Threshold solutions for the nonlinear Schrödinger equation
Autor(a) principal: | |
---|---|
Data de Publicação: | 2022 |
Outros Autores: | , |
Tipo de documento: | Artigo |
Idioma: | por |
Título da fonte: | Repositório Institucional da UFMG |
Texto Completo: | http://hdl.handle.net/1843/60391 https://orcid.org/0000-0002-7041-6031 https://orcid.org/0000-0003-1034-3480 https://orcid.org/0000-0002-7407-7639 |
Resumo: | We study the focusing NLS equation in Rᴺ in the mass-supercritical and energy-subcritical (or intercritical) regime, with H1 data at the mass-energy threshold ME.(u0)=ME(Q), where Q is the ground state. Previously, Duyckaerts–Merle studied the behavior of threshold solutions in the H¹-critical case, in dimensions N=3; 4; 5, later generalized by Li–Zhang for higher dimensions. In the intercritical case, Duyckaerts–Roudenko studied the threshold problem for the 3d cubic NLS equation. In this paper, we generalize the results of Duyckaerts–Roudenko for any dimension and any power of the nonlinearity for the entire intercritical range. We show the existence of special solutions, Q±, besides the standing wave eᶦᵗQ, which exponentially approach the standing wave in the positive time direction, but differ in its behavior for negative time. We classify solutions at the threshold level, showing either blowup occurs in finite (positive and negative) time, or scattering in both time directions, or the solution is equal to one of the three special solutions above, up to symmetries. Our proof extends to the H¹-critical case, thus, giving an alternative proof of the Li–Zhang result and unifying the critical and intercritical cases. These results are obtained by studying the linearized equation around the standing wave and some tailored approximate solutions to the NLS equation. We establish important decay properties of functions associated to the spectrum of the linearized Schrödinger operator, which, in combination with modulational stability and coercivity for the linearized operator on special subspaces, allows us to use a fixed-point argument to show the existence of special solutions. Finally, we prove the uniqueness by studying exponentially decaying solutions to a sequence of linearized equations. |
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2023-10-31T21:32:08Z2023-10-31T21:32:08Z20223851637170810.4171/RMI/13372235-0616http://hdl.handle.net/1843/60391https://orcid.org/0000-0002-7041-6031https://orcid.org/0000-0003-1034-3480https://orcid.org/0000-0002-7407-7639We study the focusing NLS equation in Rᴺ in the mass-supercritical and energy-subcritical (or intercritical) regime, with H1 data at the mass-energy threshold ME.