Etude D'une Équation Non Linéaire, Non Dispersive Et Complètement Integrable Et de Ses Perturbations PDF Download
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Author: Oana Pocovnicu
Publisher:
ISBN:
Category :
Languages : en
Pages : 0
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Book Description
In this Ph.D. thesis, we study the Szegö equation on the real lineas well as its perturbations.It was recently introduced by Gérard and Grellier as a toy model of a non-lineartotally non dispersive equation. The Szegö equation appears naturally in the study of thenon-linear Schrödinger equation (NLS) in super-critical situations where dispersion lacks,for example, when one considers NLS on the Heisenberg group. Consequently, one of themotivations of this Ph.D. thesis is fi nding new results for the Szegö equation in hope thatthey could be eventually used in the context of the non-linear Schrödinger equation.Our first result is a classification of the solitons of the Szegö equation. We show thatthey are all rational functions with one simple pole. In addition, we prove the orbitalstability of solitons.The Szegö equation has the remarkable property of being completely integrable. Thisallows us to find an explicit formula for solutions. We obtain three applications of thisformula. (A) We prove soliton resolution for solutions which are generic rational functions.(B) We construct an example of non-generic solution whose high Sobolev norms grow toinfinity over time. (C) We find generalized action-angle variables when restricting the Szegöequation to a finite dimensional sub-manifold. In particular, this yields that most of thetrajectories of the Szegö equation are spirals around toroidal cylinders.Since the Szegö equation is completely integrable, it is natural to study its perturbationsand deduce new properties of such perturbations from the known results for the Szegöequation. One perturbation of the Szegö equation is a non-linear wave equation(NLW) with small initial data.We prove that the Szegö equation is the first order approximation of NLW. More precisely,if an initial condition of NLW is small and supported only on non-negative frequencies, thenthe corresponding solution can be approximated by the solution of the Szegö equation, fora long time. We then construct a solution of NLW whose high Sobolev norms grow.On the torus T, Gérard and Grellier proved an analogous first order approximationresult for NLW. By considerning the second order approximation, we obtain an improvedresult with a smaller error.Lastly, we consider the Szegö equation perturbed by a small multiplicative potential.We study the interaction of this potential with solitons. More precisely, we show that, if theinitial condition is that of a soliton for the unperturbed Szegö equation, then the solutionpreserves the shape of a soliton for a long time. In addition, we prescribe the effectivedynamics, i.e. we derive the differential equations satisfied by the parameters of the soliton.
Author: Oana Pocovnicu
Publisher:
ISBN:
Category :
Languages : en
Pages : 0
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Book Description
In this Ph.D. thesis, we study the Szegö equation on the real lineas well as its perturbations.It was recently introduced by Gérard and Grellier as a toy model of a non-lineartotally non dispersive equation. The Szegö equation appears naturally in the study of thenon-linear Schrödinger equation (NLS) in super-critical situations where dispersion lacks,for example, when one considers NLS on the Heisenberg group. Consequently, one of themotivations of this Ph.D. thesis is fi nding new results for the Szegö equation in hope thatthey could be eventually used in the context of the non-linear Schrödinger equation.Our first result is a classification of the solitons of the Szegö equation. We show thatthey are all rational functions with one simple pole. In addition, we prove the orbitalstability of solitons.The Szegö equation has the remarkable property of being completely integrable. Thisallows us to find an explicit formula for solutions. We obtain three applications of thisformula. (A) We prove soliton resolution for solutions which are generic rational functions.(B) We construct an example of non-generic solution whose high Sobolev norms grow toinfinity over time. (C) We find generalized action-angle variables when restricting the Szegöequation to a finite dimensional sub-manifold. In particular, this yields that most of thetrajectories of the Szegö equation are spirals around toroidal cylinders.Since the Szegö equation is completely integrable, it is natural to study its perturbationsand deduce new properties of such perturbations from the known results for the Szegöequation. One perturbation of the Szegö equation is a non-linear wave equation(NLW) with small initial data.We prove that the Szegö equation is the first order approximation of NLW. More precisely,if an initial condition of NLW is small and supported only on non-negative frequencies, thenthe corresponding solution can be approximated by the solution of the Szegö equation, fora long time. We then construct a solution of NLW whose high Sobolev norms grow.On the torus T, Gérard and Grellier proved an analogous first order approximationresult for NLW. By considerning the second order approximation, we obtain an improvedresult with a smaller error.Lastly, we consider the Szegö equation perturbed by a small multiplicative potential.We study the interaction of this potential with solitons. More precisely, we show that, if theinitial condition is that of a soliton for the unperturbed Szegö equation, then the solutionpreserves the shape of a soliton for a long time. In addition, we prescribe the effectivedynamics, i.e. we derive the differential equations satisfied by the parameters of the soliton.
