Enlargement of Lie super algebra B(0, 1) was given firstly. Then nonlinear super integrable couplings of the super classical Boussinesq hierarchy based upon this enlarged matrix Lie super algebra were constructed secondly. And its super Hamiltonian structures were established by using super trace identity thirdly. As its reduction, special integrable couplings of classical Boussinesq hierarchy ...

We apply a version of the dressing method to a system of four dimensional nonlinear Partial Differential Equations (PDEs), which contains both Pohlmeyer equation (i.e. nonlinear PDE integrable by the Inverse Spectral Transform Method) and nonlinear matrix PDE integrable by the method of characteristics as particular reductions. Some other reductions are suggested.

We propose a construction of 1+1 integrable Heisenberg-Landau-Lifshitz type equations in the ${\rm gl}_N$ case. The dynamical variables are matrix elements $N\times N$ $S$ with property $S^2={\rm const}\cdot S$. Lax pair spectral parameter is constructed by means quantum $R$-matrix satisfying associative Yang-Baxter equation. Equations motion for Landau-Lifshitz model derived from Zakharov-Shab...

– Many models for chaotic systems consist of joining two integrable systems with incompatible constants of motion. The quantum counterparts of such models have a propagator which factorizes into two integrable parts. Each part can be diagonalized. The two eigenvector bases are related by an orthogonal (or unitary) transformation. We construct a random matrix ensemble that mimics this situation ...

The integrable system is introduced based on the Poisson rs-matrix structure. This is a generalization of the Gaudin magnet, and in SL(2) case isomorphic to the generalized Neumann model. The separation of variables is discussed for both classical and quantum case.

We construct the Darboux transformation with Dihedral reduction group for the 2–dimensional generalisation of the periodic Volterra lattice. The resulting Bäcklund transformation can be viewed as a nonevolutionary integrable differential difference equation. We also find its generalised symmetry and the Lax representation for this symmetry. Using formal diagonalisation of the Darboux matrix we ...

We give a reinterpretation of the matrix theory discussed by Moore, Nekrasov and Shatashivili (MNS) in terms of the second quantized operators which describes the homology class of the Hilbert scheme of points on surfaces. It naturally relate the contribution from each pole to the inner product of orthogonal basis of free boson Fock space. These basis can be related to the eigenfunctions of Cal...

In the space ?d, d ? 2, we study diffusion equation ?div (?u? +b(x/?)?u?)+u? = f ?L2(?d), u?H1(?d), where b(y) is an unbounded 1-periodic skew symmetric matrix and ? a small parameter. The assumed to be integrable with respect period exponent s, s for 3 > 2 2. Assuming that solution not necessarily unique, find asymptotics so-called approximate resolvent remainder of order ?2 as ? 0.

The Schrödinger operators with matrix rational potential, which are D-integrable, i.e. can be intertwined with the pure Laplacian, are investigated. Corresponding potentials are uniquely determined by their singular data which is a configuration of the hyperplanes in C with prescribed matrices. We describe some algebraic conditions (matrix locus equations) on these data, which are sufficient fo...

We present an inverse scattering approach to defects in classical integrable field theories. Integrability is proved systematically by constructing the generating function of the infinite set of modified integrals of motion. The contribution of the defect to all orders is explicitely identified in terms of a defect matrix. The underlying geometric picture is that those defects correspond to Bäc...