Self-Consistent-Field Method and $\tau$-Functional Method on Group Manifold in Soliton Theory: a Review and New Results

TL;DR

This paper reviews and introduces new results on the self-consistent field and τ-functional methods on group manifolds within soliton theory, emphasizing a geometric approach using curvature and collective variables.

Contribution

It presents a novel geometric framework for the self-consistent field method using curvature and collective coordinates on group manifolds, extending traditional approaches.

Findings

The curvature condition C=0 ensures integrability.

A fluid dynamics-inspired Lagrange approach describes collective coordinates.

The method explicitly incorporates group structure into the SCF framework.

Abstract

The maximally-decoupled method has been considered as a theory to apply an basic idea of an integrability condition to certain multiple parametrized symmetries. The method is regarded as a mathematical tool to describe a symmetry of a collective submanifold in which a canonicity condition makes the collective variables to be an orthogonal coordinate-system. For this aim we adopt a concept of curvature unfamiliar in the conventional time-dependent (TD) self-consistent field (SCF) theory. Our basic idea lies in the introduction of a sort of Lagrange manner familiar to fluid dynamics to describe a collective coordinate-system. This manner enables us to take a one-form which is linearly composed of a TD SCF Hamiltonian and infinitesimal generators induced by collective variable differentials of a canonical transformation on a group. The integrability condition of the system read the curvature C=0. Our method is constructed manifesting itself the structure of the group under consideration. >...