of R n with n > 1.

Lemma 4.2. Let U be an open bounded

Proof. We know that there exists an extension operator P which maps continuously

we have

This last space coincide with W 2,p (R n+1 ), by Mihlin's theorem (see Mihlin [START_REF] Mihlin | On the multipliers of Fourier integrals[END_REF]). Consequently, χ, the restriction of P χ to R × U , belongs to W 2,p (R × U ).

Therefore, we deduce that V 1 and V 2 have the same regularities on (0, ρ) × (0, ω), with ρ ∈ (0, ρ 0 ].

Conflict of interest

On behalf of all authors, the corresponding author states that there is no conflict of interest.


Filip Novkoski
email: filip.novkoski@u-paris.fr

Eric Falcon
email: eric.falcon@u-paris.fr

Chi-Tuong Pham
email: chi-tuong.pham@upsaclay.fr

A numerical direct scattering method for the periodic sine-Gordon equation

We propose a procedure for computing the direct scattering transform of the periodic sine-Gordon equation. This procedure, previously used within the periodic Korteweg-de Vries equation framework, is implemented for the case of the sine-Gordon equation and is validated numerically. In particular, we show that this algorithm works well with signals involving topological solitons, such as kink or anti-kink solitons, but also for non-topological solitons, such as breathers. It also has the ability to distinguish between these different solutions of the sine-Gordon equation within the complex plane of the eigenvalue spectrum of the scattering problem. The complex trace of the scattering matrix is made numerically accessible, and the influence of breathers on the latter is highlighted. Finally, periodic solutions of the sine-Gordon equation and their spectral signatures are explored in both the large-amplitude (cnoidal-like waves) and low-amplitude (radiative modes) limits.

Introduction

Solitons are encountered in various fields of physics, from condensed matter, optics to hydrodynamics [START_REF] Russell | Report on waves[END_REF][START_REF] Frenkel | On the theory of plastic deformation and twinning[END_REF][START_REF] Remoissenet | Waves Called Solitons[END_REF][START_REF] Dauxois | Physics of Solitons[END_REF]. It also has received considerable atten-tion in applied mathematics, in particular, in the study of integrable systems [START_REF] Ablowitz | Solitons and the Inverse Scattering Transform[END_REF][START_REF] Drazin | Solitons: An Introduction[END_REF]. Advanced tools of the inverse scattering transform (IST) have significantly changed the theoretical view of soliton solutions of nonlinear partial differential equations (PDEs), such as Korteweg-de Vries (KdV) and nonlinear Schrödinger (NLS) equations [START_REF] Osborne | Nonlinear Ocean Waves and the Inverse Scattering Transform[END_REF]. It has also recently led to experimental applications in optics [START_REF] Suret | Soliton refraction through an optical soliton gas[END_REF] and in hydrodynamics surface waves [START_REF] Redor | Experimental Evidence of a Hydrodynamic Soliton Gas[END_REF],

This work is licensed under a "CC BY-NC-SA 4.0" license. such as the experimental synthesis of soliton gas [START_REF] Suret | Nonlinear spectral synthesis of soliton gas in deepwater surface gravity waves[END_REF] or Peregrine solitons [START_REF] Tikan | Prediction and manipulation of hydrodynamic rogue waves via nonlinear spectral engineering[END_REF].

The sine-Gordon (SG) equation admits a variety of different soliton solutions, such as topological solitons (in the form of kinks or anti-kinks), non-topological breathers (i.e., solutions oscillating in time and localized in space), or multisolitons. Sine-Gordon solitons have been experimentally evidenced in the canonical example of a chain of coupled pendulums [START_REF] Scott | A nonlinear Klein-Gordon equation[END_REF], but also in Josephson tunnel junctions between superconductors [START_REF] Ustinov | Solitons in Josephson junctions[END_REF], dislocations in crystals [START_REF] Frenkel | On the theory of plastic deformation and twinning[END_REF][START_REF] Dauxois | Physics of Solitons[END_REF], or in biological cellular structures [START_REF] Ivancevic | Sine-Gordon Solitons, Kinks and Breathers as Physical Models of Nonlinear Excitations in Living Cellular Structures[END_REF].

