Long-time asymptotics for the reverse space-time nonlocal Hirota equation with decaying initial value problem: without solitons. (English) Zbl 1542.35064
Summary: In this work, we mainly consider the Cauchy problem for the reverse space-time nonlocal Hirota equation with the initial data rapidly decaying in the solitonless sector. Start from the Lax pair, we first construct the basis Riemann-Hilbert problem for the reverse space-time nonlocal Hirota equation. Furthermore, using the approach of Deift-Zhou nonlinear steepest descent, the explicit long-time asymptotics for the reverse space-time nonlocal Hirota is derived. For the reverse space-time nonlocal Hirota equation, since the symmetries of its scattering matrix are different with the local Hirota equation, the \(\vartheta (\lambda_i)\) \((i = 0, 1)\) would like to be imaginary, which results in the \(\delta_{\lambda_i}^0\) contains an increasing \(t \frac{\pm Im \vartheta (\lambda_i)}{2}\), and then the asymptotic behavior for nonlocal Hirota equation becomes differently.
MSC:
| 35B40 | Asymptotic behavior of solutions to PDEs |
| 35C15 | Integral representations of solutions to PDEs |
| 35G25 | Initial value problems for nonlinear higher-order PDEs |
| 35Q51 | Soliton equations |
Keywords:
Riemann-Hilbert problem; reverse space-time nonlocal Hirota equation; long-time asymptotics; nonlinear steepest descent methodReferences:
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