The model equation of soliton theory

TL;DR

This paper develops a new hierarchy of integrable 1+2-dimensional equations linked to Lie algebras, providing explicit solutions, a generating function equation, and applications to spectral problems, advancing the mathematical understanding of soliton theory.

Contribution

It introduces a novel hierarchy of integrable equations related to Lie algebras and derives a simple generating function equation with broad applications.

Findings

Solutions depend on arbitrary functions of one variable.

A simple generating function equation for the hierarchy is derived.

Applications to second order spectral problems are demonstrated.

Abstract

We consider an hierarchy of integrable 1+2-dimensional equations related to Lie algebra of the vector fields on the line. The solutions in quadratures are constructed depending on $n$ arbitrary functions of one argument. The most interesting result is the simple equation for the generating function of the hierarchy which defines the dynamics for the negative times and also has applications to the second order spectral problems. A rather general theory of integrable 1+1-dimensional equations can be developed by study of polynomial solutions of this equation under condition of regularity of the corresponding potentials.