Author: Sigeru Mizohata
Publisher: Academic Press
ISBN: 148326906X
Category : Mathematics
Languages : en
Pages : 186
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Book Description
Notes and Reports in Mathematics in Science and Engineering, Volume 3: On the Cauchy Problem focuses on the processes, methodologies, and mathematical approaches to Cauchy problems. The publication first elaborates on evolution equations, Lax-Mizohata theorem, and Cauchy problems in Gevrey class. Discussions focus on fundamental proposition, proof of theorem 4, Gevrey property in t of solutions, basic facts on pseudo-differential, and proof of theorem 3. The book then takes a look at micro-local analysis in Gevrey class, including proof and consequences of theorem 1. The manuscript examines Schrödinger type equations, as well as general view-points on evolution equations. Numerical representations and analyses are provided in the explanation of these type of equations. The book is a valuable reference for mathematicians and researchers interested in the Cauchy problem.
Author: V. Babelon
Publisher: Springer Science & Business Media
ISBN: 1461203155
Category : Mathematics
Languages : en
Pages : 368
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Book Description
This book constitutes the proceedings of the International Conference on Integrable Systems in memory of J.-L. Verdier. It was held on July 1-5, 1991 at the Centre International de Recherches Mathematiques (C.I.R.M.) at Luminy, near Marseille (France). This collection of articles, covering many aspects of the theory of integrable Hamiltonian systems, both finite and infinite-dimensional, with an emphasis on the algebro-geometric meth ods, is published here as a tribute to Verdier who had planned this confer ence before his death in 1989 and whose active involvement with this topic brought integrable systems to the fore as a subject for active research in France. The death of Verdier and his wife on August 25, 1989, in a car accident near their country house, was a shock to all of us who were acquainted with them, and was very deeply felt in the mathematics community. We knew of no better way to honor Verdier's memory than to proceed with both the School on Integrable Systems at the C.I.M.P.A. (Centre International de Mathematiques Pures et Appliquees in Nice), and the Conference on the same theme that was to follow it, as he himself had planned them.
Author: Angelo Guerraggio
Publisher: Springer Science & Business Media
ISBN: 9783764365554
Category : Mathematics
Languages : en
Pages : 320
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Book Description
This book describes Italian mathematics in the period between the two World Wars. It analyzes the development by focusing on both the interior and the external influences. Italian mathematics in that period was shaped by a colorful array of strong personalities who concentrated their efforts on a select number of fields and won international recognition and respect in an incredibly short time. Consequently, Italy was considered a third mathematical power after France and Germany.
