Long-time asymptotic behavior of the fifth-order modified KdV equation in low regularity spaces

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 I-method, we give its global well-posedness in lower regularity Sobolev spaces and then obtain the asymptotic behavior in these spaces with weights.