The Prolongation Structure of the Modified Nonlinear Schrödinger Equation and Its Initial-Boundary Value Problem on the Half Line via the Riemann-Hilbert Approach
In this paper, the Lax pair of the modified nonlinear Schrödinger equation (mNLS) is derived by means of the prolongation structure theory. Based on the obtained Lax pair, the mNLS equation on the half line is analyzed with the assistance of Fokas method. A Riemann-Hilbert problem is formul...
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MDPI AG
2019-02-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/7/2/170 |
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| author | Tongshuai Liu Huanhe Dong |
| author_facet | Tongshuai Liu Huanhe Dong |
| author_sort | Tongshuai Liu |
| collection | DOAJ |
| description | In this paper, the Lax pair of the modified nonlinear Schrödinger equation (mNLS) is derived by means of the prolongation structure theory. Based on the obtained Lax pair, the mNLS equation on the half line is analyzed with the assistance of Fokas method. A Riemann-Hilbert problem is formulated in the complex plane with respect to the spectral parameter. According to the initial-boundary values, the spectral function can be defined. Furthermore, the jump matrices and the global relations can be obtained. Finally, the potential <inline-formula> <math display="inline"> <semantics> <mrow> <mi>q</mi> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> can be represented by the solution of this Riemann-Hilbert problem. |
| first_indexed | 2024-12-21T08:36:37Z |
| format | Article |
| id | doaj.art-9dbdcaa5c459473b85920e1b1d498e6f |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-12-21T08:36:37Z |
| publishDate | 2019-02-01 |
| publisher | MDPI AG |
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| series | Mathematics |
| spelling | doaj.art-9dbdcaa5c459473b85920e1b1d498e6f2022-12-21T19:10:03ZengMDPI AGMathematics2227-73902019-02-017217010.3390/math7020170math7020170The Prolongation Structure of the Modified Nonlinear Schrödinger Equation and Its Initial-Boundary Value Problem on the Half Line via the Riemann-Hilbert ApproachTongshuai Liu0Huanhe Dong1College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, ChinaCollege of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, ChinaIn this paper, the Lax pair of the modified nonlinear Schrödinger equation (mNLS) is derived by means of the prolongation structure theory. Based on the obtained Lax pair, the mNLS equation on the half line is analyzed with the assistance of Fokas method. A Riemann-Hilbert problem is formulated in the complex plane with respect to the spectral parameter. According to the initial-boundary values, the spectral function can be defined. Furthermore, the jump matrices and the global relations can be obtained. Finally, the potential <inline-formula> <math display="inline"> <semantics> <mrow> <mi>q</mi> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> can be represented by the solution of this Riemann-Hilbert problem.https://www.mdpi.com/2227-7390/7/2/170prolongation structuremNLS equationRiemann-Hilbert probleminitial-boundary value problem |
| spellingShingle | Tongshuai Liu Huanhe Dong The Prolongation Structure of the Modified Nonlinear Schrödinger Equation and Its Initial-Boundary Value Problem on the Half Line via the Riemann-Hilbert Approach Mathematics prolongation structure mNLS equation Riemann-Hilbert problem initial-boundary value problem |
| title | The Prolongation Structure of the Modified Nonlinear Schrödinger Equation and Its Initial-Boundary Value Problem on the Half Line via the Riemann-Hilbert Approach |
| title_full | The Prolongation Structure of the Modified Nonlinear Schrödinger Equation and Its Initial-Boundary Value Problem on the Half Line via the Riemann-Hilbert Approach |
| title_fullStr | The Prolongation Structure of the Modified Nonlinear Schrödinger Equation and Its Initial-Boundary Value Problem on the Half Line via the Riemann-Hilbert Approach |
| title_full_unstemmed | The Prolongation Structure of the Modified Nonlinear Schrödinger Equation and Its Initial-Boundary Value Problem on the Half Line via the Riemann-Hilbert Approach |
| title_short | The Prolongation Structure of the Modified Nonlinear Schrödinger Equation and Its Initial-Boundary Value Problem on the Half Line via the Riemann-Hilbert Approach |
| title_sort | prolongation structure of the modified nonlinear schrodinger equation and its initial boundary value problem on the half line via the riemann hilbert approach |
| topic | prolongation structure mNLS equation Riemann-Hilbert problem initial-boundary value problem |
| url | https://www.mdpi.com/2227-7390/7/2/170 |
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