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Inverse scattering transform for a nonlinear lattice equation under non-vanishing boundary conditions

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Abstract

Under investigation in this paper is the inverse scattering transform for a nonlinear lattice equation, which can be used to study the fluctuation of nonlinear optics and dynamics of anharmonic lattices. Symmetries, analyticities and asymptotic behaviors of eigenfunctions will be obtained in the direct scattering analysis to establish a suitable Riemann-Hilbert problem. The Riemann-Hilbert problem of the scattering data with simple poles will be constructed. In particular, by using the Laurent expansion and the generalized residue condition to solve the Riemann-Hilbert problem, the determinant representation of N-soliton solution for the equation will be presented. One-dark-soliton under non-vanishing boundary conditions will be displayed through some representative reflectionless potentials.

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Acknowledgements

We express our sincere thanks to each member of our discussion group for their suggestions. This work has been supported by the Fund Program for the Scientific Activities of Selected Returned Overseas Scholars in Shanxi Province under Grant No. 20220008, and the Shanxi Province Science Foundation under Grant No. 202303021221031.

Funding

This work has been supported by the Fund Program for the Scientific Activities of Selected Returned Overseas Scholars in Shanxi Province under Grant No. 20220008, and the Shanxi Province Science Foundation under Grant No. 202303021221031.

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  1. College of Mathematics, Taiyuan University of Technology, Taiyuan, 030024, China

    Qin-Ling Liu & Rui Guo

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  1. Qin-Ling Liu
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  2. Rui Guo
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Correspondence to Rui Guo.

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Liu, QL., Guo, R. Inverse scattering transform for a nonlinear lattice equation under non-vanishing boundary conditions. Opt Quant Electron 56, 1017 (2024). https://doi.org/10.1007/s11082-024-06886-7

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