On the collapse arresting effects of discreteness
We examine the effects of discreteness on a prototypical example of a collapse exhibiting partial differential equation (PDE). As our benchmark example, we select the discrete nonlinear Schrödinger (DNLS) equation. We provide a number of physical settings where issues of the interplay of collapse an...
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Veröffentlicht in: | Mathematics and computers in simulation 2005-08, Vol.69 (5), p.553-566 |
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creator | Tzirakis, N. Kevrekidis, P.G. |
description | We examine the effects of discreteness on a prototypical example of a collapse exhibiting partial differential equation (PDE). As our benchmark example, we select the discrete nonlinear Schrödinger (DNLS) equation. We provide a number of physical settings where issues of the interplay of collapse and discreteness may arise and focus on the quintic, one-dimensional DNLS. We justify that collapse in the sense of continuum limit (i.e., of the
L
∞
norm becoming infinite) cannot occur in the discrete setting. We support our qualitative arguments both with numerical simulations as well as with an analysis of a quasi-continuum, pseudo-differential approximation to the discrete model. Global well-posedness is proved for the latter problem in
H
s
, for
s
>
1
/
2
. While the collapse arresting nature of discreteness can be immediately realized, our estimates elucidate the “approach” towards the collapse-bearing continuum limit and the mechanism through which focusing arises in the latter. |
doi_str_mv | 10.1016/j.matcom.2005.03.013 |
format | Article |
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L
∞
norm becoming infinite) cannot occur in the discrete setting. We support our qualitative arguments both with numerical simulations as well as with an analysis of a quasi-continuum, pseudo-differential approximation to the discrete model. Global well-posedness is proved for the latter problem in
H
s
, for
s
>
1
/
2
. While the collapse arresting nature of discreteness can be immediately realized, our estimates elucidate the “approach” towards the collapse-bearing continuum limit and the mechanism through which focusing arises in the latter.</description><identifier>ISSN: 0378-4754</identifier><identifier>EISSN: 1872-7166</identifier><identifier>DOI: 10.1016/j.matcom.2005.03.013</identifier><language>eng</language><publisher>Elsevier B.V</publisher><subject>Collapse arrest ; Discreteness ; DNLS equation ; Well-posedness</subject><ispartof>Mathematics and computers in simulation, 2005-08, Vol.69 (5), p.553-566</ispartof><rights>2005 IMACS</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c368t-b686d6126d02d66bc3ee43bf0917b6324f1c8079edb3b94481574d8e192905973</citedby><cites>FETCH-LOGICAL-c368t-b686d6126d02d66bc3ee43bf0917b6324f1c8079edb3b94481574d8e192905973</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktohtml>$$Uhttps://dx.doi.org/10.1016/j.matcom.2005.03.013$$EHTML$$P50$$Gelsevier$$H</linktohtml><link.rule.ids>314,780,784,3550,27924,27925,45995</link.rule.ids></links><search><creatorcontrib>Tzirakis, N.</creatorcontrib><creatorcontrib>Kevrekidis, P.G.</creatorcontrib><title>On the collapse arresting effects of discreteness</title><title>Mathematics and computers in simulation</title><description>We examine the effects of discreteness on a prototypical example of a collapse exhibiting partial differential equation (PDE). As our benchmark example, we select the discrete nonlinear Schrödinger (DNLS) equation. We provide a number of physical settings where issues of the interplay of collapse and discreteness may arise and focus on the quintic, one-dimensional DNLS. We justify that collapse in the sense of continuum limit (i.e., of the
L
∞
norm becoming infinite) cannot occur in the discrete setting. We support our qualitative arguments both with numerical simulations as well as with an analysis of a quasi-continuum, pseudo-differential approximation to the discrete model. Global well-posedness is proved for the latter problem in
H
s
, for
s
>
1
/
2