(u0)=ME(Q), where Q is the ground state. Previously, Duyckaerts–Merle studied the behavior of threshold solutions in the H¹-critical case, in dimensions N=3; 4; 5, later generalized by Li–Zhang for higher dimensions. In the intercritical case, Duyckaerts–Roudenko studied the threshold problem for the 3d cubic NLS equation. In this paper, we generalize the results of Duyckaerts–Roudenko for any dimension and any power of the nonlinearity for the entire intercritical range. We show the existence of special solutions, Q±, besides the standing wave eᶦᵗQ, which exponentially approach the standing wave in the positive time direction, but differ in its behavior for negative time. We classify solutions at the threshold level, showing either blowup occurs in finite (positive and negative) time, or scattering in both time directions, or the solution is equal to one of the three special solutions above, up to symmetries. Our proof extends to the H¹-critical case, thus, giving an alternative proof of the Li–Zhang result and unifying the critical and intercritical cases. These results are obtained by studying the linearized equation around the standing wave and some tailored approximate solutions to the NLS equation. We establish important decay properties of functions associated to the spectrum of the linearized Schrödinger operator, which, in combination with modulational stability and coercivity for the linearized operator on special subspaces, allows us to use a fixed-point argument to show the existence of special solutions. Finally, we prove the uniqueness by studying exponentially decaying solutions to a sequence of linearized equations.Estudamos a equação NLS focalizada em Rᴺ no regime massa supercrítica e energia subcrítica (ou intercrítica), com dados H1 no limiar massa-energia ME.(u0)=ME(Q), onde Q é o estado fundamental. Anteriormente, Duyckaerts–Merle estudou o comportamento de soluções limiares no caso H¹-crítico, nas dimensões N=3; 4; 5, posteriormente generalizado por Li – Zhang para dimensões superiores. No caso intercrítico, Duyckaerts – Roudenko estudou o problema de limite para a equação NLS cúbica 3d. Neste artigo, generalizamos os resultados de Duyckaerts–Roudenko para qualquer dimensão e qualquer potência da não linearidade para todo o intervalo intercrítico. Mostramos a existência de soluções especiais, Q±, além da onda estacionária eᶦᵗQ, que se aproximam exponencialmente da onda estacionária na direção positiva do tempo, mas diferem em seu comportamento para o tempo negativo. Classificamos as soluções no nível de limiar, mostrando que a explosão ocorre em tempo finito (positivo e negativo), ou espalhamento em ambas as direções do tempo, ou que a solução é igual a uma das três soluções especiais acima, até simetrias. Nossa prova se estende ao caso H¹-crítico, dando assim uma prova alternativa do resultado de Li–Zhang e unificando os casos críticos e intercríticos. Esses resultados são obtidos estudando a equação linearizada em torno da onda estacionária e algumas soluções aproximadas personalizadas para a equação NLS. Estabelecemos importantes propriedades de decaimento de funções associadas ao espectro do operador linearizado de Schrödinger, que, em combinação com estabilidade modulacional e coercividade para o operador linearizado em subespaços especiais, nos permite usar um argumento de ponto fixo para mostrar a existência de soluções especiais . Finalmente, provamos a unicidade estudando soluções com decaimento exponencial para uma sequência de equações linearizadas.CNPq - Conselho Nacional de Desenvolvimento Científico e TecnológicoFAPEMIG - Fundação de Amparo à Pesquisa do Estado de Minas GeraisCAPES - Coordenação de Aperfeiçoamento de Pessoal de Nível SuperiorFAPESP - Fundação de Amparo à Pesquisa do Estado de São PauloporUniversidade Federal de Minas GeraisUFMGBrasilICX - DEPARTAMENTO DE MATEMÁTICARevista matemática iberoamericanaSchrodinger, Equação de.Equações diferenciais parciaisMatemáticaThreshold solutions for the nonlinear Schrödinger equationSoluções limite para a equação não linear de Schrödingerinfo:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/articlehttps://ems.press/journals/rmi/articles/4552493Luccas CamposLuiz Gustavo Farah DiasSvetlana Roudenkoapplication/pdfinfo:eu-repo/semantics/openAccessreponame:Repositório Institucional da UFMGinstname:Universidade Federal de Minas Gerais (UFMG)instacron:UFMGLICENSELicense.txtLicense.txttext/plain; charset=utf-82042https://repositorio.ufmg.br/bitstream/1843/60391/1/License.txtfa505098d172de0bc8864fc1287ffe22MD51ORIGINALThreshold solutions for the nonlinear Schrödinger equation.pdfThreshold solutions for the nonlinear Schrödinger equation.pdfapplication/pdf755645https://repositorio.ufmg.br/bitstream/1843/60391/2/Threshold%20solutions%20for%20the%20nonlinear%20Schr%c3%b6dinger%20equation.pdf00f8ab06cacb1316366aa911a928625cMD521843/603912023-10-31 18:32:08.687oai:repositorio.ufmg.br:1843/60391TElDRU7vv71BIERFIERJU1RSSUJVSe+/ve+/vU8gTu+/vU8tRVhDTFVTSVZBIERPIFJFUE9TSVTvv71SSU8gSU5TVElUVUNJT05BTCBEQSBVRk1HCiAKCkNvbSBhIGFwcmVzZW50Ye+/ve+/vW8gZGVzdGEgbGljZW7vv71hLCB2b2Pvv70gKG8gYXV0b3IgKGVzKSBvdSBvIHRpdHVsYXIgZG9zIGRpcmVpdG9zIGRlIGF1dG9yKSBjb25jZWRlIGFvIFJlcG9zaXTvv71yaW8gSW5zdGl0dWNpb25hbCBkYSBVRk1HIChSSS1VRk1HKSBvIGRpcmVpdG8gbu+/vW8gZXhjbHVzaXZvIGUgaXJyZXZvZ++/vXZlbCBkZSByZXByb2R1emlyIGUvb3UgZGlzdHJpYnVpciBhIHN1YSBwdWJsaWNh77+977+9byAoaW5jbHVpbmRvIG8gcmVzdW1vKSBwb3IgdG9kbyBvIG11bmRvIG5vIGZvcm1hdG8gaW1wcmVzc28gZSBlbGV0cu+/vW5pY28gZSBlbSBxdWFscXVlciBtZWlvLCBpbmNsdWluZG8gb3MgZm9ybWF0b3Mg77+9dWRpbyBvdSB277+9ZGVvLgoKVm9j77+9IGRlY2xhcmEgcXVlIGNvbmhlY2UgYSBwb2zvv710aWNhIGRlIGNvcHlyaWdodCBkYSBlZGl0b3JhIGRvIHNldSBkb2N1bWVudG8gZSBxdWUgY29uaGVjZSBlIGFjZWl0YSBhcyBEaXJldHJpemVzIGRvIFJJLVVGTUcuCgpWb2Pvv70gY29uY29yZGEgcXVlIG8gUmVwb3NpdO+/vXJpbyBJbnN0aXR1Y2lvbmFsIGRhIFVGTUcgcG9kZSwgc2VtIGFsdGVyYXIgbyBjb250Ze+/vWRvLCB0cmFuc3BvciBhIHN1YSBwdWJsaWNh77+977+9byBwYXJhIHF1YWxxdWVyIG1laW8gb3UgZm9ybWF0byBwYXJhIGZpbnMgZGUgcHJlc2VydmHvv73vv71vLgoKVm9j77+9IHRhbWLvv71tIGNvbmNvcmRhIHF1ZSBvIFJlcG9zaXTvv71yaW8gSW5zdGl0dWNpb25hbCBkYSBVRk1HIHBvZGUgbWFudGVyIG1haXMgZGUgdW1hIGPvv71waWEgZGUgc3VhIHB1YmxpY2Hvv73vv71vIHBhcmEgZmlucyBkZSBzZWd1cmFu77+9YSwgYmFjay11cCBlIHByZXNlcnZh77+977+9by4KClZvY++/vSBkZWNsYXJhIHF1ZSBhIHN1YSBwdWJsaWNh77+977+9byDvv70gb3JpZ2luYWwgZSBxdWUgdm9j77+9IHRlbSBvIHBvZGVyIGRlIGNvbmNlZGVyIG9zIGRpcmVpdG9zIGNvbnRpZG9zIG5lc3RhIGxpY2Vu77+9YS4gVm9j77+9IHRhbWLvv71tIGRlY2xhcmEgcXVlIG8gZGVw77+9c2l0byBkZSBzdWEgcHVibGljYe+/ve+/vW8gbu+/vW8sIHF1ZSBzZWphIGRlIHNldSBjb25oZWNpbWVudG8sIGluZnJpbmdlIGRpcmVpdG9zIGF1dG9yYWlzIGRlIG5pbmd177+9bS4KCkNhc28gYSBzdWEgcHVibGljYe+/ve+/vW8gY29udGVuaGEgbWF0ZXJpYWwgcXVlIHZvY