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Mathematically, the limit enables a novel weakly nonlinear analysis to be performed which allows the local depth of the granular layer to vary by $O(1)$ amounts even when the pattern amplitude is small. The weakly nonlinear analysis and numerical computations provide a robust possible explanation of past experimental results.<\/jats:p>","DOI":"10.1137\/090762865","type":"journal-article","created":{"date-parts":[[2010,3,12]],"date-time":"2010-03-12T18:21:00Z","timestamp":1268418060000},"page":"238-260","source":"Crossref","is-referenced-by-count":19,"title":["Localized States in a Model of Pattern Formation in a Vertically Vibrated Layer"],"prefix":"10.1137","volume":"9","author":[{"given":"J. H. P.","family":"Dawes","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"S.","family":"Lilley","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2010,3,12]]},"reference":[{"key":"R1","doi-asserted-by":"publisher","DOI":"10.1103\/RevModPhys.78.641"},{"key":"R2","doi-asserted-by":"publisher","DOI":"10.1137\/080713306"},{"key":"R3","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevA.38.5461"},{"key":"R4","doi-asserted-by":"crossref","unstructured":"A. Bergeon, J. Burke, E. Knobloch, and I. Mercader,\n                      Eckhaus instability and homoclinic snaking\n                      , Phys. Rev. E, 78 (2008), paper 046201.","DOI":"10.1103\/PhysRevE.78.046201"},{"key":"R5","doi-asserted-by":"publisher","DOI":"10.1137\/070698191"},{"key":"R6","doi-asserted-by":"crossref","unstructured":"J. Burke and E. 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Oldeman, R. Paffenroth, B. Sandstede, X. Wang, and C. Zhang,\n                      AUTO-\n                      07\n                      p: Continuation and Bifurcation Software for Ordinary Differential Equations\n                      , 2007; software available to download from http:\/\/indy.cs.concordia.ca\/auto\/ http:\/\/indy.cs.concordia.ca\/auto\/."},{"key":"R17","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.59.4476"},{"key":"R18","doi-asserted-by":"crossref","unstructured":"W. Firth, L. Columbo, and A. J. Scroggie,\n                      Proposed resolution of theory-experiment discrepancy in homoclinic snaking\n                      , Phys. Rev. Lett., 99 (2007), paper 104503.","DOI":"10.1103\/PhysRevLett.99.104503"},{"key":"R19","unstructured":"I. S. Gradshteyn and I. M. Ryzhik,\n                      Table of Integrals, Series and Products\n                      , 6th ed., A. Jeffrey and D. 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