The SG equation has been extensively studied in the context of the IST [START_REF] Malomed | The sine-Gordon Model: General Background, Physical Motivations, Inverse Scattering, and Solitons[END_REF], and, in particular, the initial value problem can be solved using IST methods [START_REF] Drazin | Solitons: An Introduction[END_REF][START_REF] Ablowitz | Method for solving the sine-Gordon equation[END_REF]. The corresponding Lax pairs are well known [START_REF] Takhtadzhyan | Essentially nonlinear one-dimensional model of classical field theory[END_REF], whereas the spectral theory behind the SG equation with periodic boundary conditions has been developed using Floquet theory [START_REF] Forest | Spectral theory for the periodic sine-Gordon equation: A concrete viewpoint[END_REF].

Generally speaking, if an integrable equation admits soliton solutions, a specific c omplex value λ can be assigned to each of such solitons, thus fully characterizing the solution. The value λ is the solution of a linear eigenvalue problem related to the given integrable nonlinear PDE. For example, in the case of the KdV equation, the linear problem is related to the Schrödinger equation, ψ xx -(u -λ)ψ = 0, of the field ψ(x, t ) w ith an unknown potential u(x, t) coming from the solution of the KdV solution [START_REF] Gardner | Method for solving the Korteweg-de Vries equation[END_REF]. A critical property of integrable equations is isospectrality, meaning that the values of λ are constant in time. The direct oscattering problem aims to seek the values of λ (independent of time) from a given initial condition u(x, 0). For the KdV equation, the values of λ are purely real, whereas, for the focusing NLS and SG equations, λ can take also complex values as well.While on the infinite l ine, this eigenvalue is unique, as we will see, in the periodic case this corresponds to a band which is bounded by two values. Numerical simulations of the eigenvalue equations of the direct scattering in the sine-Gordon framework have been performed, as well as the IST by numerically solving the related nonlinear eigenvalue problem to achieve the soliton spectrum [START_REF] Overman | Coherence and chaos in the driven damped sine-Gordon equation: Measurement of the soliton spectrum[END_REF][START_REF] Flesch | Numerical inverse spectral transform for the periodic sine-Gordon equation: Theta function solutions and their linearized stability[END_REF]. Here, we apply a more general and straightforward method for the direct scattering problem since it comes down to a product of matrices. Such a method has been previously applied to the KdV equation [START_REF] Osborne | Nonlinear Ocean Waves and the Inverse Scattering Transform[END_REF], and used experimentally in the periodic KdV equation [START_REF] Novkoski | Experimental observation of periodic Korteweg-de 12 Vries solitons along a torus of fluid[END_REF].

While SG solitons have been experimentally observed in various fields [START_REF] Scott | A nonlinear Klein-Gordon equation[END_REF][START_REF] Ustinov | Solitons in Josephson junctions[END_REF][START_REF] Ivancevic | Sine-Gordon Solitons, Kinks and Breathers as Physical Models of Nonlinear Excitations in Living Cellular Structures[END_REF], a direct link between the direct scattering method and experimental data is still elusive so far. An experimental investigation of the integrability of physical systems described by the SG equation is yet to be made and would also provide a step towards the experimental study of integrable turbulence, in particular in the case of topological solitons, and for the synthesis of the corresponding soliton gas. Here, we present a tool and a case study of individual solutions to the SG equation, that could potentially allow for easy use of the direct scattering method on experimental data.

The article is organized as follows: in Sec. 2, we compare classic solutions of the SG equation on the real line to those satisfying periodic boundary conditions. In Section 3, we recall the scattering theory generally used in the literature and introduce a simpler numerical scattering method. We then apply the latter to well-known solutions and test its validity (Sec. 4 for one-and two-kink solitons, along with breathers ; Sec. 5 for periodic trains of kinks and radiative modes). We draw our conclusions in Sec. 6.

The sine-Gordon equation on the real line and with periodic boundary conditions 2.1 On the real line

The (1+1) sine-Gordon equation (1 dimension of space and 1 of time) on the real line as an initialvalue problem and for the unknown field ϕ(x, t), reads in nondimensional form

ϕ tt -ϕ xx + sin ϕ = 0, (1)
for (x, t) ∈ R × [0, ∞] and an initial condition ϕ(x, t = 0) = ϕ 0 (x). One set of spatial boundary conditions can be chosen as "vanishing" ones at infinity, that is ϕ(x, t) → 0 (mod 2π), as |x| → ∞ (2)

ϕ x (x, t) → 0, as |x| → ∞. (3)
In this article, we will refer to this SG equation along the real line, with vanishing boundary conditions as the iSG equation.

