An Explicit Formula for the Benjamin-Ono Hierarchy with Applications to Traveling Waves and Zero-Dispersion Limits

TL;DR

This paper extends an explicit formula for the Benjamin-Ono hierarchy, enabling classification of traveling waves and analysis of zero-dispersion limits, with convergence results and geometric characterizations.

Contribution

It provides a new explicit formula for the entire Benjamin-Ono hierarchy and applies it to classify solutions and analyze zero-dispersion limits.

Findings

Complete classification of traveling wave solutions for all hierarchy flows.

Weak convergence of solutions in $L^2$ as dispersion tends to zero.

Geometric characterization of the zero-dispersion limit as an alternating sum.

Abstract

In this paper, we first extend the explicit formula \cite{gerard2023explicit} for the classical Benjamin-Ono equation to each flow of the Benjamin-Ono hierarchy on line. We then use this representation to derive two main applications. First, we obtain a complete classification of traveling wave solutions for all higher-order flows in the hierarchy. Second, we analyze the zero-dispersion limit for the corresponding small-dispersion flows. For every fixed time $t\in\mathbb R$, we prove that, at any time, the solution converges weakly in $L^2(\mathbb R)$ as the dispersion parameter tends to $0$, and we provide a geometric characterization of the limit in terms of an alternating sum, which yields the higher-order analogue of the formula obtained in \cite{miller2011zero}, \cite{Gerard2025small} for the Benjamin-Ono equation.