A Discourse on the Benney Equation

TL;DR

This paper demonstrates that the Benney system can be reformulated as a classical dynamical problem with a potential dependent on the action function, providing insights into its general solution and specific particular solutions.

Contribution

It establishes the equivalence between the Benney system and a one-body classical dynamical problem, introducing a new approach to solving the system via Hamilton-Jacobi methods.

Findings

The Benney system is equivalent to a classical dynamical problem.

Explicit particular solutions of the Hamilton-Jacobi equation are presented.

The general solution of the Benney system is characterized through this equivalence.

Abstract

It is shown that the Benney system is equivalent to a one body classical dynamical problem in which the interaction potential is a functional of the action function. The general solution of this equation is responsible for the general solution of the Benney system. Some particular solutions of the selfconsistent Hamilton-Jacobi equation arising in this investigation are presented in explicit form.