Lax hierarchy, Solitons, Sumrules and a dual Lax hierarchy

TL;DR

This paper explores the mathematical structure of soliton solutions in integrable systems, introducing a dual hierarchy and sumrules that connect soliton potentials, eigenstates, and eigenvalues, enhancing understanding of soliton dynamics.

Contribution

It introduces a dual hierarchy of functions related to the Lax hierarchy and derives sumrules linking soliton potentials to eigenvalues, providing new insights into soliton solutions.

Findings

Representation of Lax hierarchy functions via eigenstates

Sumrules relating potentials and eigenvalues

Existence of a dual hierarchy with soliton solutions

Abstract

It is shown that a set of functions which characterise the Lax hierarchy of non-linear equations may be represented in terms of the eigenstates of the potential which satisfies the generalised KdV equation. Such a representation leads to sumrules relating integrals involving the soliton potential and its various derivatives to sums involving the boundstate eigenvalues of the Schroedinger equation for the reflectionless potential. A new hierarchy of functions, which is in a sense dual to the Lax hierarchy, is identified. It is shown that time dependent equations involving the dual functions may be established which permit solutions related to an N-soliton structure similar to that for the Lax hierarchy but with a different 'speed' for the solitons.