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

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Abstract

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.

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Authors and Affiliations

  1. State Key Laboratory of Information Photonics and Optical Communications, and School of Science, Beijing University of Posts and Telecommunications, Beijing, 100876, China

    Bo Qin, Bo Tian, Yu-Feng Wang & Ming Wang

  2. Department of Computer Science and Engineering, Hong Kong University of Science and Technology Clear Water Bay, Hong Kong, China

    Bo Qin

  3. College of Science, Minzu University of China, Beijing, 100081, China

    Yu-Feng Wang

  4. Ministry-of-Education Key Laboratory of Fluid Mechanics and National Laboratory for Computational Fluid Dynamics, Beijing University of Aeronautics and Astronautics, Beijing, 100191, China

    Yu-Jia Shen

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  1. Bo Qin
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  2. Bo Tian
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  5. Ming Wang
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Correspondence to Bo Tian.

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Qin, B., Tian, B., Wang, YF. et al. Bell-polynomial approach and Wronskian determinant solutions for three sets of differential–difference nonlinear evolution equations with symbolic computation. Z. Angew. Math. Phys. 68, 111 (2017). https://doi.org/10.1007/s00033-017-0853-1

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  • DOI: https://doi.org/10.1007/s00033-017-0853-1

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