Bell-polynomial approach and Wronskian determinant solutions for three sets of differential–difference nonlinear evolution equations with symbolic computation

  • Bo Qin
  • Bo Tian
  • Yu-Feng Wang
  • Yu-Jia Shen
  • Ming Wang


Under investigation in this paper are the Belov–Chaltikian (BC), Leznov and Blaszak–Marciniak (BM) lattice equations, which are associated with the conformal field theory, UToda\((m_1,m_2)\) system and r-matrix, respectively. With symbolic computation, the Bell-polynomial approach is developed to directly bilinearize those three sets of differential–difference nonlinear evolution equations (NLEEs). This Bell-polynomial approach does not rely on any dependent variable transformation, which constitutes the key step and main difficulty of the Hirota bilinear method, and thus has the advantage in the bilinearization of the differential–difference NLEEs. Based on the bilinear forms obtained, the N-soliton solutions are constructed in terms of the \(N \times N\) Wronskian determinant. Graphic illustrations demonstrate that those solutions, more general than the existing results, permit some new properties, such as the solitonic propagation and interactions for the BC lattice equations, and the nonnegative dark solitons for the BM lattice equations.


Binary Bell polynomials Differential–difference nonlinear evolution equations Solitonic propagation and interactions Symbolic computation 

Mathematics Subject Classification

35C08 35F20 35G20 


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Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Bo Qin
    • 1
    • 2
  • Bo Tian
    • 1
  • Yu-Feng Wang
    • 1
    • 3
  • Yu-Jia Shen
    • 4
  • Ming Wang
    • 1
  1. 1.State Key Laboratory of Information Photonics and Optical Communications, and School of ScienceBeijing University of Posts and TelecommunicationsBeijingChina
  2. 2.Department of Computer Science and EngineeringHong Kong University of Science and Technology Clear Water BayHong KongChina
  3. 3.College of ScienceMinzu University of ChinaBeijingChina
  4. 4.Ministry-of-Education Key Laboratory of Fluid Mechanics and National Laboratory for Computational Fluid DynamicsBeijing University of Aeronautics and AstronauticsBeijingChina

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