Journal
JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL
Volume 50, Issue 39, Pages -Publisher
IOP PUBLISHING LTD
DOI: 10.1088/1751-8121/aa825b
Keywords
integrable system; coupled modified Korteweg-de Vries equation; Riemann-Hilbert problem; initial-boundary value problem; Dirichlet to Neumann map
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Funding
- Qinglan Engineering project of Jiangsu Universities
- National Natural Science Foundation of China [11301527]
- China Postdoctoral Science Foundation [2015M570498, 2017T100413]
- China University of Mining and Technology [YC150003]
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In this paper, we implement the Fokas method in order to study initial boundary value problems of the coupled modified Korteweg-de Vries equation formulated on the half-line, with Lax pairs involving 3 x 3 matrices. This equation can be considered as a generalization of the modified KdV equation. We show that the solution {p(x, t), q(x, t)} 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 matrix-value spectral functions s(k) and S(k), which are respectively determined by the initial values and boundary values at x = 0. Finally, the associated Dirichlet to Neumann map of the equation is analyzed in detail. Some of these boundary values are unknown; however, using the fact that these specific functions satisfy a certain global relation, the unknown boundary values can be expressed in terms of the given initial and boundary data.
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