We begin with a review of the antiferromagnetic spin 1/2 Heisenberg chain. In particular, we show that the model has particle-like excitations with spin 1/2, and we compute the exact bulk S matrix. We then review our recent work which generalizes these results. We first consider an integrable alternating spin 1/2 spin 1 chain. In addition to having excitations with spin 1/2, this model also has...

The matrix sine-Gordon theory, a matrix generalization of the well-known sine-Gordon theory, is studied. In particular, the A3-generalization where fields take value in SU(2) describes integrable deformations of conformal field theory corresponding to the coset SU(2)× SU(2)/SU(2). Various classical aspects of the matrix sine-Gordon theory are addressed. We find exact solutions, solitons and bre...

The first integral method is an efficient method for obtaining exact solutions of some nonlinear partial differential equations. This method can be applied to non integrable equations as well as to integrable ones. In this paper, the first integral method is used to construct exact solutions of the 2D Ginzburg-Landau equation.

For quantum integrable models with elliptic R-matrix, we construct the Baxter Q-operator in infinite-dimensional representations of the algebra of observables.

We describe the unconventional infinite-dimensional Hopf superalgebra related to the integrable S-matrix of the AdS/CFT correspondence, and discuss its typical and atypical representations.

We use the braid-monoid algebra to construct integrable mixed vertex models. The transfer matrix of a mixed SU (N) model is diagonalized by nested Bethe ansatz approach.

In these notes we review the method to construct integrable deformations of the compactified c = 1 bosonic string theory by primary fields (momentum or winding modes), developed recently in collaboration with S. Alexandrov and V. Kazakov. The method is based on the formulation of the string theory as a matrix model. The flows generated by either momentum or winding modes (but not both) are inte...

A standard-form Wadati-Konno-Ichikawa(WKI) type integrable hierarchy is derived from a corresponding matrix spectral problem associated with the Lie algebra sl(2, R). Each equation in resulting has bi-Hamiltonian structure furnished by trace identity. Then, higher grading affine algebraic construction of some special cases proposed. We also show that eneralized short pulse arises naturally nega...