Recent Progress on Nonlinear Schrödinger Systems with Quadratic Interactions
The study of nonlinear Schrödinger systems with quadratic interactions has attracted much attention in the recent years. In this paper, we summarize time decay estimates of small solutions to the systems under the mass resonance condition in 2-dimensional space. We show the existence of wave operato...
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| Format: | Article |
| Language: | English |
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Wiley
2014-01-01
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| Series: | The Scientific World Journal |
| Online Access: | http://dx.doi.org/10.1155/2014/214821 |
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| _version_ | 1832552333751353344 |
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| author | Chunhua Li Nakao Hayashi |
| author_facet | Chunhua Li Nakao Hayashi |
| author_sort | Chunhua Li |
| collection | DOAJ |
| description | The study of nonlinear Schrödinger systems with quadratic interactions has attracted much attention in the recent years. In this paper, we summarize time decay estimates of small solutions to the systems under the mass resonance condition in 2-dimensional space. We show the existence of wave operators and modified wave operators of the systems under some mass conditions in n-dimensional space, where n≥2. The existence of scattering operators and finite time blow-up of the solutions for the systems in higher space dimensions is also shown. |
| format | Article |
| id | doaj-art-ea906156431a4397bfd696d0d97163c1 |
| institution | Kabale University |
| issn | 2356-6140 1537-744X |
| language | English |
| publishDate | 2014-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | The Scientific World Journal |
| spelling | doaj-art-ea906156431a4397bfd696d0d97163c12025-02-03T05:59:00ZengWileyThe Scientific World Journal2356-61401537-744X2014-01-01201410.1155/2014/214821214821Recent Progress on Nonlinear Schrödinger Systems with Quadratic InteractionsChunhua Li0Nakao Hayashi1Department of Mathematics, College of Science, Yanbian University, No.977 Gongyuan Road, Yanji, Jilin 133002, ChinaDepartment of Mathematics, Graduate School of Science, Osaka University, Toyonaka, Osaka 560-0043, JapanThe study of nonlinear Schrödinger systems with quadratic interactions has attracted much attention in the recent years. In this paper, we summarize time decay estimates of small solutions to the systems under the mass resonance condition in 2-dimensional space. We show the existence of wave operators and modified wave operators of the systems under some mass conditions in n-dimensional space, where n≥2. The existence of scattering operators and finite time blow-up of the solutions for the systems in higher space dimensions is also shown.http://dx.doi.org/10.1155/2014/214821 |
| spellingShingle | Chunhua Li Nakao Hayashi Recent Progress on Nonlinear Schrödinger Systems with Quadratic Interactions The Scientific World Journal |
| title | Recent Progress on Nonlinear Schrödinger Systems with Quadratic Interactions |
| title_full | Recent Progress on Nonlinear Schrödinger Systems with Quadratic Interactions |
| title_fullStr | Recent Progress on Nonlinear Schrödinger Systems with Quadratic Interactions |
| title_full_unstemmed | Recent Progress on Nonlinear Schrödinger Systems with Quadratic Interactions |
| title_short | Recent Progress on Nonlinear Schrödinger Systems with Quadratic Interactions |
| title_sort | recent progress on nonlinear schrodinger systems with quadratic interactions |
| url | http://dx.doi.org/10.1155/2014/214821 |
| work_keys_str_mv | AT chunhuali recentprogressonnonlinearschrodingersystemswithquadraticinteractions AT nakaohayashi recentprogressonnonlinearschrodingersystemswithquadraticinteractions |