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Nonlocal Kundu–Eckhaus equation: integrability, Riemann–Hilbert approach and Cauchy problem with step-like initial data

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Abstract

The main purpose of this paper is to discuss the Cauchy problem of integrable nonlocal (reverse-space-time) Kundu–Eckhaus (KE) equation through the Riemann–Hilbert (RH) method. Firstly, based on the zero-curvature equation, we present an integrable nonlocal KE equation and its Lax pair. Then, we discuss the properties of eigenfunctions and scattering matrix, such as analyticity, asymptotic behavior, and symmetry. Finally, for the prescribed step-like initial value: \(u(z,t)=o(1)\), \(z\rightarrow -\infty \) and \(u(z,t)=R+o(1)\), \(z\rightarrow +\infty \), where \(R>0\) is an arbitrary constant, we consider the initial value problem of the nonlocal KE equation. The paramount techniques is the asymptotic analysis of the associated RH problem.

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Acknowledgements

The authors would like to express our sincere thanks to every member in our discussion group for their valuable comments.

Funding

This study was funded by National Natural Science Foundation of China (Grant Number 12147115), by Natural Science Foundation of Anhui Province (Grant Number 2108085QA09), by University Natural Science Research Project of Anhui Province (Grant Numbers 2022AH051109 and 2023AH040225), by Discipline (Subject) Leader Cultivation Project of Anhui Province (Grant Number DTR2023052), by Scientific Research Foundation Funded Project of Chuzhou University (Grant Numbers 2022qd022 and 2022qd038).

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Authors and Affiliations

  1. School of Mathematics and Finance, Chuzhou University, Anhui, 239000, China

    Bei-Bei Hu, Zu-Yi Shen & Ling Zhang

  2. Applied Mathematics Research Center, Chuzhou University, Anhui, 239000, China

    Bei-Bei Hu & Ling Zhang

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  1. Bei-Bei Hu
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  2. Zu-Yi Shen
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Hu, BB., Shen, ZY. & Zhang, L. Nonlocal Kundu–Eckhaus equation: integrability, Riemann–Hilbert approach and Cauchy problem with step-like initial data. Lett Math Phys 114, 55 (2024). https://doi.org/10.1007/s11005-024-01802-2

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