Solitary Waves in the Nonlinear Dirac Equation at the Continuum Limit: Stability and Dynamics

In the present work, we give a comparative summary of different recent contributions to the theme of the linear stability and nonlinear dynamics of solitary waves in the nonlinear Dirac equation in the form of the Gross-Neveu model. We indicate some of the key controversial statements in publication...

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Detalhes bibliográficos
Autores: Cuevas-Maraver, Jesús, Kevrekidis, Panayotis G., Saxena, Avadh, Cooper, Fred, Mertens, Franz G.
Tipo de documento: capítulo de livro
Estado:Versão publicada
Data de publicação:2015
País:España
Recursos:Universidad de Sevilla (US)
Repositório:idUS. Depósito de Investigación de la Universidad de Sevilla
OAI Identifier:oai:idus.us.es:11441/146913
Acesso em linha:https://hdl.handle.net/11441/146913
Access Level:Acceso aberto
Palavra-chave:Solitons
nonlinear Dirac equation
stability
dynamics
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spelling Solitary Waves in the Nonlinear Dirac Equation at the Continuum Limit: Stability and DynamicsCuevas-Maraver, JesúsKevrekidis, Panayotis G.Saxena, AvadhCooper, FredMertens, Franz G.Solitonsnonlinear Dirac equationstabilitydynamicsIn the present work, we give a comparative summary of different recent contributions to the theme of the linear stability and nonlinear dynamics of solitary waves in the nonlinear Dirac equation in the form of the Gross-Neveu model. We indicate some of the key controversial statements in publications within the past few years and we attempt to address them to the best of our current understanding. The conclusion that we reach is that the solitary wave solution of the model is spectrally stable in the cubic nonlinearity case, however, it may become unstable through an instability amounting to the violation of the Vakhitov-Kolokolov criterion for higher exponents. We find that for the Dirac model, the interval of instability is narrower. A fundamental numerical finding of our work is that, contrary to what is the case in the nonlinear Schrodinger analogue of the model, the unstable dynamical evolution, does not lead to collapse (blowup) and hence it appears that the relativistic nature of the model mitigates the collapse instability. Various issues associated with different numerical schemes are highlighted and some possibilities for future alleviation of these is suggested.NovaFísica Aplicada I2015info:eu-repo/semantics/bookPartinfo:eu-repo/semantics/publishedVersionapplication/pdfapplication/pdfhttps://hdl.handle.net/11441/146913reponame:idUS. Depósito de Investigación de la Universidad de Sevillainstname:Universidad de Sevilla (US)InglésOrdinary and Partial Differential Equationshttps://novapublishers.com/shop/ordinary-and-partial-differential-equations/info:eu-repo/semantics/openAccessoai:idus.us.es:11441/1469132026-06-17T12:51:07Z
dc.title.none.fl_str_mv Solitary Waves in the Nonlinear Dirac Equation at the Continuum Limit: Stability and Dynamics
title Solitary Waves in the Nonlinear Dirac Equation at the Continuum Limit: Stability and Dynamics
spellingShingle Solitary Waves in the Nonlinear Dirac Equation at the Continuum Limit: Stability and Dynamics
Cuevas-Maraver, Jesús
Solitons
nonlinear Dirac equation
stability
dynamics
title_short Solitary Waves in the Nonlinear Dirac Equation at the Continuum Limit: Stability and Dynamics
title_full Solitary Waves in the Nonlinear Dirac Equation at the Continuum Limit: Stability and Dynamics
title_fullStr Solitary Waves in the Nonlinear Dirac Equation at the Continuum Limit: Stability and Dynamics
title_full_unstemmed Solitary Waves in the Nonlinear Dirac Equation at the Continuum Limit: Stability and Dynamics
title_sort Solitary Waves in the Nonlinear Dirac Equation at the Continuum Limit: Stability and Dynamics
dc.creator.none.fl_str_mv Cuevas-Maraver, Jesús
Kevrekidis, Panayotis G.
Saxena, Avadh
Cooper, Fred
Mertens, Franz G.
author Cuevas-Maraver, Jesús
author_facet Cuevas-Maraver, Jesús
Kevrekidis, Panayotis G.
Saxena, Avadh
Cooper, Fred
Mertens, Franz G.
author_role author
author2 Kevrekidis, Panayotis G.
Saxena, Avadh
Cooper, Fred
Mertens, Franz G.
author2_role author
author
author
author
dc.contributor.none.fl_str_mv Física Aplicada I
dc.subject.none.fl_str_mv Solitons
nonlinear Dirac equation
stability
dynamics
topic Solitons
nonlinear Dirac equation
stability
dynamics
description In the present work, we give a comparative summary of different recent contributions to the theme of the linear stability and nonlinear dynamics of solitary waves in the nonlinear Dirac equation in the form of the Gross-Neveu model. We indicate some of the key controversial statements in publications within the past few years and we attempt to address them to the best of our current understanding. The conclusion that we reach is that the solitary wave solution of the model is spectrally stable in the cubic nonlinearity case, however, it may become unstable through an instability amounting to the violation of the Vakhitov-Kolokolov criterion for higher exponents. We find that for the Dirac model, the interval of instability is narrower. A fundamental numerical finding of our work is that, contrary to what is the case in the nonlinear Schrodinger analogue of the model, the unstable dynamical evolution, does not lead to collapse (blowup) and hence it appears that the relativistic nature of the model mitigates the collapse instability. Various issues associated with different numerical schemes are highlighted and some possibilities for future alleviation of these is suggested.
publishDate 2015
dc.date.none.fl_str_mv 2015
dc.type.none.fl_str_mv info:eu-repo/semantics/bookPart
info:eu-repo/semantics/publishedVersion
format bookPart
status_str publishedVersion
dc.identifier.none.fl_str_mv https://hdl.handle.net/11441/146913
url https://hdl.handle.net/11441/146913
dc.language.none.fl_str_mv Inglés
language_invalid_str_mv Inglés
dc.relation.none.fl_str_mv Ordinary and Partial Differential Equations
https://novapublishers.com/shop/ordinary-and-partial-differential-equations/
dc.rights.none.fl_str_mv info:eu-repo/semantics/openAccess
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv application/pdf
application/pdf
dc.publisher.none.fl_str_mv Nova
publisher.none.fl_str_mv Nova
dc.source.none.fl_str_mv reponame:idUS. Depósito de Investigación de la Universidad de Sevilla
instname:Universidad de Sevilla (US)
instname_str Universidad de Sevilla (US)
reponame_str idUS. Depósito de Investigación de la Universidad de Sevilla
collection idUS. Depósito de Investigación de la Universidad de Sevilla
repository.name.fl_str_mv
repository.mail.fl_str_mv
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score 15.300719

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