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A Type I Defect and New Integrable Boundary Conditions for the Coupled Nonlinear Schrödinger Equation

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Abstract

We study two integrable systems associated with the coupled NLS equation: the integrable defect system and the integrable boundary systems. Regarding the first one, we present a type I defect condition, which is described by a Bäcklund transformation frozen at the defect location. For the resulting defect system, we prove its integrability both by showing the existence of an infinite set of conserved quantities and by implementing the classical r-matrix method. Regarding the second one, we present some new integrable boundary conditions for the coupled NLS equation by imposing suitable reductions on the defect conditions. Our new boundary conditions, unlike the usual boundary conditions (such as the Robin boundary), involve time derivatives of the coupled NLS fields and are characterised by non constant \(K(\lambda )\) matrices. We prove the integrability of our new boundary conditions by using Sklyanin’s approach.

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Acknowledgements

The author would like to express his sincerest gratitude to the referees for their valuable comments, which have helped him to improve this paper. This work was supported by the National Natural Science Foundation of China (Grant No. 11771186) and 333 Project of Jiangsu Province (No. BRA2020246).

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  1. School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou, 221116, Jiangsu, People’s Republic of China

    Baoqiang Xia

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Correspondence to Baoqiang Xia.

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Communicated by Robert Buckingham.

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Xia, B. A Type I Defect and New Integrable Boundary Conditions for the Coupled Nonlinear Schrödinger Equation. J Nonlinear Sci 32, 53 (2022). https://doi.org/10.1007/s00332-022-09809-6

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