Polynomial Bundles and Generalised Fourier Transforms for Integrable Equations on A.III-type Symmetric Spaces

TL;DR

This paper explores integrable nonlinear equations on A.III-type symmetric spaces using inverse scattering, constructing soliton solutions, establishing a generalized Fourier transform framework, and analyzing Hamiltonian structures of related models.

Contribution

It introduces a novel approach to analyze integrable equations on A.III symmetric spaces via polynomial bundles and generalized Fourier transforms, including soliton solutions and Hamiltonian structures.

Findings

Constructed two classes of soliton solutions.

Established the inverse scattering method as a generalized Fourier transform.

Analyzed Hamiltonian structures of multi-component models.

Abstract

A special class of integrable nonlinear differential equations related to A.III-type symmetric spaces and having additional reductions are analyzed via the inverse scattering method (ISM). Using the dressing method we construct two classes of soliton solutions associated with the Lax operator. Next, by using the Wronskian relations, the mapping between the potential and the minimal sets of scattering data is constructed. Furthermore, completeness relations for the 'squared solutions' (generalized exponentials) are derived. Next, expansions of the potential and its variation are obtained. This demonstrates that the interpretation of the inverse scattering method as a generalized Fourier transform holds true. Finally, the Hamiltonian structures of these generalized multi-component Heisenberg ferromagnetic (MHF) type integrable models on A.III-type symmetric spaces are briefly analyzed.