An Explicit Formula for the Solution of the Benjamin-Ono Equation and its Applications
Abstract
We summarize recent developments in the theory of the Benjamin-Ono equation, a well-known longwave 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.
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BibTeX
@misc{gassot2026explicit,
title = {{An Explicit Formula for the Solution of the Benjamin-Ono Equation and its Applications}},
author = {Louise Gassot and Patrick Gérard and Peter D. Miller},
howpublished = {HAL},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2610.00770},
eprint = {2610.00770},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05775087v1},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - An Explicit Formula for the Solution of the Benjamin-Ono Equation and its Applications
AU - Louise Gassot
AU - Patrick Gérard
AU - Peter D. Miller
PY - 2026
DA - 2026/09/30
PB - HAL
DO - 10.48550/arxiv.2610.00770
UR - https://hal.science/hal-05775087v1
AB - We summarize recent developments in the theory of the Benjamin-Ono equation, a well-known longwave 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.
ER -