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Estrutura hamiltoniana da hierarquia PIVRuy, Danilo Virges [UNESP] 18 February 2011 (has links) (PDF)
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ruy_dv_me_ift.pdf: 500107 bytes, checksum: fef6f049175c290422f569aa7ad7e26e (MD5) / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / Esta dissertação trata da construção de hierarquias compatíveis com a equação PIV a partir dos modelos: AKNS, dois bósons e dois bósons quadráticos. Também são construidos os problema linear de Jimbo-Miwa dos três modelos e discutimos a hamiltoniana correspondente a equação PIV a partir do formalismo lagrangiano / This dissertation contains the construction of compatible hierarchies with the PIV equation from the models: AKNS, two-boson and quadratic two-boson. Also it is build the Jimbo-Miwa linear problem for the three models and we discuss the hamiltonian corresponding to fouth Painlevé equation from the Lagrangian formalism
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Estrutura hamiltoniana da hierarquia PIV /Ruy, Danilo Virges. January 2011 (has links)
Orientador: Abraham Hirsz Zimerman / Banca: Iberê Luiz Caldas / Banca: Roberto André Kraenkel / Resumo: Esta dissertação trata da construção de hierarquias compatíveis com a equação PIV a partir dos modelos: AKNS, dois bósons e dois bósons quadráticos. Também são construidos os problema linear de Jimbo-Miwa dos três modelos e discutimos a hamiltoniana correspondente a equação PIV a partir do formalismo lagrangiano / Abstract: This dissertation contains the construction of compatible hierarchies with the PIV equation from the models: AKNS, two-boson and quadratic two-boson. Also it is build the Jimbo-Miwa linear problem for the three models and we discuss the hamiltonian corresponding to fouth Painlevé equation from the Lagrangian formalism / Mestre
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Ανάλυση ιδιομορφιών και μελέτη της κίνησης ατόμου υδρογόνου σε δυναμικό Van der WaalsΑντωνόπουλος, Χρήστος 31 August 2009 (has links)
Στην παρούσα διπλωματική εργασία μελετάμε την κλασική δυναμική
ατόμου υδρογόνου σε γενικευμένο δυναμικό Van der Waals. Το πρόβλημα ανήκει
στην ευρύτερη κατηγορία των μη γραμμικών Χαμιλτώνιων δυναμικών
συστημάτων. Σκοπός της μελέτης είναι η ανάλυση των ιδιομορφιών της
κανονικής και χαοτικής κίνησης του συστήματος στο μιγαδικό πεδίο του
χρόνου και η εξαγωγή συμπερασμάτων σχετικά με την ολοκληρωσιμότητα
και επιλυσιμότητά του. Εκείνο που θέλουμε να κατανοήσουμε, επίσης,
είναι τον ρόλο που παίζει η εμφάνιση ιδιομορφιών σε κάποια σημεία
του χώρου των φάσεων και κατά πόσο μπορούν αυτές να επηρεάσουν
συνολικά τις ιδιότητες των λύσεων.
Για πρώτη φορά, επίσης, στην διπλωματική αυτή εργασία εφαρμόζεται σε
ένα Χαμιλτώνιο δυναμικό σύστημα μία νέα αριθμητική μέθοδος διάκρισης
μεταξύ κανονικής και χαοτικής συμπεριφοράς σε διαφορετικές περιοχές του χώρου φάσεων, η μέθοδος των Μικρότερων Δεικτών
Ευθυγράμμισης (Smaller Alignment Indices method ή SALI).
Η μέθοδος αυτή έχει χρησιμοποιηθεί κατά το πρόσφατο παρελθόν σε απεικονίσεις δύο, τεσσάρων και έξι
διαστάσεων με πολύ ενδιαφέροντα αποτελέσματα. Χαρακτηριστικά της
είναι η αποτελεσματικότητα και η δυνατότητα εξαγωγής χρήσιμων
συμπερασμάτων ως προς την κανονική και χαοτική φύση των τροχιών ενός
δυναμικού συστήματος με μεγαλύτερη ταχύτητα και αξιοπιστία από την
μέθοδο των χαρακτηριστικών εκθετών Lyapunov καθώς και άλλων
νεότερων μεθόδων στην σύγχρονη βιβλιογραφία.
