Journal
JOURNAL OF DIFFERENTIAL EQUATIONS
Volume 262, Issue 1, Pages 506-558Publisher
ACADEMIC PRESS INC ELSEVIER SCIENCE
DOI: 10.1016/j.jde.2016.09.033
Keywords
Riemann-Hilbert problem; Integrable system; Coupled nonlinear Schrodinger equation; Initial-boundary value problem; Dirichlet-to-Neumann map
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Funding
- Fundamental Research Funds for Talents Cultivation Project of China University of Mining and Technology [YC150003]
- Fundamental Research Funds for key discipline construction [XZD201602]
- Fundamental Research Funds for the Central Universities of China [2015QNA53, 2015XKQY14]
- General Financial Grant from the China Postdoctoral Science Foundation [2015M570498]
- Natural Sciences Foundation of China [11301527, 11371361]
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Bouncy value problems for integrable nonlinear differential equations can be analyzed via the Fokas method. In this paper, this method is employed in order to study initial boundary value problems of the general coupled nonlinear Schrodinger equation formulated on the finite interval with 3 x 3 Lax pairs. The solution can be written in terms of the solution of a 3 x 3 Riemann-Hilbert problem. The relevant jump matrices are explicitly expressed in terms of the three matrix-value spectral functions s(k), S(k), and S-L(k). The associated general Dirichlet to Neumann map is also analyzed via the global relation. It is interesting that the relevant formulas can be reduced to the analogous formulas derived for boundary value problems formulated on the half-line in the limit when the length of the interval tends to infinity. It is shown that the formulas characterizing the Dirichlet to Neumann map coincide with the analogous formulas obtained via a Gelfand-Levitan-Marchenko representation. (C) 2016 Elsevier Inc. All rights reserved.
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