Constraints and Soliton Solutions for the KdV Hierarchy and AKNS Hierarchy

TL;DR

This paper introduces a method to constrain integrable systems like the KdV and AKNS hierarchies using special Lax pairs, simplifying their solutions to lower-dimensional systems and revealing linearizability under constraints.

Contribution

It generalizes finite-gap solution generation to a new class of Lax pairs, enabling reduction of complex hierarchies to simpler, solvable forms.

Findings

n-soliton solutions described by ODEs

AKNS hierarchy constrained to univariate integrable hierarchies

All constrained AKNS equations are linearizable

Abstract

It is well-known that the finite-gap solutions of the KdV equation can be generated by its recursion operator.We generalize the result to a special form of Lax pair, from which a method to constrain the integrable system to a lower-dimensional or fewer variable integrable system is proposed. A direct result is that the $n$-soliton solutions of the KdV hierarchy can be completely depicted by a series of ordinary differential equations (ODEs), which may be gotten by a simple but unfamiliar Lax pair. Furthermore the AKNS hierarchy is constrained to a series of univariate integrable hierarchies. The key is a special form of Lax pair for the AKNS hierarchy. It is proved that under the constraints all equations of the AKNS hierarchy are linearizable.