The Benjamin-Ono equation in the long-time limit: linearized self-similar universality
Louise Gassot, Patrick Gérard, and Peter D. Miller
LG: CNRS and Department of Mathematics, University of Rennes, Rennes, France
PG: Laboratoire de Mathématiques d'Orsay, Université Paris-Saclay, Orsay, France
PDM: Department of Mathematics, University of Michigan, Ann Arbor
Abstract:
We obtain the leading term in the solution of the Cauchy problem for the Benjamin-Ono equation in the limit \(t \to +\infty\) with \(x = O(t^{1/2})\). We show that the rate of decay exceeds that of self-similar solutions and obtain an explicit universal profile for the decaying solution, relating it to the linearization of the profile equation for self-similar solutions. The proof assumes a class of rational initial data \(u_0\) in \(L^2(\mathbb{R})\cap L^1(\mathbb{R})\) that exhibit generic behavior of the reflection coefficient at the origin.
