Non-autonomous soliton hierarchies

TL;DR

This paper introduces a formalism for constructing integrable non-autonomous deformations of soliton hierarchies using a Lie algebra-based initial value problem, with explicit solutions for key hierarchies like KdV.

Contribution

It presents a systematic method to generate non-autonomous soliton hierarchies via a Frobenius integrability condition on Lie algebras, including explicit solutions for important cases.

Findings

Formal solution exists for the IVP in the Lie algebra framework.

Explicit forms of solutions are derived for KdV, water waves, and AKNS hierarchies.

The formalism unifies the construction of non-autonomous soliton hierarchies.

Abstract

A formalism of systematic construction of integrable non-autonomous deformations of soliton hierarchies is presented. The theory is formulated as an initial value problem (IVP) for an associated Frobenius integrability condition on a Lie algebra. It is showed that this IVP has a formal solution and within the framewrok of two particular subalgebras of the hereditary Lie algebra the explicit forms of this formal solution are presented. Finally, this formalism is applied to Korteveg-de Vries, dispersive water waves and Ablowitz-Kaup-Newell-Segur soliton hierarchies.