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

Abstract

AbstractBased 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.