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Solving the KdV hierarchy with self-consistent sources by inverse scattering method

Author

Listed:
  • Lin, Runliang
  • Zeng, Yunbo
  • Ma, Wen-Xiu

Abstract

The evolution of the eigenfunctions in the Lax representation of the KdV hierarchy with self-consistent sources possesses singularity. By proposing a method to treat the singularity to determine the evolution of scattering data, the KdV hierarchy with self-consistent sources is integrated by the inverse scattering method. The soliton solutions of these equations are obtained. It is shown that the insertion of a source may cause the variation of the speed of soliton. This approach can be applied to other (1+1)-dimensional soliton hierarchies.

Suggested Citation

  • Lin, Runliang & Zeng, Yunbo & Ma, Wen-Xiu, 2001. "Solving the KdV hierarchy with self-consistent sources by inverse scattering method," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 291(1), pages 287-298.
    • Handle: RePEc:eee:phsmap:v:291:y:2001:i:1:p:287-298
    DOI: 10.1016/S0378-4371(00)00519-7
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    References listed on IDEAS

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    1. Zeng, Yunbo, 1998. "Constructing hierarchy of sinh-Gordon-type equations from soliton hierarchy with self-consistent source," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 259(3), pages 278-290.
    2. Zeng, Yunbo, 1999. "Bi-hamiltonian structure of JM hierarchy with self-consistent sources," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 262(3), pages 405-419.
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    Cited by:

    1. Gegenhasi, & Hu, Xing-Biao, 2007. "Integrability of a differential-difference KP equation with self-consistent sources," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 74(2), pages 145-158.
    2. Babadjanova, Aygul & Kriecherbauer, Thomas & Urazboev, Gayrat, 2020. "The periodic solutions of the discrete modified KdV equation with a self-consistent source," Applied Mathematics and Computation, Elsevier, vol. 376(C).
    3. Wang, Hong-Yan, 2009. "Commutativity of source generation procedure and Bäcklund transformation: A BKP equation," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 80(4), pages 779-785.
    4. Xiao, Ting & Zeng, Yunbo, 2005. "A new constrained mKP hierarchy and the generalized Darboux transformation for the mKP equation with self-consistent sources," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 353(C), pages 38-60.
    5. Li, Qi & Chen, Deng-yuan & Zhang, Jian-bing & Chen, Shou-ting, 2012. "Solving the non-isospectral Ablowitz–Ladik hierarchy via the inverse scattering transform and reductions," Chaos, Solitons & Fractals, Elsevier, vol. 45(12), pages 1479-1485.
    6. Bondarenko, N. & Freiling, G. & Urazboev, G., 2013. "Integration of the matrix KdV equation with self-consistent source," Chaos, Solitons & Fractals, Elsevier, vol. 49(C), pages 21-27.

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