Abstract
In this study, we present the rigorous theory of the robust inverse scattering method for the discrete high-order nonlinear Schrödinger (HNLS) equation with a nonzero boundary condition (NZBC). Using the direct scattering problem, we deduce the analyticity, symmetries, and asymptotic behaviors of the Jost solutions and scattering matrix. We also formulate the inverse scattering problem using the matrix Riemann–Hilbert problem (RHP). Furthermore, utilizing the loop group theory, we construct the multi-fold Darboux transformation (DT) within the framework of the robust inverse scattering transform. Additionally, we develop the corresponding Bäcklund transformation (BT) to obtain the multi-fold lattice soliton solutions. To derive the high-order rational solutions, we further construct the high-order DT. Finally, we theoretically and graphically analyze these solutions, which exhibit lattice breather waves, W-shape lattice solitons, high-order lattice rogue waves (RW), and their interactions.
Funding source: Postdoctoral Research Foundation of China
Award Identifier / Grant number: 2022M710969
Funding source: National Natural Science Foundation of China
Award Identifier / Grant number: 42204119
Funding source: Fundamental Research Funds for the Central Universities
Award Identifier / Grant number: 2022FRFK060015
Funding source: National Natural Science Foundation of China
Award Identifier / Grant number: 11971133
Acknowledgments
We are grateful to the reviewers for their encouraging suggestions that were helpful in improving this paper further.
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Research ethics: Not applicable
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Author contributions: The authors have accepted responsibility for the entire content of this manuscript and approved its submission.
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Competing interests: The authors state no competing interests.
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Research funding: The work of X.W. Yan was supported by the China Postdoctoral Science Foundation (No. 2022M710969) and National Natural Science Foundation of China (No. 42204119). The work of Y. Chen was supported by the National Natural Science Foundation of China (No. 11971133). This work was also supported by Fundamental Research Funds for the Central Universities (http://doi.org/10.13039/501100012226, No. 2022FRFK060015).
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Data availability: Not applicable.
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