An explicit formula for the Benjamin--Ono equation

TL;DR

This paper derives an explicit formula for the general solution of the Benjamin-Ono equation on both the torus and the line, providing a significant analytical tool for understanding this nonlinear wave equation.

Contribution

It presents the first explicit solution formula for the Benjamin-Ono equation on the torus and line, advancing analytical methods for this integrable system.

Findings

Explicit solution formula on the torus

Explicit solution formula on the line

Enhanced understanding of Benjamin-Ono dynamics

Abstract

We establish an explicit formula for the general solution of the Benjamin-Ono equation on the torus and on the line. Contents 1. Introduction 1 1.1. The Benjamin-Ono equation 1 1.2. The Lax pair 2 1.3. The explicit formula on the torus 3 1.4. The explicit formula on the line 3 1.5. Organization of the paper 4 2. Proof of the explicit formula on the torus 4 3. Proof of the explicit formula on the line 6 4. Final remarks 8 References 9