Singular sectors of the 1-layer Benney and dToda systems and their interrelations

TL;DR

This paper thoroughly analyzes the singular sectors of the 1-layer Benney and dToda systems using Euler-Poisson-Darboux equations, providing explicit solutions and exploring their interrelations.

Contribution

It offers a complete description of the singular sectors and solutions of the 1-layer Benney system, and investigates the connections between related Euler-Poisson-Darboux equations.

Findings

Explicit solutions depending on two parameters are provided.

The relation between Euler-Poisson-Darboux equations with opposite signs of parameter a is discussed.

A comprehensive classification of singular sectors is achieved.

Abstract

Complete description of the singular sectors of the 1-layer Benney system (classical long wave equation) and dToda system is presented. Associated Euler-Poisson-Darboux equations E(1/2,1/2) and E(-1/2,-1/2) are the main tool in the analysis. A complete list of solutions of the 1-layer Benney system depending on two parameters and belonging to the singular sector is given. Relation between Euler-Poisson-Darboux equations E(a,a) with opposite sign of a is discussed.