Os modelos de Toda admitem uma representação de suas equações de movimento em termos da curvatura nula, isto é, existem potenciais que são funcionais dos campos da teoria e pertencem a uma álgebra de Kac-Moody tal que a condição de curvatura nula seja equivalente às equações de movimento. Para a construção das soluções solitônicas e cargas conservadas são necessários a gradação inteira da álgebra de Kac-Moody e a existência de soluções de vácuo, de forma que os potenciais assumam valores em uma subálgebra abeliana quando calculados nestas soluções de vácuo. A gradação da álgebra é de extrema importância pois garante que o potencial transformado tenha a mesma estrutura que o potencial de vácuo. As cargas conservadas são então construídas partindo de soluções da órbita do vácuo por meio de transformações de dressing, que consistem na aplicação da decomposição de Gauss para a produção de um potencial transformado a partir de duas transformações de Gauge. Nesta dissertação calculamos as infinitas cargas conservadas dos modelos de Toda sl(3) e também sl(N), avaliadas nas soluções pertencentes à órbita do vácuo sob transformações de dressing. As soluções de interesse físico, como sólitons e breathers pertencem a esta órbita, e as cargas conservadas para tais soluções são escritas como uma soma sobre os sólitons. Mostramos que a energia e o momento proveem de termos de superfície. / The Toda models admit a zero curvature representation of their equations of motion, i.e. there exist potentials, (A), wich are functionals of the fields of the theory and which belong to a Kac-Moody algebra G such that the zero curvature condition is equivalent to the equations of motion. For the construction of the solitons solutions and conserved charges is required an integer gradation of the Kac-Moody algebra and a ``vacuum solution\'\', such that the potentials evaluated on it belong to an abelian subalgebra. The gradation of the algebra is of extreme importance since it guarantees that the transformed potential have the same structure as the vacuum potential. The conserved charges are then constructed using the dressing method, that through the Gauss decomposition, leads to the transformed potentials by two gauge transformations. In this dissertation we calculate the infinite conserved charges of models Toda sl (3) and also sl (N) evaluated on the solutions belonging to the orbit of the vacuum under dressing transformations. The solutions of physical interest, like solitons and breathers belong to this orbit and the conserved charges for such solutions are written as a sum over the number the solitons. We show that the energy and momentum are boundary terms.

Os modelos de Toda admitem uma representação de suas equações de movimento em termos da curvatura nula, isto é, existem potenciais que são funcionais dos campos da teoria e pertencem a uma álgebra de Kac-Moody tal que a condição de curvatura nula seja equivalente às equações de movimento. Para a construção das soluções solitônicas e cargas conservadas são necessários a gradação inteira da álgebra de Kac-Moody e a existência de soluções de vácuo, de forma que os potenciais assumam valores em uma subálgebra abeliana quando calculados nestas soluções de vácuo. A gradação da álgebra é de extrema importância pois garante que o potencial transformado tenha a mesma estrutura que o potencial de vácuo. As cargas conservadas são então construídas partindo de soluções da órbita do vácuo por meio de transformações de dressing, que consistem na aplicação da decomposição de Gauss para a produção de um potencial transformado a partir de duas transformações de Gauge. Nesta dissertação calculamos as infinitas cargas conservadas dos modelos de Toda sl(3) e também sl(N), avaliadas nas soluções pertencentes à órbita do vácuo sob transformações de dressing. As soluções de interesse físico, como sólitons e breathers pertencem a esta órbita, e as cargas conservadas para tais soluções são escritas como uma soma sobre os sólitons. Mostramos que a energia e o momento proveem de termos de superfície. / The Toda models admit a zero curvature representation of their equations of motion, i.e. there exist potentials, (A), wich are functionals of the fields of the theory and which belong to a Kac-Moody algebra G such that the zero curvature condition is equivalent to the equations of motion. For the construction of the solitons solutions and conserved charges is required an integer gradation of the Kac-Moody algebra and a ``vacuum solution\'\', such that the potentials evaluated on it belong to an abelian subalgebra. The gradation of the algebra is of extreme importance since it guarantees that the transformed potential have the same structure as the vacuum potential. The conserved charges are then constructed using the dressing method, that through the Gauss decomposition, leads to the transformed potentials by two gauge transformations. In this dissertation we calculate the infinite conserved charges of models Toda sl (3) and also sl (N) evaluated on the solutions belonging to the orbit of the vacuum under dressing transformations. The solutions of physical interest, like solitons and breathers belong to this orbit and the conserved charges for such solutions are written as a sum over the number the solitons. We show that the energy and momentum are boundary terms.