A well-known solitonic solution of this problem reads

ϕ σ kink (x, t) = 4 arctan exp σ x -vt -x 0 √ 1 -v 2 . (4)
The case σ = +1 corresponds to the so-called kink solution, whereas the case σ = -1 corresponds to the so-called anti-kink solution, both moving at speed v ∈] -1, +1[. Note that throughout the rest of the article, we will refer to the term kink (resp. anti-kink) as a topological state connecting values of the function ϕ(x, t) going from 2nπ to 2(n + 1)π (resp. 2mπ to 2(m -1)π) as x increases, with m, n integers.

If the vanishing boundary conditions (2, 3) are disregarded, another set of solutions, moving at speed v ∈ ] -1, +1[, can be found as an infinite periodic train of kinks (σ = +1) or anti-kinks (σ = -1). They read

ϕ σ am (x, t) = π + σ am x -vt -x 0 m(1 -v 2 ) , m , (5)
where am(x, m) is the Jacobi amplitude function, for m ∈ ]0, +1[. They form a two-parameter family of solutions, parameterized by their speed v and the elliptic parameter m. σ = +1 is related to the increasing solution (kinks), whereas σ = -1 corresponds to the decreasing one (anti-kinks). Note that the limit-case m = 1 corresponds to the kink or the anti-kink solutions given by ( 4).

For such a solution, the period Λ between two neighboring periodic kinks is given by

Λ(m, v) = 2 m(1 -v 2 ) K(m), ( 6
)
where K is the complete elliptic function of the first kind.

Periodic boundary conditions

In the rest of the article, we study the (1+1) SG equation (1) but we impose "periodic" boundary conditions with a spatial period L :

ϕ(x + L, t) = ϕ(x, t) (mod 2π), (7)
ϕ x (x + L, t) = ϕ x (x, t), (8)
We assume the initial condition ϕ(x, t = 0) = u(x), as well as ϕ x (x, t = 0) and ϕ t (t = 0) are known. We will call this set of equations the periodic sine-Gordon equation (pSG). The classic sine-Gordon equation is known as the continuous limit of a discrete mechanical chain of torsionnally coupled pendula. Here, the periodic conditions [START_REF] Osborne | Nonlinear Ocean Waves and the Inverse Scattering Transform[END_REF][START_REF] Redor | Experimental Evidence of a Hydrodynamic Soliton Gas[END_REF] turns the usual straight pendulum chain into a circle chain and includes possible non-zero topological charges, hence the possible presence of kinks.

The length L will be chosen to be larger than the typical length scale equal to 1 in the case of the nondimensional SG equation [START_REF] Russell | Report on waves[END_REF].

Fig. 1 Scheme in the complex plane of the energies, E = λ 2 , of different types of soliton that can take real (topological soliton in red) or complex (breather soliton in blue) values.

The equivalent of the 1-kink solution for such periodic boundary conditions and moving at speed v corresponds to the solution given by ( 5), provided the distance between two successive periodic kinks is L, namely we have the relationship between L, v, m

L = Λ(m, v) = 2 m(1 -v 2 ) K(m). (9)
In turn, the N -periodic kink solutions (in the sense of a 2N π-jump in ϕ) for such periodic boundary conditions are given by ( 5), with the relationship between L, v, m, N

L = N Λ(m, v) = 2N m(1 -v 2 ) K(m). ( 10
)
3 Direct scattering method

From the solutions of the pSG equation, we wish to obtain the nonlinear spectral content of our system. To that end, the SG equation is viewed as a compatibility condition of two matrix equations [START_REF] Malomed | The sine-Gordon Model: General Background, Physical Motivations, Inverse Scattering, and Solitons[END_REF] Ψ

x = i λ 2 σ z + ϕ x -ϕ t 4 σ x - σ z cos ϕ + σ y sin ϕ 8λ Ψ, iΨ t = λ 2 σ z + ϕ x -ϕ t 4 σ x + σ z cos ϕ + σ y sin ϕ 8λ Ψ, (11)
where Ψ(x, t) is a two-component complex Jost function, and σ j are the Pauli matrices with j =

x, y or z

σ x = 0 +1 -1 0 ; σ y = 0 -i +i 0 ; σ z = +1 0 0 -1 .
(12) Unlike the KdV or NLS equation, where the problem of finding λ is essentially linear, here we are dealing with an eigenvalue problem which is linear in Ψ but nonlinear in λ. We will be interested in finding the value E, which we refer to as energy, such that λ = √ E. Because of the periodic boundary conditions, it is possible to gain more insight of the problem by applying Floquet theory [START_REF] Forest | Spectral theory for the periodic sine-Gordon equation: A concrete viewpoint[END_REF]. After fixing a given point x 0 , we consider a basis of solutions

