Long-time Asymptotic Behavior of the Fifth-order Modified KdV Equation in Low Regularity Spaces

Abstract

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

58 pages, 11 figures

References in corpus (3)

• Decay estimates for a class of wave equations
• Long-time Asymptotics for the NLS equation via dbar methods
• Global well-posedness for the fifth-order KdV equation in $H^{-1}(\mathbb{R})$