Similarity solutions to evolution equations in one-dimensional interfaces
In this note, we study the evolution equation $$ partial_t h = -upartial^2_x h-Kpartial^4_x h +lambda_1(partial_x h)^2-lambda_2partial^2_x(partial_x h)^2. $$ which was introduced by Munoz-Garcia [9] in the context of erosion by ion beam sputtering. We obtain an analytic solution that has the...
| Main Authors: | , , , |
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| Format: | Article |
| Language: | English |
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Texas State University
2011-05-01
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| Series: | Electronic Journal of Differential Equations |
| Subjects: | |
| Online Access: | http://ejde.math.txstate.edu/Volumes/2011/68/abstr.html |
| _version_ | 1811302937592332288 |
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| author | Mohammed Benlahsen Ayman Eldoussouki Mohammed Guedda Mustapha Jazar |
| author_facet | Mohammed Benlahsen Ayman Eldoussouki Mohammed Guedda Mustapha Jazar |
| author_sort | Mohammed Benlahsen |
| collection | DOAJ |
| description | In this note, we study the evolution equation $$ partial_t h = -upartial^2_x h-Kpartial^4_x h +lambda_1(partial_x h)^2-lambda_2partial^2_x(partial_x h)^2. $$ which was introduced by Munoz-Garcia [9] in the context of erosion by ion beam sputtering. We obtain an analytic solution that has the similarity form, which is used in obtaining the coarsening behavior. This solution has amplitude and wavelength that increase like $ln(t)$ and $ sqrt{tln(t)}$, respectively. |
| first_indexed | 2024-04-13T07:39:25Z |
| format | Article |
| id | doaj.art-a5fa9af2f2b64da2b8222b123bf13a10 |
| institution | Directory Open Access Journal |
| issn | 1072-6691 |
| language | English |
| last_indexed | 2024-04-13T07:39:25Z |
| publishDate | 2011-05-01 |
| publisher | Texas State University |
| record_format | Article |
| series | Electronic Journal of Differential Equations |
| spelling | doaj.art-a5fa9af2f2b64da2b8222b123bf13a102022-12-22T02:55:59ZengTexas State UniversityElectronic Journal of Differential Equations1072-66912011-05-01201168,15Similarity solutions to evolution equations in one-dimensional interfacesMohammed BenlahsenAyman EldoussoukiMohammed GueddaMustapha JazarIn this note, we study the evolution equation $$ partial_t h = -upartial^2_x h-Kpartial^4_x h +lambda_1(partial_x h)^2-lambda_2partial^2_x(partial_x h)^2. $$ which was introduced by Munoz-Garcia [9] in the context of erosion by ion beam sputtering. We obtain an analytic solution that has the similarity form, which is used in obtaining the coarsening behavior. This solution has amplitude and wavelength that increase like $ln(t)$ and $ sqrt{tln(t)}$, respectively.http://ejde.math.txstate.edu/Volumes/2011/68/abstr.htmlNonlinear dynamicinstabilitysimilarity solutioncoarsening |
| spellingShingle | Mohammed Benlahsen Ayman Eldoussouki Mohammed Guedda Mustapha Jazar Similarity solutions to evolution equations in one-dimensional interfaces Electronic Journal of Differential Equations Nonlinear dynamic instability similarity solution coarsening |
| title | Similarity solutions to evolution equations in one-dimensional interfaces |
| title_full | Similarity solutions to evolution equations in one-dimensional interfaces |
| title_fullStr | Similarity solutions to evolution equations in one-dimensional interfaces |
| title_full_unstemmed | Similarity solutions to evolution equations in one-dimensional interfaces |
| title_short | Similarity solutions to evolution equations in one-dimensional interfaces |
| title_sort | similarity solutions to evolution equations in one dimensional interfaces |
| topic | Nonlinear dynamic instability similarity solution coarsening |
| url | http://ejde.math.txstate.edu/Volumes/2011/68/abstr.html |
| work_keys_str_mv | AT mohammedbenlahsen similaritysolutionstoevolutionequationsinonedimensionalinterfaces AT aymaneldoussouki similaritysolutionstoevolutionequationsinonedimensionalinterfaces AT mohammedguedda similaritysolutionstoevolutionequationsinonedimensionalinterfaces AT mustaphajazar similaritysolutionstoevolutionequationsinonedimensionalinterfaces |