Εδώ θα παρουσιασθεί η μέθοδος αυτή με ορισμένες βελτιώσεις ώστε να
μπορεί να εφαρμοσθεί σε συστήματα μη γραμμικών διαφορικών εξισώσεων
οποιασδήποτε διάστασης ελέγχοντας συστηματικά ένα όσο πυκνό πλέγμα
αρχικών συνθηκών του χώρου φάσεων επιθυμούμε, αντιστοιχώντας σε κάθε
μία από αυτές ένα χρώμα. Κάθε χρώμα αντιστοιχεί και σε ένα
διαφορετικό εύρος τάξεων του SALI δημιουργώντας έτσι μία
συνολική εικόνα στο χώρο φάσεων που μας επιτρέπει να γνωρίζουμε τη
φύση της τροχιάς κάθε συγκεκριμένης αρχικής συνθήκης. Σχηματίζεται
με αυτόν τον τρόπο το "μωσαϊκό" του χώρου φάσεων και
αποκαλύπτονται περιοχές κανονικής κίνησης, χαοτικής κίνησης καθώς
και νησίδες ή περιοχές στις οποίες δεν αντιστοιχεί καθόλου κίνηση. / In this master thesis we study the classical dynamics of hydrogen atoms in a generalized Van der Waals potential. The problem belongs to the class of non linear Hamiltonian systems. Our aim is the singularity analysis of the ordered and chaotic motion of the system in the complex plain of time and the extraction of valuable conclusions concerning its integrability and solvability. What we want to understand, also, is the role of the emergence of singularities in some points of the phase space of the aforementioned system and how the singularities can affect globally the properties of the solutions.
For the first time, in this master thesis, we introduce and apply in a Hamiltonian system a new numerical method for the fast and efficient discrimination between ordered and chaotic motion in different parts of phase space, namely the method of the Smaller Alignment Index (SALI). The method has been introduced and applied recently in mappings of two, four and six dimensions with very satisfactory results. Its main characteristics are the effectiveness and the ability of extracting valuable conclusions about the ordered and chaotic nature of trajectories of a dynamical system faster than the traditional method of Lyapunov exponents as well as of other indices in the bibliography.
We will introduce SALI with appropriate modifications that help using it in non linear systems of differential equations of arbitrary dimensions checking systematically a dense grid of initial conditions and corresponding in every orbit a different color. Every color corresponds to a different range of SALI values creating by that way a global picture of the phase space that allows us to know the dynamic nature of initial conditions. By that way, we construct a “mosaic” of the phase space and reveal parts of ordered motion as well as parts of chaotic motion and islands of stability.
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Painlevé Integrability and mixed P_III-P_V system solutions /Alves, Victor César Costa January 2017 (has links)
Orientador: Abraham Hirsz Zimerman / Abstract: The current work aims at applications of mathematical methods of Painlevé integrability in physics, on the other side it also approaches the integrable hierarchies formalism and the 2M-bose model where differential equations methods are used as well as a method for solutions using Padé approximants. / Resumo: O presente trabalho trata de um abordagem de aplicações em física dos métodos matemáticos de integrabilidade de Painlevé, por outro lado também aborda o formalismo de hierarquias integráveis e o modelo de 2M-bosons onde são usados métodos de equações diferenciais bem como um método para soluções usando aproximantes de Padé. / Mestre
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Black Hole Formation in Lovelock GravityTaves, Timothy Mark January 2012 (has links)
Some branches of quantum gravity demand the existence of higher dimensions and the addition of higher curvature terms to the gravitational Lagrangian in the form of the Lovelock polynomials. In this thesis we investigate some of the classical properties of Lovelock gravity.
We first derive the Hamiltonian for Lovelock gravity and find that it takes the same form as in general relativity when written in terms of the Misner-Sharp mass function. We then minimally couple the action to matter fields to find Hamilton’s equations of motion. These are gauge fixed to be in the Painleve-Gullstrand co–ordinates and are well suited to numerical studies of black hole formation.
We then use these equations of motion for the massless scalar field to study the formation of general relativistic black holes in four to eight dimensions and Einstein-Gauss-Bonnet black holes in five and six dimensions. We study Choptuik scaling, a phenomenon which relates the initial conditions of a matter distribution to the final observables of small black holes.
