An explicit formula for the solution of the Benjamin-Ono equation and its applications
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 summarize recent developments in the theory of the Benjamin-Ono equation, a well-known long-wave asymptotic model for internal waves in deep water, focusing on results obtained by means of an explicit formula for the solution of the Cauchy problem in terms of given initial data. We give a full description of this formula, and provide simple examples, with the aim of bringing it into the realm of applied mathematics. We also show how the formula simplifies further upon restriction to rational initial data. We then show how the formula and its rational restriction explain in great detail, and in a strikingly simple fashion, the asymptotic behavior of solutions of the Benjamin-Ono equation in the small-dispersion and long-time limits. In the setting of long-time asymptotics, we prove some new results related to soliton resolution yielding pointwise convergence with a concrete decay rate under the assumption of rational initial data. The first half of the paper is a stand-alone general survey and user's guide, while the appendices constituting the second half are intended for readers who want to explore the topics in greater depth, and it includes the proofs of our new results.