Akns Hierarchy, Self-Similarity, String Equations and the Grassmannian

TL;DR

This paper explores the geometric structure of the AKNS hierarchy using the infinite dimensional Grassmannian, linking self-similarity conditions, string equations, and rational solutions of related integrable systems.

Contribution

It characterizes self-similar solutions of the AKNS hierarchy within the Grassmannian framework and connects string equations to Galilean self-similarity.

Findings

Identifies the moduli space of self-similar solutions in the Sato Grassmannian.

Characterizes points in the Segal--Wilson Grassmannian related to rational solutions.

Provides explicit families of Galilean self-similar solutions for AKNS and NLS equations.

Abstract

In this paper the Galilean, scaling and translational self--similarity conditions for the AKNS hierarchy are analysed geometrically in terms of the infinite dimensional Grassmannian. The string equations found recently by non--scaling limit analysis of the one--matrix model are shown to correspond to the Galilean self--similarity condition for this hierarchy. We describe, in terms of the initial data for the zero--curvature 1--form of the AKNS hierarchy, the moduli space of these self--similar solutions in the Sato Grassmannian. As a byproduct we characterize the points in the Segal--Wilson Grassmannian corresponding to the Sachs rational solutions of the AKNS equation and to the Nakamura--Hirota rational solutions of the NLS equation. An explicit 1--parameter family of Galilean self--similar solutions of the AKNS equation and the associated solution to the NLS equation is determined.