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Nonlinear differential-difference hierarchy relevant to the Ablowitz-Ladik equation: Lax pair, conservation laws, N-fold Darboux transformation and explicit exact solutions

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  • Shen, Yuan
  • Tian, Bo
  • Zhou, Tian-Yu
  • Gao, Xiao-Tian

Abstract

Nonlinear differential-difference equations appear in optics, condensed matter physics, plasma physics and other fields. In this paper, we investigate a nonlinear differential-difference hierarchy relevant, in the case of θ=0, to the Ablowitz-Ladik equation, where θ=0,1. That hierarchy is obtained via a discrete spectral problem and the associated discrete spectral problem. When θ=1, Lax pair of the first nonlinear differential-difference system in that hierarchy is obtained. When θ=1, conservation laws and N-fold Darboux transformation of the first nonlinear differential-difference system in that hierarchy are derived with the aid of that Lax pair, where N is a positive integer. When θ=1, explicit exact solutions of that system are determined via that N-fold Darboux transformation. Discrete one soliton and interaction between the discrete one soliton and one breather-like wave are graphically depicted.

Suggested Citation

  • Shen, Yuan & Tian, Bo & Zhou, Tian-Yu & Gao, Xiao-Tian, 2022. "Nonlinear differential-difference hierarchy relevant to the Ablowitz-Ladik equation: Lax pair, conservation laws, N-fold Darboux transformation and explicit exact solutions," Chaos, Solitons & Fractals, Elsevier, vol. 164(C).
    • Handle: RePEc:eee:chsofr:v:164:y:2022:i:c:s0960077922006701
    DOI: 10.1016/j.chaos.2022.112460
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    Cited by:

    1. Sang, Xue & Dong, Huanhe & Fang, Yong & Liu, Mingshuo & Kong, Yuan, 2024. "Soliton, breather and rogue wave solutions of the nonlinear Schrödinger equation via Darboux transformation on a time–space scale," Chaos, Solitons & Fractals, Elsevier, vol. 184(C).
    2. Yuan, Cuilian & Yang, Hujiang & Meng, Xiankui & Tian, Ye & Zhou, Qin & Liu, Wenjun, 2023. "Modulational instability and discrete rogue waves with adjustable positions for a two-component higher-order Ablowitz–Ladik system associated with 4 × 4 Lax pair," Chaos, Solitons & Fractals, Elsevier, vol. 168(C).

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