{Ψ + (x, x 0 , E), Ψ -(x, x 0 , E)} such that Ψ + (x = x 0 , x 0 , E) = 1 0 , (13)
Ψ -(x = x 0 , x 0 , E) = 0 1 . ( 14
)
Due to the periodicity, the functions Ψ + (x + L, x 0 , E) also have to be solutions of Eq. ( 11), and are expressible in terms of the basis functions. This leads us to write

Ψ + (x + L, x 0 , E) Ψ -(x + L, x 0 , E) = S(E) Ψ + (x, x 0 , E) Ψ -(x, x 0 , E) (15)
where S is the scattering matrix (or transfer matrix, also called the monodromy matrix). In order for the solutions to be bounded, certain conditions need to be imposed onto the matrix. In addition, the spectrum of the problem in Eq. ( 11) is divided into its discrete and continuous parts. The eigenvalues of the discrete spectrum correspond to solitons and are such that they satisfy [START_REF] Forest | Spectral theory for the periodic sine-Gordon equation: A concrete viewpoint[END_REF] |tr(S)| = 2, ( 16)

ℑ(tr S) = 0, ( 17
)
where ℑ stands for the imaginary part. In addition, for real potentials, the eigenvalues are negative. Thus, the solitons will lie in the E < 0 plane as shown in Fig. 1, for the case of kinks (for which E ∈ R) and breathers (E ∈ C \ R).

On the other hand, the continuous spectrum consists of eigenvalues such that E > 0 with

|tr(S)| ≤ 2, (18)
ℑ(tr S) = 0. ( 19
)
This part of the spectrum is associated with the radiation of phonons, i.e., the dispersive component of the wave field. Thus, in order to spectrally detect the solitons (in the E-plane), numerical computations of the trace of the monodromy matrix are needed.

In this article, we tackle the problem by following a different approach, using a method similar to that first used in Refs. [START_REF] Osborne | Nonlinear Ocean Waves and the Inverse Scattering Transform[END_REF][START_REF] Osborne | Automatic algorithm for the numerical inverse scattering transform of the Korteweg-de Vries equation[END_REF] for the periodic KdV case. Expanding the Jost function at x + ∆x yields

Ψ(x + ∆x) = e T∆x Ψ(x) = Q(x)Ψ(x), (20)
where the matrix T is inferred from Eq. ( 11) and Q is defined as Q(x) ≡ e T∆x . By iterating the above process over a piecewise constant signal containing N points, we find the scattering matrix

M(E) = 0 n=N -1 Q(x n , E), (21)
which relates now Ψ(x + L) to Ψ(x), with x n = n∆x and ∆x = L/N . By choosing x = x 0 , we have that tr(M) = tr(S). Note that our calculation of M(E) consists in a mere product of matrices, whereas the computations of S(E) is based on non-straightforward numerical schemes [START_REF] Overman | Coherence and chaos in the driven damped sine-Gordon equation: Measurement of the soliton spectrum[END_REF].

In order to simplify computations, we are able to reduce the matrix Q of Eq. ( 20) to

Q = cos(∆x)I + i sin(∆x)⃗ v • ⃗ σ σ σ, (22)
with

⃗ v = ϕ x -ϕ t 4 , - sin ϕ 8λ , λ 2 - cos ϕ 8λ , (23)
speeding calculations up around two times. In practice, in order to compute Q, we assume that ϕ(x, t = t 0 ) as well as ϕ(x, t = t 0 + ∆t) are known, since one only has to have information of the time derivative at t 0 . Knowing the full temporal evolution of ϕ is unnecessary. Despite this simplification, computation using complex values is still required. A decomposition of Eq. ( 21)

into real and imaginary parts would be a further improvement, but is not obvious. In the following, using well-known solutions along with their spectral characteristics, we will test this numerical method and compare its results to those of the literature (for instance those studied in Ref. [START_REF] Forest | Spectral theory for the periodic sine-Gordon equation: A concrete viewpoint[END_REF] using the S matrix).

