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

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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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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Keywords

  • Focusing Hirota equation
  • Non-zero boundary conditions
  • Inverse scattering
  • Oscillatory Riemann-Hilbert problem
  • Nonlinear steepest-descent method
  • Long-time asymptotics

Mathematics Subject Classification (2010)

  • 35Q15
  • 37K40
  • 37K15
  • 14K25
  • 35Q60