We start with a Riemann-Hilbert problem (RHP) related to a BD.I-type symmetric spaces $SO(2r+1)/S(O(2r-2s +1)\otimes O(2s))$, $s\geq 1$. We consider two Riemann-Hilbert problems: the first formulated on the real axis $\mathbb{R}$ in the complex $\lambda$-plane; the second one is formulated on $\mathbb{R} \oplus i\mathbb{R}$. The first RHP for $s=1$ allows one to solve the Kulish-Sklyanin (KS) model; the second RHP is relevant for a new type of KS model. An important example for nontrivial deep reductions of KS model is given. Its effect on the scattering matrix is formulated. In particular we obtain new 2-component NLS equations. Finally, using the Wronskian relations we demonstrate that the inverse scattering method for KS models may be understood as a generalized Fourier transforms. Thus we have a tool to derive all their fundamental properties, including the hierarchy of equations and the hierarchy of their Hamiltonian structures.

Kulish-Sklyanin type models: integrability and reductions

We start with a Riemann–Hilbert problem (RHP) related to BD.I-type symmetric spaces SO(2r + 1)/S(O(2r − 2s+1) ⊗ O(2s)), s ≥ 1. We consider two RHPs: the first is formulated on the real axis R in the complex-λ plane; the second, on R ⊗ iR. The first RHP for s = 1 allows solving the Kulish–Sklyanin (KS) model; the second RHP is related to a new type of KS model. We consider an important example of nontrivial deep reductions of the KS model and show its effect on the scattering matrix. In particular, we obtain new two-component nonlinear Schrodinger equations. Finally, using the Wronski relations, we show that the inverse scattering method for KS models can be understood as generalized Fourier transforms. We thus find a way to characterize all the fundamental properties of KS models including the hierarchy of equations and the hierarchy of their Hamiltonian structures.

Kulish–Sklyanin-type models: Integrability and reductions

Multi-component integrable generalizations of the Fokas-Lenells equation, associated with each irreducible Hermitian symmetric space are formulated. Description of the underlying structures associated to the integrability, such as the Lax representation and the bi-Hamiltonian formulation of the equations is provided. Two reductions are considered as well, one of which leads to a nonlocal integrable model. Examples with Hermitian symmetric spaces of all classical series of types A.III, BD.I, C.I and D.III are presented in details, as well as possibilities for further reductions in a general form.

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Multicomponent Fokas–Lenells equations on Hermitian symmetric spaces

Multicomponent Fokas-Lenells equations on Hermitian symmetric spaces

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Модели типа Кулиша - Склянина: интегрируемость и редукции@@@Kulish - Sklyanin-type models: Integrability and reductions

We study MNLS related to the DIII-type symmetric spaces Applying to them Mikhailov reduction groups of the type $\mathbb{Z}_r\times \mathbb{Z}_2$ we derive new types of 2-component NLS equations These are {\bf not} counterexamples to the Zakharov-Schulman theorem because the corresponding interaction Hamiltonians depend not only on $|q_k|^2$, but also on $q_1q_2^* +q_1^* q_2$

Alexander A. Stefanov

Papers

New types of two component NLS-type equations

Real Hamiltonian forms of affine Toda field theories: Spectral aspects