In both higher dimensional general relativity and Einstein-Gauss-Bonnet gravity we confirm the existence of cusps in the mass scaling relation which had previously only been observed in four dimensional general relativity. In the general relativistic case we then calculate the critical exponents for four to eight dimensions and find agreement with previous calculations by Bland et al but not Sorkin et al who both worked in null co–ordinates.
For the Einstein-Gauss-Bonnet case we find that the self-similar behaviour seen in the general relativistic case is destroyed. We find that it is replaced by some other form of scaling structure. In five dimensions we find that the period of the critical solution at the origin is proportional to roughly the cube root of the Gauss-Bonnet parameter and that there is evidence for a minimum black hole radius. In six dimensions we see evidence for a new type of scaling. We also show, from the equations of motion, that there is reason to expect qualitative differences between five and higher dimensions.
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Black Hole Formation in Lovelock GravityTaves, Timothy Mark January 2012 (has links)
Some branches of quantum gravity demand the existence of higher dimensions and the addition of higher curvature terms to the gravitational Lagrangian in the form of the Lovelock polynomials. In this thesis we investigate some of the classical properties of Lovelock gravity.
We first derive the Hamiltonian for Lovelock gravity and find that it takes the same form as in general relativity when written in terms of the Misner-Sharp mass function. We then minimally couple the action to matter fields to find Hamilton’s equations of motion. These are gauge fixed to be in the Painleve-Gullstrand co–ordinates and are well suited to numerical studies of black hole formation.
We then use these equations of motion for the massless scalar field to study the formation of general relativistic black holes in four to eight dimensions and Einstein-Gauss-Bonnet black holes in five and six dimensions. We study Choptuik scaling, a phenomenon which relates the initial conditions of a matter distribution to the final observables of small black holes.
In both higher dimensional general relativity and Einstein-Gauss-Bonnet gravity we confirm the existence of cusps in the mass scaling relation which had previously only been observed in four dimensional general relativity. In the general relativistic case we then calculate the critical exponents for four to eight dimensions and find agreement with previous calculations by Bland et al but not Sorkin et al who both worked in null co–ordinates.
For the Einstein-Gauss-Bonnet case we find that the self-similar behaviour seen in the general relativistic case is destroyed. We find that it is replaced by some other form of scaling structure. In five dimensions we find that the period of the critical solution at the origin is proportional to roughly the cube root of the Gauss-Bonnet parameter and that there is evidence for a minimum black hole radius. In six dimensions we see evidence for a new type of scaling. We also show, from the equations of motion, that there is reason to expect qualitative differences between five and higher dimensions.
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Stokes' Phenomenon arising from the confluence of two simple polesHorrobin, Calum January 2018 (has links)
We study certain confluences of equations with two Fuchsian singularities which produce an irregular singularity of Poincaré rank one. We demonstrate a method to understand how to pass from solutions with power-like behavior which are analytic in neighbourhoods to solutions with exponential behavior which are analytic in sectors and have divergent asymptotic behavior. We explicitly calculate the Stokes' matrices of the confluent system in terms of the monodromy data, specifically the connection matrices, of the original system around the merging singularities. The confluence of Gauss' hypergeometric equation gives an excellent opportunity to show our approach with a concrete example. We explicitly show how the Stokes' data arise in the confluences of the isomonodromic deformation problems for the Painlevé equations PVI to PV and PV to PIII(D6).
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Prolongation Structures, Backlund Transformations And Painleve Analysis Of Nonlinear Evolution EquationsYurdusen, Ismet 01 November 2004 (has links) (PDF)
The Wahlquist-Estabrook prolongation
technique and the Painleve analysis, used for testing
the integrability
of nonlinear evolution equations, are
considered and
applied both to the Drinfel' / d-Sokolov system of
equations, indeed known to be one of the coupled
Korteweg-de Vries (KdV)
systems, and Kersten-Krasil' / shchik coupled KdV-mKdV equations. Some new Backlund
transformations for the Drinfel' / d-Sokolov system of equations are also found.
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Algebraic Curves and Flag Varieties in Solutions of the KP Hierarchy and the Full Kostant-Toda HierarchyXie, Yuancheng January 2021 (has links)
No description available.
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Geometry of moduli spaces of meromorphic connections on curves, Stokes data, wild nonabelian Hodge theory, hyperkahler manifolds, isomonodromic deformations, Painleve equations, and relations to Lie theory.Boalch, Philip 12 December 2012 (has links) (PDF)
Short summary of main work since 1999
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