++/vSBu77+9byBwb3NzdWkgYSB0aXR1bGFyaWRhZGUgZG9zIGRpcmVpdG9zIGF1dG9yYWlzLCB2b2Pvv70gZGVjbGFyYSBxdWUgb2J0ZXZlIGEgcGVybWlzc++/vW8gaXJyZXN0cml0YSBkbyBkZXRlbnRvciBkb3MgZGlyZWl0b3MgYXV0b3JhaXMgcGFyYSBjb25jZWRlciBhbyBSZXBvc2l077+9cmlvIEluc3RpdHVjaW9uYWwgZGEgVUZNRyBvcyBkaXJlaXRvcyBhcHJlc2VudGFkb3MgbmVzdGEgbGljZW7vv71hLCBlIHF1ZSBlc3NlIG1hdGVyaWFsIGRlIHByb3ByaWVkYWRlIGRlIHRlcmNlaXJvcyBlc3Tvv70gY2xhcmFtZW50ZSBpZGVudGlmaWNhZG8gZSByZWNvbmhlY2lkbyBubyB0ZXh0byBvdSBubyBjb250Ze+/vWRvIGRhIHB1YmxpY2Hvv73vv71vIG9yYSBkZXBvc2l0YWRhLgoKQ0FTTyBBIFBVQkxJQ0Hvv73vv71PIE9SQSBERVBPU0lUQURBIFRFTkhBIFNJRE8gUkVTVUxUQURPIERFIFVNIFBBVFJPQ++/vU5JTyBPVSBBUE9JTyBERSBVTUEgQUfvv71OQ0lBIERFIEZPTUVOVE8gT1UgT1VUUk8gT1JHQU5JU01PLCBWT0Pvv70gREVDTEFSQSBRVUUgUkVTUEVJVE9VIFRPRE9TIEUgUVVBSVNRVUVSIERJUkVJVE9TIERFIFJFVklT77+9TyBDT01PIFRBTULvv71NIEFTIERFTUFJUyBPQlJJR0Hvv73vv71FUyBFWElHSURBUyBQT1IgQ09OVFJBVE8gT1UgQUNPUkRPLgoKTyBSZXBvc2l077+9cmlvIEluc3RpdHVjaW9uYWwgZGEgVUZNRyBzZSBjb21wcm9tZXRlIGEgaWRlbnRpZmljYXIgY2xhcmFtZW50ZSBvIHNldSBub21lKHMpIG91IG8ocykgbm9tZXMocykgZG8ocykgZGV0ZW50b3IoZXMpIGRvcyBkaXJlaXRvcyBhdXRvcmFpcyBkYSBwdWJsaWNh77+977+9bywgZSBu77+9byBmYXLvv70gcXVhbHF1ZXIgYWx0ZXJh77+977+9bywgYWzvv71tIGRhcXVlbGFzIGNvbmNlZGlkYXMgcG9yIGVzdGEgbGljZW7vv71hLgo=Repositório de PublicaçõesPUBhttps://repositorio.ufmg.br/oaiopendoar:2023-10-31T21:32:08Repositório Institucional da UFMG - Universidade Federal de Minas Gerais (UFMG)false |
dc.title.pt_BR.fl_str_mv |
Threshold solutions for the nonlinear Schrödinger equation |
dc.title.alternative.pt_BR.fl_str_mv |
Soluções limite para a equação não linear de Schrödinger |
title |
Threshold solutions for the nonlinear Schrödinger equation |
spellingShingle |
Threshold solutions for the nonlinear Schrödinger equation Luccas Campos Schrodinger, Equação de. Equações diferenciais parciais Matemática |
title_short |
Threshold solutions for the nonlinear Schrödinger equation |
title_full |
Threshold solutions for the nonlinear Schrödinger equation |
title_fullStr |
Threshold solutions for the nonlinear Schrödinger equation |
title_full_unstemmed |
Threshold solutions for the nonlinear Schrödinger equation |
title_sort |
Threshold solutions for the nonlinear Schrödinger equation |
author |
Luccas Campos |
author_facet |
Luccas Campos Luiz Gustavo Farah Dias Svetlana Roudenko |
author_role |
author |
author2 |
Luiz Gustavo Farah Dias Svetlana Roudenko |
author2_role |
author author |
dc.contributor.author.fl_str_mv |
Luccas Campos Luiz Gustavo Farah Dias Svetlana Roudenko |
dc.subject.other.pt_BR.fl_str_mv |
Schrodinger, Equação de. Equações diferenciais parciais Matemática |
topic |
Schrodinger, Equação de. Equações diferenciais parciais Matemática |
description |
