On the derivative nonlinear Schrodinger equation related to symmetric spaces

Georgi Grahovski, University of Essex

We will present multi-component generalizations of derivative nonlinear Schrodinger (DNLS) type of equations having quadratic bundle Lax pairs related to Z2-graded Lie algebras and A.III symmetric spaces. The Jost solutions and the minimal set of scattering data for the case of local and nonlocal reductions are constructed.

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.

The infinite set of integrals of motion for these models is briefly described at the end. Based on a joint work with Vladimir Gerdjikov and Rossen Ivanov.