4 One-and two-soliton solutions

Single kink solitons for periodic boundary conditions

Let us begin with the topological soliton solutions of the pSG equation, i.e, periodic kinks and anti-kinks (see Eqs. ( 5) and ( 9)). We examine three solitons, with the same velocity v = 0.3, but different values o f t he e lliptic p arameter m = 0.5, 0.8, 0.99, described by Eq. ( 5). Using the algorithm prescribed above, based on the calculation of the matrix M(E), we use these three soliton solutions to compute their corresponding traces of the scattering matrix. We focus on kinks, i.e. σ = +1, and the space domain is chosen so that it corresponds to one period leading to one single soliton. As a consequence, each spatial domain has a different l ength L.

The results for the traces are shown in Fig. 2. We observe that for all the three solitons, there is one intersection with +1 and one with -1 as expected theoretically. These intersections corresponding to solitons, we see that the method permits their detection. As the elliptic parameter is varied, the two corresponding energies move closer and closer together (and in the limit of m = 1, they would coincide). The velocity v and elliptic parameter m are linked to the energies of the soliton and are given by

m = 2 1 + 1 2 E 2 /E 1 + E 1 /E 2 , ( 24
) v = 4(E 1 E 2 ) 1/2 -1 4(E 1 E 2 ) 1/2 + 1 , (25)
as computed using expressions found in [START_REF] Forest | Spectral theory for the periodic sine-Gordon equation: A concrete viewpoint[END_REF]. It is worth mentioning that previous references dealing with this type of solutions do not state that the elliptic parameter m is fixed by the energies in this way. Using a bisection method, we numerically given by Eq. ( 5) and used as the initial condition for the direct scattering method with m = 0.8 and m = 0.99 (same color as in the main figure). Note that, in both cases, we have used three periods L(m) (see Eq. ( 9)) as an illustration of their periodicity (hence a 6π-jump). determine the intersections with tr(M)/2 = ±1, allowing us to obtain the energies E 1 and E 2 . Given this statement, along with Eq. ( 24), we are able to confirm that the method correctly identifies the solitons, with the numerical error being within 0.3%.

Truncated kink soliton of the whole line

Here, we will truncate the kink solution on the real line. We want to show in what extent the numerical procedure still holds when the input signal is not an exact solution (here, the boundary conditions (equations ( 7) and ( 8)) are not satisfied). Let us begin with the topological soliton solutions of the iSG equation, i.e, kinks and anti-kinks.

A single soliton solution with energy E = -K is given by ( 4) and its velocity v satisfies

v = 4K -σ 1 + 4Kσ , (26)
for σ = ±1 discriminating kinks (+) from antikinks (-) [START_REF] Kivshar | Creation of sine-Gordon solitons by a pulse force[END_REF] . We apply the algorithm proposed in Sec. 3 in order to compute the trace of the scattering matrix M and to find the eigenvalues λ of the following signal. The input signal is chosen to be a kink (displayed in the inset of Fig. 3), for which we take K = 1 in Eq. ( 26), σ = +1, and u(x) = ϕ(x, 0) as in Eq. ( 4) consisting of N = 1000 points, on an interval of length L = 20. While the signal is a solution for the infinite line, and does not satisfy perfectly the periodic boundary condition, we consider that the errors due to the mismatch at the two ends will be negligible and use the given form of the field nevertheless. By iterating over both the signal and different values for the energy E, we find the half-trace of the matrix M(E) as shown in Fig. 3. The imaginary part of the trace is zero, and the real part of tr(M)/2 takes values ±1 at E 1 = E 2 = -1.0024, as expected. These values of the crossings are obtained using the bisection method with the cutoff precision set to 10 -12 . We do not notice a numerical difference between the two values. Close E = 0 the trace tends to -∞ which is expected theoretically [START_REF] Forest | Spectral theory for the periodic sine-Gordon equation: A concrete viewpoint[END_REF]. In conclusion, the algorithm has detected the presence of a kink soliton on a line, without imposition of periodic boundary conditions.

