Reductions of integrable equations onA.III-type symmetric spaces

Abstract

Abstract. We study a class of integrable non-linear differential equations related to the A.III-type symmetric spaces. These spaces are realized as factor groups of the form SU (N )/S(U (N − k) × U (k)). We use the Cartan involution corresponding to this symmetric space as an element of the reduction group and restrict generic Lax operators to this symmetric space. The symmetries of the Lax operator are inherited by the fundamental analytic solutions and give a characterization of the corresponding Riemann-Hilbert data.

CBC Systems with Mikhailov Reductions by Coxeter Automorphism: I. Spectral Theory of the Recursion Operators

“…The Inverse Scattering Method then consists of recovering the potential from the scattering data. The relations (22) could be regarded as Riemann-Hilbert problem related to the bunch of rays l ν with canonical normalization at λ = ∞. Without going into details, on the possibility to solve that problem is based the so-called dressing method for finding the soliton solutions.…”

“…The implications of the Mikhailov‐type reductions to the theory of Recursion Operators in the case been considered recently in several papers, . In fact, the system that has been studied is a CBC system in pole gauge, but the theory have been developed independently of this fact.…”

Geometry of the Recursion Operators for the GMV System

“…We shall call this system GMV system. As described in [10,11] the GMV system arises naturally when one looks for integrable system having a Lax representation [L, A] = 0 with L of the form i∂ x + λL 1 and L, A subject to Mikhailov-type reduction requirements, see [20,26,27]. In this particular case the Mikhailov reduction group G 0 is generated by the two elements g 1 and g 2 acting in the following way on the fundamental solutions of the system (1.6):…”

“…In the papers [10,11] the study of the spectral properties of (1.6) has been started and the generating operators for the system have been calculated using two different techniquesassuming they are operators for which the adjoint solutions of (1.6) should be eigenfunctions, and a symmetry approach based on technique developed in the work [18]. The Recursion Operators obtained raise several interesting questions.…”

“…For this reason we shall use the variable S(x) = −L 1 (x) and shall consider GMV system as GZS system in pole gauge subject to some restrictions. As already mentioned, in [10,11] have been considered the spectral theory aspects of the generating operators related to (1.6). The geometric aspects of the theory has not been developed yet, an issue that we shall address in this paper.…”