Exponential Asymptotics for Line Solitons in Two-Dimensional Periodic Potentials
As a first step toward a fully two-dimensional asymptotic theory for the bifurcation of solitons from infinitesimal continuous waves, an analytical theory is presented for line solitons, whose envelope varies only along one direction, in general two-dimensional periodic potentials. For this two-dime...
| Main Authors: | , , |
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| Other Authors: | |
| Format: | Article |
| Language: | en_US |
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Wiley Blackwell
2015
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| Online Access: | http://hdl.handle.net/1721.1/97216 https://orcid.org/0000-0002-5246-4574 |
| _version_ | 1826198910110007296 |
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| author | Nixon, Sean D. Yang, Jianke Akylas, Triantaphyllos R. |
| author2 | Massachusetts Institute of Technology. Department of Mechanical Engineering |
| author_facet | Massachusetts Institute of Technology. Department of Mechanical Engineering Nixon, Sean D. Yang, Jianke Akylas, Triantaphyllos R. |
| author_sort | Nixon, Sean D. |
| collection | MIT |
| description | As a first step toward a fully two-dimensional asymptotic theory for the bifurcation of solitons from infinitesimal continuous waves, an analytical theory is presented for line solitons, whose envelope varies only along one direction, in general two-dimensional periodic potentials. For this two-dimensional problem, it is no longer viable to rely on a certain recurrence relation for going beyond all orders of the usual multiscale perturbation expansion, a key step of the exponential asymptotics procedure previously used for solitons in one-dimensional problems. Instead, we propose a more direct treatment which not only overcomes the recurrence-relation limitation, but also simplifies the exponential asymptotics process. Using this modified technique, we show that line solitons with any rational line slopes bifurcate out from every Bloch-band edge; and for each rational slope, two line-soliton families exist. Furthermore, line solitons can bifurcate from interior points of Bloch bands as well, but such line solitons exist only for a couple of special line angles due to resonance with the Bloch bands. In addition, we show that a countable set of multiline-soliton bound states can be constructed analytically. The analytical predictions are compared with numerical results for both symmetric and asymmetric potentials, and good agreement is obtained. |
| first_indexed | 2024-09-23T11:11:52Z |
| format | Article |
| id | mit-1721.1/97216 |
| institution | Massachusetts Institute of Technology |
| language | en_US |
| last_indexed | 2024-09-23T11:11:52Z |
| publishDate | 2015 |
| publisher | Wiley Blackwell |
| record_format | dspace |
| spelling | mit-1721.1/972162022-10-01T01:56:51Z Exponential Asymptotics for Line Solitons in Two-Dimensional Periodic Potentials Nixon, Sean D. Yang, Jianke Akylas, Triantaphyllos R. Massachusetts Institute of Technology. Department of Mechanical Engineering Akylas, Triantaphyllos R. As a first step toward a fully two-dimensional asymptotic theory for the bifurcation of solitons from infinitesimal continuous waves, an analytical theory is presented for line solitons, whose envelope varies only along one direction, in general two-dimensional periodic potentials. For this two-dimensional problem, it is no longer viable to rely on a certain recurrence relation for going beyond all orders of the usual multiscale perturbation expansion, a key step of the exponential asymptotics procedure previously used for solitons in one-dimensional problems. Instead, we propose a more direct treatment which not only overcomes the recurrence-relation limitation, but also simplifies the exponential asymptotics process. Using this modified technique, we show that line solitons with any rational line slopes bifurcate out from every Bloch-band edge; and for each rational slope, two line-soliton families exist. Furthermore, line solitons can bifurcate from interior points of Bloch bands as well, but such line solitons exist only for a couple of special line angles due to resonance with the Bloch bands. In addition, we show that a countable set of multiline-soliton bound states can be constructed analytically. The analytical predictions are compared with numerical results for both symmetric and asymmetric potentials, and good agreement is obtained. United States. Air Force Office of Scientific Research (Grant USAF 9550-12-1-0244) 2015-06-08T15:36:50Z 2015-06-08T15:36:50Z 2013-03 2013-01 Article http://purl.org/eprint/type/JournalArticle 00222526 1467-9590 http://hdl.handle.net/1721.1/97216 Nixon, Sean D., T. R. Akylas, and Jianke Yang. “Exponential Asymptotics for Line Solitons in Two-Dimensional Periodic Potentials.” Studies in Applied Mathematics 131, no. 2 (March 19, 2013): 149–178. https://orcid.org/0000-0002-5246-4574 en_US http://dx.doi.org/10.1111/sapm.12006 Studies in Applied Mathematics Creative Commons Attribution-Noncommercial-Share Alike http://creativecommons.org/licenses/by-nc-sa/4.0/ application/pdf Wiley Blackwell arXiv |
| spellingShingle | Nixon, Sean D. Yang, Jianke Akylas, Triantaphyllos R. Exponential Asymptotics for Line Solitons in Two-Dimensional Periodic Potentials |
| title | Exponential Asymptotics for Line Solitons in Two-Dimensional Periodic Potentials |
| title_full | Exponential Asymptotics for Line Solitons in Two-Dimensional Periodic Potentials |
| title_fullStr | Exponential Asymptotics for Line Solitons in Two-Dimensional Periodic Potentials |
| title_full_unstemmed | Exponential Asymptotics for Line Solitons in Two-Dimensional Periodic Potentials |
| title_short | Exponential Asymptotics for Line Solitons in Two-Dimensional Periodic Potentials |
| title_sort | exponential asymptotics for line solitons in two dimensional periodic potentials |
| url | http://hdl.handle.net/1721.1/97216 https://orcid.org/0000-0002-5246-4574 |
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