Two-kink solution

We apply now the procedure for a signal consisting of a pair of two kinks with two different energies K 1 and K 2 and their corresponding velocities v 1 and v 2 . The input signal is calculated using the formula for a two-kink solution that can be obtained through a Bäcklund transform [START_REF] Barone | Theory and applications of the sine-Gordon equation[END_REF] and reads

u(x, t) = 4 arctan k ϕ 1 -ϕ 2 1 + ϕ1 ϕ2 , (27)
ϕ j = exp   x -v j t 1 -v 2 j   , v j = 4K j -1 1 + 4K j , (28)
k = β 1 -β 2 β 1 + β 2 , β j = 1 -v j 1 + v j , j = 1, 2. ( 29
)
We here stress that the link between the eigenvalues, i.e. K 1 and K 2 , and the velocities (v 1 and v 2 ) is the same as for the one-kink solution. This link has not been clearly stated in the literature, to our knowledge. The initial condition used is shown in the inset of Fig. 4 for L = 100, with the energies of the two solitons fixed a t values K 1 = 1 a nd K 2 = 1 .5. In response to this initial condition, the numerically computed half-trace M(E) is shown in Fig. 4. The energies found using the bisection method lead to E 1 = E 2 = -1.5009 and E 3 = E 4 = -1.0024, which is in excellent agreement with the prescribed initial soliton energies. The application of the algorithm to arbitrary numbers of kink or antikink solitons is thus achievable as long as their energies are different enough. Due to the fact that their energies are real numbers, the detection of kinks and anti-kinks is numerically easy, and the computation time is relatively short. The case of a kink train will be tested below in section 5.

Breathers

The breather is a time-oscillating and spatially localized soliton of the SG equation which can be described by an angle 0 < µ < π/2 in the complex plane of λ and is associated with two complex values of λ 1,2 = √ E = ± 1 2 e ±iµ [START_REF] Kivshar | Creation of sine-Gordon solitons by a pulse force[END_REF]. Its shape is given by ϕ(x, t) = 4 arctan tan µ sin(t cos µ + ϕ 0 ) cosh(x sin µ) . (30)

Our calculation of the trace must be extended to complex values of the energy and we shall be looking for the points where ℑ(tr M) = 0 and still ℜ[tr(M)/2] = 1. Nevertheless, we find t hat the presence of breathers is easily identified b y simple inspection of the imaginary part of the trace, from which we numerically find and confirm their energies.

We take the value of µ = π/3 and apply the algorithm to a single breather. The search for the eigenvalue is now extended into the complex plane by calculating the trace on a mesh of energies. The real part of the trace of M is shown in Fig. 5(a), alongside the imaginary part in Fig. 5(b). We observe an entangled pattern of values, which oscillates from positive to negative values. The focus should be put on the imaginary part of the trace, which gives a qualitative hint of the presence of a breather.

The breather changes the spectrum by "pinching" together two regions of negative (and positive) values of the imaginary part of the trace. Without the presence of the soliton, the regions are well separated and smooth (see Appendix A). The accurate location of the "pinch" in the Espace does not correspond exactly to its eigenvalue, and depends on the resolution of the mesh, but it serves as a good qualitative indicator.

In order to locate the exact position of the breather eigenvalue we use the pinch as a first guess, and focus on the function

G = |ℜ(tr M)| -2 2 + ℑ(tr M) 2 (31)
which will be identically zero when the real part is equal to 2 and the imaginary part is zero. We plot this function G(E) in a zoomed-in region around the pinch in Fig. 6. As we can see, there is a single minimum, which corresponds to the breather eigenvalue. By applying a root-finding algorithm, we find the location of the root to be E = -0.1268 + 0.2116i, which up to our precision, is close to the expected value of E = -1 8 + √ 3 8 i. For the root-finding procedure, the position of the minimum found in Fig. 6 was taken as the initial guess.

In order to better quantify the detection of breathers instead of just relying on visual inspection, we use Cauchy's argument principle which states that

I = 1 2iπ γ f ′ (z) f (z) dz, (32)
for a holomorphic function f(z) and a closed contour γ, is an integer equal to Z, the number of zeros of f enclosed by γ and counted with their multiplicities. Here we compute the value of I with f(z) = tr M(z) -2. For the sake of simplicity and efficiency, we use a regular Cartesian grid of point of the complex E-plane, a rectangle contour γ based on the grid points and centered differ-ences for the calculation of the complex derivatives along γ. We use the trapezoidal rule in order to com-pute the integral. The value of the index is found to be I = 0.9995 ≈ 1 (the numerical error stem-ming from the discretization). It corresponds to one breather, since, much like the kink, the order of the zero of tr M -2 is 1. This method has also been tested in the case of multiple breathers and found to be reliable: for instance, four breathers lead to I = 4, provided γ encloses the four zeros of f (data not shown).

