Nonlinear Dynamics

, Volume 93, Issue 4, pp 1799–1808 | Cite as

Coherent structure of Alice–Bob modified Korteweg de-Vries equation

  • Congcong Li
  • S. Y. LouEmail author
  • Man Jia
Original Paper


To describe two-place events, Alice–Bob systems have been established by means of the shifted parity and delayed time reversal in the preprint arXiv:1603.03975v2 [nlin.SI], (2016). In this paper, we mainly study exact solutions of the integrable Alice–Bob modified Korteweg de-Vries (AB-mKdV) system. The general Nth Darboux transformation for the AB-mKdV equation is constructed. By using the Darboux transformation, some types of shifted parity and time reversal symmetry breaking solutions including one-soliton, two-soliton, and rogue wave solutions are explicitly obtained. In addition to the similar solutions of the mKdV equation (group invariant solutions), there are abundant new localized structures for the AB-mKdV systems.


Nonlinear nonlocal partial differential equations Darboux transformations Exact solutions solitons Rogue waves 



The authors are grateful to thank Professors D. J. Zhang, Z. N. Zhu, Q. P. Liu, X. B. Hu, and Y. Chen for their helpful discussions. The work was sponsored by the Global Change Research Program of China (No. 2015CB953904), Shanghai Knowledge Service Platform for Trustworthy Internet of Things (No. ZF1213), the National Natural Science Foundations of China (No. 11435005), and K. C. Wong Magna Fund in Ningbo University.


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© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Center for Nonlinear Science and Department of PhysicsNingbo UniversityNingboChina
  2. 2.Shanghai Key Laboratory of Trustworthy ComputingEast China Normal UniversityShanghaiChina

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