Abstract
To get more insight into the relation between discrete model and continuous counterpart, a new integrable semi-discrete Kundu–Eckhaus equation is derived from the reduction in an extended Ablowitz–Ladik hierarchy. The integrability of the semi-discrete model is confirmed by showing the existence of Lax pair and infinite number of conservation laws. The dynamic characteristics of the breather and rational solutions have been analyzed in detail for our semi-discrete Kundu–Eckhaus equation to reveal some new interesting phenomena which was not found in continuous one. It is shown that the theory of the discrete system including Lax pair, Darboux transformation and explicit solutions systematically yields their continuous counterparts in the continuous limit.
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Acknowledgements
The work of HQZ is supported by National Natural Science Foundation of China under Grant 11301331, Natural Science Foundation of Shanghai under Grant 17ZR1411600, and Innovation Program of Shanghai Municipal Education Commission under Grant 14YZ135, that of JYY by CAPES and CNPq of Brazil, that of ZNZ by National Natural Science Foundation of China under Grants 11271254, 11428102 and 11671255, and by the Ministry of Economy and Competitiveness of Spain under grands MTM2012-37070 and MTM2016-80276-P(AEI/FEDER,EU). We sincerely thank the referees for their very useful and constructive comments.
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Communicated by Ferdinand Verhulst.
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Zhao, Hq., Yuan, J. & Zhu, Zn. Integrable Semi-discrete Kundu–Eckhaus Equation: Darboux Transformation, Breather, Rogue Wave and Continuous Limit Theory. J Nonlinear Sci 28, 43–68 (2018). https://doi.org/10.1007/s00332-017-9399-9
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DOI: https://doi.org/10.1007/s00332-017-9399-9