Periodic solutions: trains of kinks and radiative modes

We now turn to the periodic solutions of the SG equation. We will be interested only in stable solutions [START_REF] Scott | A nonlinear Klein-Gordon equation[END_REF]. We will first t est t he m ethod for nonlinear periodic solutions (cnoidal-like waves as a periodic train of kinks) and then linear periodic solutions as radiative modes.

Periodic train of kinks

We begin with the monotonically increasing kink train, which is characterized by two energies E 1 < E 2 < 0, and requires that (E 1 E 2 ) 1/2 > 1. Its form is given by Eq. ( 5), and the number of kinks that are contained in the domain sets the size of the latter through Eq. [START_REF] Suret | Nonlinear spectral synthesis of soliton gas in deepwater surface gravity waves[END_REF]. The velocity v and elliptic parameter m are given by Eq. ( 24). If we let E 1 and E 2 approach each other, the distance between the kinks increases and we recover the single kink solution of Eq. ( 4).

In order to test our direct scattering method on this solution, let us choose the kink train velocity to be v = 0.3 and the elliptic parameter m = 0.8. The half-trace M(E) is shown in Fig. 7 for two cases of a N -soliton train. We clearly observe that the half-trace is equal to 1 at a low energy value E 1 , then is equal to -1 at a higher energy value E 2 (Fig. 7(a)). Conversely, when N is even, the half-trace is equal to +1 at two energy values (Fig. 7(b)). In both cases, the numerically obtained values of the energies are found to be E 1 = -1.2222 and E 2 = -0.1757, giving v = 0.2991 and m = 0.797. Moreover, and more interestingly, we observe that the two eigenvalues are both located at the intersection with +1 for an even number of kinks, and that the value of the half-trace remains within ±1 between the two and oscillates N -1 times. This is what distinguishes the kink train from the pure kink solution.

Radiative modes

Let us now turn to small-amplitude periodic solutions, which will illustrate the dispersive or radiative components of the spectrum. This solution is governed by two values of the energy E 1 = E * 2 = |E 1 |e iα located near the positive real axis [START_REF] Scott | A nonlinear Klein-Gordon equation[END_REF][START_REF] Forest | Spectral theory for the periodic sine-Gordon equation: A concrete viewpoint[END_REF]. The form of the waves is given as respect to that of Eq. ( 24) and read

u = 2 arcsin √ m sn x -vt √ 1 -v 2 , m , (33)
m = 1 -cos α 2 , ( 34
) v = 1 + 4(E 1 E 2 ) 1/2 1 -4(E 1 E 2 ) 1/2 . ( 35
)
The amplitude of the wave is controlled through the angle of the energy α. We select a value of the energy with α = π/4 and |E 1 | = 1 and focus on the part of the complex plane with positive real values of tr(M). The inset of Fig. 8 shows the waveform of Eq. (33) at t = 0. In Fig. 8, we only show the real part of the trace of the matrix M since there is no significant qualitative difference with its imaginary part. Similarly to the breather of Sec. 4, a distortion of the trace in the complex plane is observed, and a similar pinching is found close to the energy of the traveling kink-train, in particular when compared to the trace of M corresponding to a low-amplitude sine wave (see Appendix A). In order to evidence the presence of the prescribed energies in a clearer way, we once more make use of the function G(E) of Eq. ( 31), which is plotted in Fig. 9. We clearly see that there is a zero at the position of the prescribed energy, circled in red, as predicted by Floquet theory. Other zeros, corresponding to similar radiative modes, are also present. of the SG equation, for which the associated eigenvalues are either real (topological solitons as kinks, anti-kinks, and kink trains) or complex in the energy complex plane in the case of non-topological solitons solutions (breathers) and periodic solutions. This method avoids the direct and tedious solving of the nonlinear eigenvalue problem and can be easily applied to a given time series. This could potentially provide meaningful information in experiments, as in the cases of both the NLS and KdV equations. Furthermore, the dynamics of a large number of solitons, such as soliton gas, is a field of growing interest in the study of integrable turbulence [START_REF] Suret | Soliton gas: Theory, numerics and experiments[END_REF], and the SG equation could provide richer dynamics with its topological soliton solutions.

