Asymptotic Behavior of Solutions of the Dispersion Generalized Benjamin–Ono Equation

Abstract

We show that for any uniformly bounded in time H 1 ∩ L 1 solution of the dispersive generalized Benjamin-Ono equation, the limit infimum, as time t goes to infinity, converges to zero locally in an increasing-in-time region of space of order t/ log t. This result is in accordance with the one established by Muñoz and Ponce [20] for solutions of the Benjamin-Ono equation. Similar to solutions of the Benjamin-Ono equation, for a solution of the dispersive generalized Benjamin-Ono equation, with a mild L 1 -norm growth in time, its limit infimum must converge to zero, as time goes to infinity, locally in an increasing on time region of space of order depending on the rate of growth of its L 1 -norm. As a consequence, the existence of breathers or any other solution for the dispersive generalized Benjamin-Ono equation moving with a speed "slower" than a soliton is discarded. In our analysis the use of commutators expansions is essential.

Long time asymptotics of large data in the Kadomtsev–Petviashvili models

“…Some virial identities are independent of the integrability of the equation, consequently, are valid for plenty of dispersive models, see e.g. [9,29,35,36,[40][41][42]44] and references therein.…”

On Local Energy decay for solution of the Benjamin-Ono equation

“…The argument of the proof in [33] was based on virial identities (or weighted energy estimate) first appearing in [32] in the study of the long time behavior of solution of the generalized Kortewegde Vries (KdV) equation. In [34] and [25] this was extended, adapted and generalized to others one dimensional dispersive nonlinear systems under an assumption similar to that in (1.11).…”

Benjamin-Ono Soliton Dynamics in a slowly varying potential revisited

“…Saut [39] provides an overview of the derivations from physical models and the mathematical literature. Muñoz & Ponce [33] and Linares, Mendez, & Ponce [26] obtained local L ∞ estimates on an expanding spatial window as t → ∞. A normal forms procedure in the format of the "quasilinear modified energy method" was developed by Ifrim & Tataru [16] resulting in a new dispersive decay estimate for L 2 weighted initial data and its application to a new proof of L 2 global well-posedness.…”