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Justification of the NLS Approximation for the Euler–Poisson Equation
The nonlinear Schrödinger (NLS) equation can be derived as a formal approximation equation describing the envelopes of slowly modulated spatially and temporarily oscillating wave packet-like solutions to the ion Euler–Poisson equation. In this paper, we rigorously justify such approximation by givin...
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Published in: | Communications in mathematical physics 2019-10, Vol.371 (2), p.357-398 |
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description | The nonlinear Schrödinger (NLS) equation can be derived as a formal approximation equation describing the envelopes of slowly modulated spatially and temporarily oscillating wave packet-like solutions to the ion Euler–Poisson equation. In this paper, we rigorously justify such approximation by giving error estimates in Sobolev norms between exact solutions of the ion Euler–Poisson system and the formal approximation obtained via the NLS equation. The justification consists of several difficulties such as the resonances and loss of regularity, due to the quasilinearity of the problem. These difficulties are overcome by introducing normal form transformation and cutoff functions and carefully constructed energy functional of the equation. |
doi_str_mv | 10.1007/s00220-019-03576-4 |
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Math. Phys</addtitle><description>The nonlinear Schrödinger (NLS) equation can be derived as a formal approximation equation describing the envelopes of slowly modulated spatially and temporarily oscillating wave packet-like solutions to the ion Euler–Poisson equation. In this paper, we rigorously justify such approximation by giving error estimates in Sobolev norms between exact solutions of the ion Euler–Poisson system and the formal approximation obtained via the NLS equation. The justification consists of several difficulties such as the resonances and loss of regularity, due to the quasilinearity of the problem. These difficulties are overcome by introducing normal form transformation and cutoff functions and carefully constructed energy functional of the equation.</description><subject>Approximation</subject><subject>Canonical forms</subject><subject>Classical and Quantum Gravitation</subject><subject>Complex Systems</subject><subject>Functionals</subject><subject>Mathematical analysis</subject><subject>Mathematical and Computational Physics</subject><subject>Mathematical Physics</subject><subject>Nonlinearity</subject><subject>Norms</subject><subject>Physics</subject><subject>Physics and Astronomy</subject><subject>Poisson equation</subject><subject>Quantum Physics</subject><subject>Relativity Theory</subject><subject>Theoretical</subject><issn>0010-3616</issn><issn>1432-0916</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2019</creationdate><recordtype>article</recordtype><recordid>eNp9kMFKAzEQhoMoWKsv4GnBc3Qm2U2aYym1VYoK6jms24luqU2b7ILefAff0Ccx7grePA3MfP_MPz9jpwjnCKAvIoAQwAENB1loxfM9NsBcCg4G1T4bACBwqVAdsqMYVwBghFIDNrtuY1O7uiqb2m8y77LmhbKbxX023m6Df6tf-4HzoZtM2zWFr4_PO1_HmPrTXdsBx-zAletIJ791yB4vpw-TOV_czq4m4wWvJJqGa5S5hkItc6cLEiVQoQmrkTKlGcmy0q6oiOgpT67NEsVIL7UhVKBMKs7IITvr9yZzu5ZiY1e-DZt00gqZKNQai0SJnqqCjzGQs9uQPgnvFsH-BGb7wGwKzHaB2TyJZC-KCd48U_hb_Y_qG70SbdY</recordid><startdate>20191001</startdate><enddate>20191001</enddate><creator>Liu, Huimin</creator><creator>Pu, Xueke</creator><general>Springer Berlin Heidelberg</general><general>Springer Nature B.V</general><scope>AAYXX</scope><scope>CITATION</scope></search><sort><creationdate>20191001</creationdate><title>Justification of the NLS Approximation for the Euler–Poisson Equation</title><author>Liu, Huimin ; Pu, Xueke</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c319t-71347056d4f75e2a0e57e1c869a983ac7f5ceeeb49169d1287d79e160699e1f93</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2019</creationdate><topic>Approximation</topic><topic>Canonical forms</topic><topic>Classical and Quantum Gravitation</topic><topic>Complex Systems</topic><topic>Functionals</topic><topic>Mathematical analysis</topic><topic>Mathematical and Computational Physics</topic><topic>Mathematical Physics</topic><topic>Nonlinearity</topic><topic>Norms</topic><topic>Physics</topic><topic>Physics and Astronomy</topic><topic>Poisson equation</topic><topic>Quantum Physics</topic><topic>Relativity Theory</topic><topic>Theoretical</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Liu, Huimin</creatorcontrib><creatorcontrib>Pu, Xueke</creatorcontrib><collection>CrossRef</collection><jtitle>Communications in mathematical physics</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Liu, Huimin</au><au>Pu, Xueke</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Justification of the NLS Approximation for the Euler–Poisson Equation</atitle><jtitle>Communications in mathematical physics</jtitle><stitle>Commun. Math. Phys</stitle><date>2019-10-01</date><risdate>2019</risdate><volume>371</volume><issue>2</issue><spage>357</spage><epage>398</epage><pages>357-398</pages><issn>0010-3616</issn><eissn>1432-0916</eissn><abstract>The nonlinear Schrödinger (NLS) equation can be derived as a formal approximation equation describing the envelopes of slowly modulated spatially and temporarily oscillating wave packet-like solutions to the ion Euler–Poisson equation. In this paper, we rigorously justify such approximation by giving error estimates in Sobolev norms between exact solutions of the ion Euler–Poisson system and the formal approximation obtained via the NLS equation. The justification consists of several difficulties such as the resonances and loss of regularity, due to the quasilinearity of the problem. These difficulties are overcome by introducing normal form transformation and cutoff functions and carefully constructed energy functional of the equation.</abstract><cop>Berlin/Heidelberg</cop><pub>Springer Berlin Heidelberg</pub><doi>10.1007/s00220-019-03576-4</doi><tpages>42</tpages></addata></record> |
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subjects | Approximation Canonical forms Classical and Quantum Gravitation Complex Systems Functionals Mathematical analysis Mathematical and Computational Physics Mathematical Physics Nonlinearity Norms Physics Physics and Astronomy Poisson equation Quantum Physics Relativity Theory Theoretical |
title | Justification of the NLS Approximation for the Euler–Poisson Equation |
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