Fig. 2

 2 Fig.2Half-trace of the monodromy matrix M(E) for three kink solitons with p.b.c, m = 0.5, 0.8, 0.99. The trace crosses the lines ±1 only once respectively for all three. We observe how the energies move closer together as m moves towards 1. Inset: Single kink waveform ϕ(x, 0) = u(x) given by Eq. (5) and used as the initial condition for the direct scattering method with m = 0.8 and m = 0.99 (same color as in the main figure). Note that, in both cases, we have used three periods L(m) (see Eq. (9)) as an illustration of their periodicity (hence a 6π-jump).

- 1 .Fig. 3

 13 Fig.3Half-trace of the monodromy matrix M(E) for a kink soliton. Only one crossing of the trace at ±1 is found for a negative value of the energy, E = -1, as expected theoretically. Inset: The kink-soliton waveform ϕ(x, 0) = u(x) used as the initial condition for the direct scattering method (Eq. (4), with L = 20).

Fig. 4

 4 Fig.4Half-trace M(E) for a pair of kink solitons with different energies. As predicted theoretically, two crossings at the negative energies are found, corresponding well numerically to those prescribed. L = 100. Inset: Signal used as the initial condition u(x) in the direct scattering, with energies E = -1 and E = -1.5 (see Eq. (27)).

Fig. 5 Fig. 6

 56 Fig. 5 (a) Real part of the trace of M(E) showing only a slight deformation close to the complex value of the breather energy. (b) Imaginary part of the trace of M(E) in the complex plane of energy E demonstrating a strong "pinch" close to the breather energy E = 1 4 e 2iπ/3 (red circle).

Fig. 7

 7 Fig. 7 Half-trace of the monodromy matrix M for a train of kinks (as in the initial condition in the inset), defined by two different energies. In addition to the crossings at the expected energies (blue dots), we observe that the trace is indeed zero between these two values. (a) Case of a train of N = 5 kinks, (b) Case of a train of N = 8 kinks. Depending on the parity of N , the value of the half-trace of the monodromy matrix M only crosses the value +1 of both the values +1 and -1.

Fig. 8 Fig. 9

 89 Fig. 8 Real part of the trace of M for an oscillatory radiation excitation, as described by Eq. (33). The presence of such a mode distorts the trace at positive values of the real part of the energy, especially close to the prescribed mode energy. Inset: Waveform of Eq. (33) used at t = 0.

Fig

  Fig. A1 (a) Real part of the trace of M(E) in the complex plane of energy, for negative ℜ(E), for a low-amplitude sine wave. (b) Imaginary part of the trace of M(E) for the same wave.

Fig

  Fig. A2 (a) Real part of the trace of M(E) in the complex plane of energy, for positive ℜ(E), for a low-amplitude sine wave. (b) Imaginary part of the trace of M(E) for the same wave.

Fig

  Fig. B3 (a) The eigenvalues of the truncated kink with energy E = 1 for different values of the domain size. (b) Eigenvalues of the truncated kink on a domain with fixed L = 20 for different values of the discretization ∆x.

Acknowledgments. This work is supported by the French National Research Agency (ANR SOGOOD project No. ANR-21-CE30-0061-04), and by the Simons Foundation MPS No 651463-Wave Turbulence.

Data Availability Statement. No Data associated in the manuscript.

Appendix A

As a comparison with the results on the different solitons and periodic solutions to the pSG equation discussed in the main text, we here include the half-trace obtained for a sine wave of amplitude A = 0.01, with k = 2 and ω = 1 on a domain of length L = 60. The real and imaginary parts of the trace are shown in Fig. A1 for the negative real part of the energy plane, where we observe that there is no pinching. In Fig. A2 we show the same quantity for the positive real part of the energy plane.

Appendix B

In this appendix, we discuss how the domain size L and the spatial discretization ∆x affect the eigenvalues of a single infinite-line truncated kink soliton. We start varying L while keeping ∆x fixed at 0.02. Its energy has been chosen to be E = 1. We can see in Fig. B3(a) that as the domain size L is increased, the results become more accurate. This is due to the fact that the errors due to periodicity (i.e. the mismatch at the domain ends) decreases. In Fig. B3(b), we have kept the domain size value at L = 20 and changed the discretization ∆x. As expected, by decreasing the value of ∆x, the energies converge to a single value.