Based on the nonlinear steepest descent method of Deift and Zhou for oscillatory Riemann–Hilbert problems and the Dbar approach, the long-time asymptotic behavior of solutions to the fifth-order modified KdV (Korteweg–de Vries) equation on the line is studied in the case of initial conditions that belong to some weighted Sobolev spaces. Using techniques in Fourier analysis and the idea of the -method, we give its global well-posedness in lower regularity Sobolev spaces and then obtain the asymptotic behavior in these spaces with weights.

中文翻译：

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低正则空间中五阶修正 KdV 方程的长时间渐近行为

基于Deift和Zhou非线性最速下降法求解振荡Riemann-Hilbert问题和Dbar方法，研究了五阶修正KdV（Korteweg-de Vries）方程解在直线上的长时间渐近行为属于某些加权 Sobolev 空间的初始条件的情况。使用傅里叶分析中的技术和- 方法的思想，我们在较低正则 Sobolev 空间中给出其全局适定性，然后通过权重获得这些空间中的渐近行为。