On integrable wave interactions and Lax pairs on symmetric spaces

TL;DR

This paper develops a comprehensive analytical framework for multi-component integrable wave equations related to symmetric spaces, including Lax pairs, scattering data, and explicit soliton solutions, with applications to local and nonlocal reductions.

Contribution

It introduces a modified dressing method to explicitly derive soliton solutions for multi-component Gerdjikov-Ivanov equations with various reductions.

Findings

Constructed Jost solutions and scattering data for multi-component DNLS equations.

Derived Riemann-Hilbert problems for these equations.

Explicitly obtained soliton solutions with regular behavior for specific reductions.

Abstract

Multi-component generalizations of derivative nonlinear Schrodinger (DNLS) type of equations having quadratic bundle Lax pairs related to Z_2-graded Lie algebras and A.III symmetric spaces are studied. The Jost solutions and the minimal set of scattering data for the case of local and nonlocal reductions are constructed. The latter lead to multi-component integrable equations with CPT-symmetry. Furthermore, the fundamental analytic solutions (FAS) are constructed and the spectral properties of the associated Lax operators are briefly discussed. The Riemann-Hilbert problem (RHP) for the multi-component generalizations of DNLS equation of Kaup-Newell (KN) and Gerdjikov-Ivanov (GI) types is derived. A modification of the dressing method is presented allowing the explicit derivation of the soliton solutions for the multi-component GI equation with both local and nonlocal reductions. It is shown that for specific choices of the reduction these solutions can have regular behavior for all finite x and t. The fundamental properties of the multi-component GI equations are briefly discussed at the end.