We study the focusing NLS equation in Rᴺ in the mass-supercritical and energy-subcritical (or intercritical) regime, with H1 data at the mass-energy threshold ME.(u0)=ME(Q), where Q is the ground state. Previously, Duyckaerts–Merle studied the behavior of threshold solutions in the H¹-critical case, in dimensions N=3; 4; 5, later generalized by Li–Zhang for higher dimensions. In the intercritical case, Duyckaerts–Roudenko studied the threshold problem for the 3d cubic NLS equation. In this paper, we generalize the results of Duyckaerts–Roudenko for any dimension and any power of the nonlinearity for the entire intercritical range. We show the existence of special solutions, Q±, besides the standing wave eᶦᵗQ, which exponentially approach the standing wave in the positive time direction, but differ in its behavior for negative time. We classify solutions at the threshold level, showing either blowup occurs in finite (positive and negative) time, or scattering in both time directions, or the solution is equal to one of the three special solutions above, up to symmetries. Our proof extends to the H¹-critical case, thus, giving an alternative proof of the Li–Zhang result and unifying the critical and intercritical cases. These results are obtained by studying the linearized equation around the standing wave and some tailored approximate solutions to the NLS equation. We establish important decay properties of functions associated to the spectrum of the linearized Schrödinger operator, which, in combination with modulational stability and coercivity for the linearized operator on special subspaces, allows us to use a fixed-point argument to show the existence of special solutions. Finally, we prove the uniqueness by studying exponentially decaying solutions to a sequence of linearized equations. |
publishDate |
2022 |
dc.date.issued.fl_str_mv |
2022 |
dc.date.accessioned.fl_str_mv |
2023-10-31T21:32:08Z |
dc.date.available.fl_str_mv |
2023-10-31T21:32:08Z |
dc.type.status.fl_str_mv |
info:eu-repo/semantics/publishedVersion |
dc.type.driver.fl_str_mv |
info:eu-repo/semantics/article |
format |
article |
status_str |
publishedVersion |
dc.identifier.uri.fl_str_mv |
http://hdl.handle.net/1843/60391 |
dc.identifier.doi.pt_BR.fl_str_mv |
10.4171/RMI/1337 |
dc.identifier.issn.pt_BR.fl_str_mv |
2235-0616 |
dc.identifier.orcid.pt_BR.fl_str_mv |
https://orcid.org/0000-0002-7041-6031 https://orcid.org/0000-0003-1034-3480 https://orcid.org/0000-0002-7407-7639 |
identifier_str_mv |
10.4171/RMI/1337 2235-0616 |
url |
http://hdl.handle.net/1843/60391 https://orcid.org/0000-0002-7041-6031 https://orcid.org/0000-0003-1034-3480 https://orcid.org/0000-0002-7407-7639 |
dc.language.iso.fl_str_mv |
por |
language |
por |
dc.relation.ispartof.pt_BR.fl_str_mv |
Revista matemática iberoamericana |
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info:eu-repo/semantics/openAccess |
eu_rights_str_mv |
openAccess |
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application/pdf |
dc.publisher.none.fl_str_mv |
Universidade Federal de Minas Gerais |
dc.publisher.initials.fl_str_mv |
UFMG |
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Brasil |
dc.publisher.department.fl_str_mv |
ICX - DEPARTAMENTO DE MATEMÁTICA |
publisher.none.fl_str_mv |
Universidade Federal de Minas Gerais |
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reponame:Repositório Institucional da UFMG instname:Universidade Federal de Minas Gerais (UFMG) instacron:UFMG |
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