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.

我们关注具有无限大对称对称非零边界条件（NZBC）的聚焦Hirota方程（也称为三阶非线性Schrödinger方程）的解的长期渐近行为。首先，基于具有NZBC的Lax对，使用正向和反向散射问题来建立具有明显跳跃曲线的振荡Riemann-Hilbert（RH）问题。其次，采用Deift-Zhou非线性最速下降法来分析振动RH问题，从而在时空平面的两个不同域（即平面波和调制椭圆波）中提出了长期渐近解。域）。最后，还研究了所考虑的Hirota方程的调制不稳定性。