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Modulational instability and dynamics of rational soliton solutions for the coupled Volterra lattice equation associated with \(4\times 4\) Lax pair

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Abstract

The coupled Volterra lattice equation associated with \(4\times 4\) Lax pair is under investigation, which is an integrable discrete form of a coupled KdV equation applied widely in fluids, Bose–Einstein condensation and atmospheric dynamics. First, we explore the conditions for modulational instability (MI) of the constant seed background for this equation. Secondly, we present the discrete Darboux transformation (DT) and generalised DT based on the new \(4\times 4\) Lax pair. Through the resulting discrete DT, the bell-shaped and anti-N-shaped soliton solutions of the coupled Volterra lattice equation are derived. Moreover, we derive the M-shaped and N-shaped rational solitons and bell-shaped and N-shaped semirational soliton solutions of the coupled Volterra lattice equation via the discrete generalised DT. Finally, we numerically study the dynamical behaviours of such soliton solutions and find that the rational and semirational soliton solutions have better numerical stability than the usual soliton solution, although three types of solutions are robust against a small noise. The results may be helpful for understanding the two-layered fluid waves near ocean shores described by the coupled Korteweg–de Vries (KdV) equation.

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Acknowledgements

This work was partially supported by Qin Xin Talents Cultivation Program of Beijing Information Science and Technology University (QXTCP B201704), the NSFC under Grant Nos 11375030 and 61471406, the Beijing Natural Science Foundation under Grant No. 1153004.

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Authors and Affiliations

  1. School of Applied Science, Beijing Information Science and Technology University, Beijing, 100192, China

    Nan Liu & Xiao-Yong Wen

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  1. Nan Liu
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  2. Xiao-Yong Wen
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Correspondence to Xiao-Yong Wen.

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Liu, N., Wen, XY. Modulational instability and dynamics of rational soliton solutions for the coupled Volterra lattice equation associated with \(4\times 4\) Lax pair. Pramana - J Phys 93, 23 (2019). https://doi.org/10.1007/s12043-019-1784-5

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  • DOI: https://doi.org/10.1007/s12043-019-1784-5

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