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Studies on a Three-Field Lattice System: N-Fold Darboux Transformation, Conservation Laws and Analytic Solutions

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Abstract

Researches on the nonlinear lattice equations are active, with the applications in nonlinear optics, condensed matter physics, plasma physics, etc. What we study in this paper is a three-field lattice system, which can be reduced to a modified Toda lattice system and a coupled lattice system. Based on a known Lax pair, we present an N-fold Darboux matrix, and then construct an N-fold Darboux transformation for that system, where N is a positive integer. The first three conservation laws of that system are determined via the Lax pair. Utilizing that N-fold Darboux transformation with \(N=1\) and 2, we obtain the one-fold solutions and two-fold solutions of that system. Those solutions can be used to describe the discrete solitons. Via the one-fold solutions, we present a combination of the kink-shaped discrete one soliton and bell-shaped discrete one soliton. Amplitude, shape and velocity of that combination remain unchanged during the propagation.

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Notes

  1. Investigations on certain nonlinear evolution equations have been shown, e.g., in Refs. [4,5,6].

  2. We use Mathematica for the work in this paper.

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Acknowledgements

We express our sincere thanks to the Editors and Reviewers for their valuable comments. This work has been supported by the BUPT Excellent Ph.D. Students Foundation under Grant No. CX2022156, by the National Natural Science Foundation of China under Grant Nos. 11772017, 11272023 and 11471050, by the Fund of State Key Laboratory of Information Photonics and Optical Communications (Beijing University of Posts and Telecommunications), China (IPOC: 2017ZZ05) and by the Fundamental Research Funds for the Central Universities of China under Grant No. 2011BUPTYB02.

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

    Yuan Shen, Bo Tian, Dan-Yu Yang & Tian-Yu Zhou

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  1. Yuan Shen
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Shen, Y., Tian, B., Yang, DY. et al. Studies on a Three-Field Lattice System: N-Fold Darboux Transformation, Conservation Laws and Analytic Solutions. Qual. Theory Dyn. Syst. 22, 74 (2023). https://doi.org/10.1007/s12346-022-00730-7

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