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Long-Time Asymptotics for the Focusing Hirota Equation with Non-Zero Boundary Conditions at Infinity Via the Deift-Zhou Approach

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Abstract

We are concerned with the long-time asymptotic behavior of the solution for the focusing Hirota equation (also called third-order nonlinear Schrödinger equation) with symmetric, non-zero boundary conditions (NZBCs) at infinity. Firstly, based on the Lax pair with NZBCs, the direct and inverse scattering problems are used to establish the oscillatory Riemann-Hilbert (RH) problem with distinct jump curves. Secondly, the Deift-Zhou nonlinear steepest-descent method is employed to analyze the oscillatory RH problem such that the long-time asymptotic solutions are proposed in two distinct domains of space-time plane (i.e., the plane-wave and modulated elliptic-wave domains), respectively. Finally, the modulation instability of the considered Hirota equation is also investigated.

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Acknowledgements

The authors thank the referee for the valuable suggestions and comments. This work was partially supported by the NSFC under Grants Nos. 11731014 and 11925108.

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Authors and Affiliations

  1. Institute of Applied Physics and Computational Mathematics, 100088, Beijing, China

    Shuyan Chen & Boling Guo

  2. KLMM, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, 100190, Beijing, China

    Zhenya Yan

  3. School of Mathematical Sciences, University of Chinese Academy of Sciences, 100049, Beijing, China

    Zhenya Yan

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  1. Shuyan Chen
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Correspondence to Zhenya Yan.

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Chen, S., Yan, Z. & Guo, B. Long-Time Asymptotics for the Focusing Hirota Equation with Non-Zero Boundary Conditions at Infinity Via the Deift-Zhou Approach. Math Phys Anal Geom 24, 17 (2021). https://doi.org/10.1007/s11040-021-09388-0

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