Όπως μπορεί κανείς να καταλάβει και από τον τίτλο, η εργασία έχει να κάνει με μία μέθοδο επίλυσης μη γραμμικών μερικών διαφορικών εξισώσεων και, συγκεκριμένα, μιας οικογένειας τέτοιων εξισώσεων, που ονομάζονται εξισώσεις εξέλιξης. Πολλές από αυτές, μάλιστα, επιδέχονται ειδικού τύπου λύσεις που είναι γνωστές με το όνομα σολιτόνια (solitons). Αρχικά, μας απασχολεί η έννοια της ολοκληρωσιμότητας, για την οποία όμως δεν υπάρχει κάποιος σαφής ορισμός. Παρ' όλα αυτά, μπορούμε να πούμε ότι μία διαφορική εξίσωση καλείται ολοκληρώσιμη όταν μπορεί να γραμμικοποιηθεί άμεσα ή έμμεσα. Ο όρος έμμεση γραμμικοποίηση συνδέεται με την έννοια της ύπαρξης ζευγαριού Lax, την οποία εξηγούμε χρησιμοποιώντας εργαλεία της θεωρίας τελεστών. Για τις μη γραμμικές εξισώσεις εξέλιξης, έχει αναπτυχθεί πλέον πλήθος μεθόδων ανάλυσης, στα πλαίσια της ολοκληρωσιμότητας, και υπάρχει πλούσια σχετική βιβλιογραφία. Αναφέρουμε συνοπτικά μερικές από αυτές χρησιμοποιώντας κάποια παραδείγματα, ενώ επικεντρωνόμαστε στην αναλυτική περιγραφή μιας μεθόδου που πρώτοι παρουσίασαν οι Zakharov και Shabat το 1974. Η μέθοδος αυτή, η οποία αναπτύχθηκε λίγο μετά τη μέθοδο της αντίστροφης σκέδασης, ονομάζεται μέθοδος ένδυσης (dressing method) ή σχήμα των ZS. Για την παρουσίασή της, χρησιμοποιούμε μόνο τελεστές χωρίς να αναφερόμαστε πουθενά στα δεδομένα σκέδασης του προβλήματος. Εισάγουμε, με τη βοήθεια διαφορικών και ολοκληρωτικών τελεστών, το γυμνό (undressed) και το ντυμένο (dressed) τελεστή και, έπειτα, δείχνουμε πώς από αυτούς προκύπτει η γενικευμένη εξίσωση Lax. Παραθέτουμε κάποια παραδείγματα εξισώσεων στις οποίες εφαρμόζεται η μέθοδος και, τέλος, κατασκευάζουμε σολιτονικές λύσεις για τη μη γραμμική εξίσωση του Schrödinger, με τη βοήθεια της ολοκληρωτικής εξίσωσης των Gelfand-Levitan-Marchenko. Πέρα από την περιγραφή της μεθόδου ένδυσης στην αρχική της μορφή, βλέπουμε και πώς αυτή εμφανίζεται στη σύγχρονη βιβλιογραφία. Με την πάροδο του χρόνου εξελίχθηκε αρκετά και συνδέθηκε με προβλήματα της μιγαδικής ανάλυσης και, πιο συγκεκριμένα, με τα προβλήματα Riemann-Hilbert (RH) και dbar που, με τη σειρά τους, προκύπτουν σε πολλές εφαρμογές των μαθηματικών. Από ένα μεγάλο πλήθος πρόσφατα δημοσιευμένων άρθρων, παρουσιάζουμε αναλυτικότερα ένα, αυτό των Bogdanov και Zakharov (2002), που αφορά στην εξίσωση Boussinesq. Περιγράφουμε μια ειδικότερη μορφή της μεθόδου ένδυσης, η οποία ονομάζεται ένδυση dbar (dbar-dressing) και αναλύουμε, μέσω αυτής, τις σολιτονικές λύσεις και το συνεχές φάσμα της εξίσωσης Boussinesq. Οι σολιτονικές λύσεις της εξίσωσης παρουσιάζουν μία πολύ ιδιαίτερη συμπεριφορά, η οποία έρχεται σε αντίθεση με τον ευσταθή χαρακτήρα των σολιτονίων. / As one can understand from the title, our main subject is a method for solving nonlinear partial differential equations and in particular a family of such equations, called evolution equations. Many of them admit a special kind of solutions, known as solitons. One of our basic interests is the integrability of a nonlinear evolution equation, although a specific definition for that does not exist in the bibliography. However, a partial differential equation is considered to be integrable when it can be linearized directly or indirectly. By indirect linearization we mean the existence of a Lax pair for the initial equation and this connection is explained in terms of operator theory. In the frame of integrability, a large number of methods dealing with the study and analysis of nonlinear evolution equations has been developed. We briefly mention some of them and present some examples, while we focus on the analytic description of a method which was introduced by Zakharov and Shabat, in 1974. This method was developed right after the Inverse Scattering Method and it is known as dressing method or ZS scheme. In order to present it, a dressed and undressed operator are introduced, by the use of operators only whithout refering to the scattering data. Based on those operators the generalized Lax equation is produced. Then we present a number of examples of evolution equations which can be solved via the dressing method and finally we constract soliton solutions for the nonlinear Schrödinger equation by solving the Gelfand-Levitan-Marchenko integral equation. Appart from the description of dressing method in its initial form, a quick review of recent papers and results is considered. The method evolved through time and was connected with some problems of complex analysis and specifically the Riemann-Hilbert (RH) and dbar problems. Those two problems arise in many mathematical and physical applications. From a wide range of recent published articles, we analytically present one which was written by Bogdanov and Zakharov (2002) and deals with Boussinesq equation. The continuous spectrum and soliton solutions are investigated, using a special form of dressind method called dbar-dressing. Soliton solutions for the Boussinesq equations demonstrate a quite extraordinary behaviour destroying the stereotype of usual solitons which are considered to be stable objects.