Author: Lars Hörmander
Publisher: Springer Science & Business Media
ISBN: 1461207754
Category : Mathematics
Languages : en
Pages : 384
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Book Description
On March 17-19 and May 19-21,1995, analysis seminars were organized jointly at the universities of Copenhagen and Lund, under the heading "Danish-Swedish Analysis Seminar". The main topic was partial differen tial equations and related problems of mathematical physics. The lectures given are presented in this volume, some as short abstracts and some as quite complete expositions or survey papers. They span over a large vari ety of topics. The most frequently occurring theme is the use of microlocal analysis which is now important also in the study of non-linear differential equations although it originated entirely within the linear theory. Perhaps it is less surprising that microlocal analysis has proved to be useful in the study of mathematical problems of classical quantum mechanics, for it re ceived a substantial input of ideas from that field. The scientific committee for the invitation of speakers consisted of Gerd Grubb in Copenhagen, Lars Hormander and Anders MeHn in Lund, and Jo hannes Sjostrand in Paris. Lars Hormander and Anders Melin have edited the proceedings. They were hosts of the seminar days in Lund while Gerd Grubb was the host in Copenhagen. Financial support was obtained from the mathematics departments in Copenhagen and Lund, CNRS in France, the Danish and Swedish Na tional Research Councils, Gustaf Sigurd Magnuson's foundation at the Royal Swedish Academy of Sciences, and the Wenner-Gren foundation in Stockholm. We want to thank all these organisations for their support
Author: Cécile Dewitt-Morette
Publisher: Springer Science & Business Media
ISBN: 1489903194
Category : Science
Languages : en
Pages : 436
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Book Description
The program of the Institute covered several aspects of functional integration -from a robust mathematical foundation to many applications, heuristic and rigorous, in mathematics, physics, and chemistry. It included analytic and numerical computational techniques. One of the goals was to encourage cross-fertilization between these various aspects and disciplines. The first week was focused on quantum and classical systems with a finite number of degrees of freedom; the second week on field theories. During the first week the basic course, given by P. Cartier, was a presentation of a recent rigorous approach to functional integration which does not resort to discretization, nor to analytic continuation. It provides a definition of functional integrals simpler and more powerful than the original ones. Could this approach accommodate the works presented by the other lecturers? Although much remains to be done before answering "Yes," there seems to be no major obstacle along the road. The other courses taught during the first week presented: a) a solid introduction to functional numerical techniques (A. Sokal) and their applications to functional integrals encountered in chemistry (N. Makri). b) integrals based on Poisson processes and their applications to wave propagation (S. K. Foong), in particular a wave-restorer or wave-designer algorithm yielding the initial wave profile when one can only observe its distortion through a dissipative medium. c) the formulation of a quantum equivalence principle (H. Kleinert) which. given the flat space theory, yields a well-defined quantum theory in spaces with curvature and torsion.
Author: Jean-Louis Barrat
Publisher: Cambridge University Press
ISBN: 9780521789530
Category : Science
Languages : en
Pages : 410
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Book Description
Presenting a unified approach, this book focusses on the concepts and theoretical methods that are necessary for an understanding of the physics and chemistry of the fluid state. The authors do not attempt to cover the whole field in an encyclopedic manner. Instead, important ideas are presented in a concise and rigorous style, and illustrated with examples from both simple molecular liquids and more complex soft condensed matter systems such as polymers, colloids, and liquid crystals.
Author: H. A. Thurston
Publisher: Courier Corporation
ISBN: 0486154947
Category : Mathematics
Languages : en
Pages : 146
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Book Description
This book explores arithmetic's underlying concepts and their logical development, in addition to a detailed, systematic construction of the number systems of rational, real, and complex numbers. 1956 edition.
Author: Marius Tucsnak
Publisher: Springer Science & Business Media
ISBN: 3764389931
Category : Mathematics
Languages : en
Pages : 488
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Book Description
This book studies observation and control operators for linear systems where the free evolution of the state can be described by an operator semigroup on a Hilbert space. It includes a large number of examples coming mostly from partial differential equations.
Author: J. D. Van Der Waals
Publisher: Courier Corporation
ISBN: 9780486495934
Category : Science
Languages : en
Pages : 336
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Book Description
This much-cited thesis by J. D. van der Waals, the recipient of the 1910 Nobel Prize in physics, is accompanied by an introductory essay by J. S. Rowlinson and another work by van der Waals on the theory of liquid mixtures. 1988 edition.