. While the collapse arresting nature of discreteness can be immediately realized, our estimates elucidate the “approach” towards the collapse-bearing continuum limit and the mechanism through which focusing arises in the latter.</description><subject>Collapse arrest</subject><subject>Discreteness</subject><subject>DNLS equation</subject><subject>Well-posedness</subject><issn>0378-4754</issn><issn>1872-7166</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2005</creationdate><recordtype>article</recordtype><recordid>eNqNkM1OwzAQhC0EEqXwBhxy4pawGzu2c0FCFX9SpV7gbCX2BlI1cbFTJN4eV-GMOO0cZkazH2PXCAUCytttMTST9UNRAlQF8AKQn7AFalXmCqU8ZQvgSudCVeKcXcS4BYCkqwXDzZhNH5RZv9s1-0hZEwLFqR_fM-o6slPMfJe5PtpAE40U4yU765pdpKvfu2Rvjw-vq-d8vXl6Wd2vc8ulnvJWaukkltJB6aRsLScSvO2gRtVKXooOrQZVk2t5WwuhsVLCacK6rKGqFV-ym7l3H_znIW0yQ1pBaeZI_hBNqYWsNBf_MIIERExGMRtt8DEG6sw-9EMTvg2COYI0WzODNEeQBrhJIFPsbo5R-varp2Ci7Wm05PqQABnn-78LfgDA1XxC</recordid><startdate>20050801</startdate><enddate>20050801</enddate><creator>Tzirakis, N.</creator><creator>Kevrekidis, P.G.</creator><general>Elsevier B.V</general><scope>AAYXX</scope><scope>CITATION</scope><scope>7SC</scope><scope>8FD</scope><scope>JQ2</scope><scope>L7M</scope><scope>L~C</scope><scope>L~D</scope><scope>H8D</scope></search><sort><creationdate>20050801</creationdate><title>On the collapse arresting effects of discreteness</title><author>Tzirakis, N. ; Kevrekidis, P.G.</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c368t-b686d6126d02d66bc3ee43bf0917b6324f1c8079edb3b94481574d8e192905973</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2005</creationdate><topic>Collapse arrest</topic><topic>Discreteness</topic><topic>DNLS equation</topic><topic>Well-posedness</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Tzirakis, N.</creatorcontrib><creatorcontrib>Kevrekidis, P.G.</creatorcontrib><collection>CrossRef</collection><collection>Computer and Information Systems Abstracts</collection><collection>Technology Research Database</collection><collection>ProQuest Computer Science Collection</collection><collection>Advanced Technologies Database with Aerospace</collection><collection>Computer and Information Systems Abstracts Academic</collection><collection>Computer and Information Systems Abstracts Professional</collection><collection>Aerospace Database</collection><jtitle>Mathematics and computers in simulation</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Tzirakis, N.</au><au>Kevrekidis, P.G.</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>On the collapse arresting effects of discreteness</atitle><jtitle>Mathematics and computers in simulation</jtitle><date>2005-08-01</date><risdate>2005</risdate><volume>69</volume><issue>5</issue><spage>553</spage><epage>566</epage><pages>553-566</pages><issn>0378-4754</issn><eissn>1872-7166</eissn><abstract>We examine the effects of discreteness on a prototypical example of a collapse exhibiting partial differential equation (PDE). As our benchmark example, we select the discrete nonlinear Schrödinger (DNLS) equation. We provide a number of physical settings where issues of the interplay of collapse and discreteness may arise and focus on the quintic, one-dimensional DNLS. We justify that collapse in the sense of continuum limit (i.e., of the
L
∞
norm becoming infinite) cannot occur in the discrete setting. We support our qualitative arguments both with numerical simulations as well as with an analysis of a quasi-continuum, pseudo-differential approximation to the discrete model. Global well-posedness is proved for the latter problem in
H
s
, for
s
>
1
/
2
. While the collapse arresting nature of discreteness can be immediately realized, our estimates elucidate the “approach” towards the collapse-bearing continuum limit and the mechanism through which focusing arises in the latter.</abstract><pub>Elsevier B.V</pub><doi>10.1016/j.matcom.2005.03.013</doi><tpages>14</tpages></addata></record> |
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subjects | Collapse arrest Discreteness DNLS equation Well-posedness |
title | On the collapse arresting effects of discreteness |
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