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Nonlinear Dirac Equation: Spectral Stability of Solitary Waves

2019-11-21 Education
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Release : 2019-11-21
Genre : Education
Kind : eBook
Book Rating : 953/5 ( reviews)

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Book Synopsis Nonlinear Dirac Equation: Spectral Stability of Solitary Waves by : Nabile Boussaïd

Download or read book Nonlinear Dirac Equation: Spectral Stability of Solitary Waves written by Nabile Boussaïd. This book was released on 2019-11-21. Available in PDF, EPUB and Kindle. Book excerpt: This monograph gives a comprehensive treatment of spectral (linear) stability of weakly relativistic solitary waves in the nonlinear Dirac equation. It turns out that the instability is not an intrinsic property of the Dirac equation that is only resolved in the framework of the second quantization with the Dirac sea hypothesis. Whereas general results about the Dirac-Maxwell and similar equations are not yet available, we can consider the Dirac equation with scalar self-interaction, the model first introduced in 1938. In this book we show that in particular cases solitary waves in this model may be spectrally stable (no linear instability). This result is the first step towards proving asymptotic stability of solitary waves. The book presents the necessary overview of the functional analysis, spectral theory, and the existence and linear stability of solitary waves of the nonlinear Schrödinger equation. It also presents the necessary tools such as the limiting absorption principle and the Carleman estimates in the form applicable to the Dirac operator, and proves the general form of the Dirac-Pauli theorem. All of these results are used to prove the spectral stability of weakly relativistic solitary wave solutions of the nonlinear Dirac equation.

Nonlinear Dirac Equation

1920 Differential equations, Partial
Download Nonlinear Dirac Equation PDF Online Free

Author :
Release : 1920
Genre : Differential equations, Partial
Kind : eBook
Book Rating : 227/5 ( reviews)

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Book Synopsis Nonlinear Dirac Equation by : Nabile Boussaïd

Download or read book Nonlinear Dirac Equation written by Nabile Boussaïd. This book was released on 1920. Available in PDF, EPUB and Kindle. Book excerpt: This monograph gives a comprehensive treatment of spectral (linear) stability of weakly relativistic solitary waves in the nonlinear Dirac equation. It turns out that the instability is not an intrinsic property of the Dirac equation that is only resolved in the framework of the second quantization with the Dirac sea hypothesis. Whereas general results about the Dirac-Maxwell and similar equations are not yet available, we can consider the Dirac equation with scalar self-interaction, the model first introduced in 1938. In this book we show that in particular cases solitary waves in this model may be spectrally stable (no linear instability). This result is the first step towards proving asymptotic stability of solitary waves. The book presents the necessary overview of the functional analysis, spectral theory, and the existence and linear stability of solitary waves of the nonlinear Schrödinger equation. It also presents the necessary tools such as the limiting absorption principle and the Carleman estimates in the form applicable to the Dirac operator, and proves the general form of the Dirac-Pauli theorem. All of these results are used to prove the spectral stability of weakly relativistic solitary wave solutions of the nonlinear Dirac equation

Solitary Waves in Nonlinear Dirac Equation

2015
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Release : 2015
Genre :
Kind : eBook
Book Rating : /5 ( reviews)

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Book Synopsis Solitary Waves in Nonlinear Dirac Equation by :

Download or read book Solitary Waves in Nonlinear Dirac Equation written by . This book was released on 2015. Available in PDF, EPUB and Kindle. Book excerpt: This report describes the implementation of nonlinear Dirac equations in the calculation of solitary waves. Conclusions and comments on quantum elasticity are also included.

Nonlinear Systems, Vol. 1

2018-09-15 Science
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Release : 2018-09-15
Genre : Science
Kind : eBook
Book Rating : 661/5 ( reviews)

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Book Synopsis Nonlinear Systems, Vol. 1 by : Victoriano Carmona

Download or read book Nonlinear Systems, Vol. 1 written by Victoriano Carmona. This book was released on 2018-09-15. Available in PDF, EPUB and Kindle. Book excerpt: This book is part of a two volume set which presents the analysis of nonlinear phenomena as a long-standing challenge for research in basic and applied science as well as engineering. It discusses nonlinear differential and differential equations, bifurcation theory for periodic orbits and global connections. The integrability and reversibility of planar vector fields and theoretical analysis of classic physical models are sketched. This first volume concentrates on the mathematical theory and computational techniques that are essential for the study of nonlinear science, a second volume deals with real-world nonlinear phenomena in condensed matter, biology and optics.

Spectral Methods in Soliton Equations

1994-11-21 Mathematics
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Release : 1994-11-21
Genre : Mathematics
Kind : eBook
Book Rating : 630/5 ( reviews)

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Book Synopsis Spectral Methods in Soliton Equations by : I D Iliev

Download or read book Spectral Methods in Soliton Equations written by I D Iliev. This book was released on 1994-11-21. Available in PDF, EPUB and Kindle. Book excerpt: Soliton theory as a method for solving some classes of nonlinear evolution equations (soliton equations) is one of the most actively developing topics in mathematical physics. This book presents some spectral theory methods for the investigation of soliton equations ad the inverse scattering problems related to these equations. The authors give the theory of expansions for the Sturm-Liouville operator and the Dirac operator. On this basis, the spectral theory of recursion operators generating Korteweg-de Vries type equations is presented and the Ablowitz-Kaup-Newell-Segur scheme, through which the inverse scattering method could be understood as a Fourier-type transformation, is considered. Following these ideas, the authors investigate some of the questions related to inverse spectral problems, i.e. uniqueness theorems, construction of explicit solutions and approximative methods for solving inverse scattering problems. A rigorous investigation of the stability of soliton solutions including solitary waves for equations which do not allow integration within inverse scattering method is also presented.

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