Solitary and Periodic Wave Solutions for Several Short Wave Model Equations

We study the periodic and solitary wave solutions to several short wave model equations arising from a so-called $\beta$-family equation for $\beta=1,2,4$. These are integrable cases which possess Lax pair and multi-soliton solutions. By phase plane analysis, either the loop or cuspon type solutions are predicted. Then, by introducing a hodograph, or reciprocal, transformation, a coupled system is derived for each $\beta$. Applying a travelling wave setting, we are able to find the periodic solutions exactly expressed in terms of Jacobi Elliptic functions. In the limiting cases of modulus k=1, they all converge to the known solitary waves

Yangian of the Strange Lie Superalgebra of Qn₋₁ Type, Drinfel'd Approach

The Yangian of the strange Lie superalgebras in Drinfel'd realization is defined. The current system generators and defining relations are described

Solutions to graded reflection equation of GL-type

We list solutions of the graded reflection equation associated with the fundamental vector representation of the quantum supergroup of GL-type.Comment: arXiv admin note: text overlap with arXiv:math/020429

Structure of the string R-matrix

By requiring invariance directly under the Yangian symmetry, we rederive Beisert's quantum R-matrix, in a form that carries explicit dependence on the representation labels, the braiding factors, and the spectral parameters u_i. In this way, we demonstrate that there exist a rewriting of its entries, such that the dependence on the spectral parameters is purely of difference form. Namely, the latter enter only in the combination u_1-u_2, as indicated by the shift automorphism of the Yangian. When recasted in this fashion, the entries exhibit a cleaner structure, which allows to spot new interesting relations among them. This permits to package them into a practical tensorial expression, where the non-diagonal entries are taken care by explicit combinations of symmetry algebra generators.Comment: 9 pages, LaTeX; typos correcte

Weakly coupled N=4 Super Yang-Mills and N=6 Chern-Simons theories from u(2|2) Yangian symmetry

In this paper we derive the universal R-matrix for the Yangian Y(u(2|2)), which is an abstract algebraic object leading to rational solutions of the Yang-Baxter equation on representations. We find that on the fundamental representation the universal R-matrix reduces to the standard rational R-matrix R = R_0(1 + P/u), where the scalar prefactor is surprisingly simple compared to prefactors one finds e.g. for sl(n) R-matrices. This leads precisely to the S-matrix giving the Bethe Ansatz of one-loop N = 4 Super Yang-Mills theory and two-loop N = 6 Chern-Simons theory.Comment: 16 page

Secret Symmetries in AdS/CFT

We discuss special quantum group (secret) symmetries of the integrable system associated to the AdS/CFT correspondence. These symmetries have by now been observed in a variety of forms, including the spectral problem, the boundary scattering problem, n-point amplitudes, the pure-spinor formulation and quantum affine deformations.Comment: 20 pages, pdfLaTeX; Submitted to the Proceedings of the Nordita program `Exact Results in Gauge-String Dualities'; Based on the talk presented by A.T., Nordita, 15 February 201

Two-loop Integrability of Planar N=6 Superconformal Chern-Simons Theory

Bethe ansatz equations have been proposed for the asymptotic spectral problem of AdS_4/CFT_3. This proposal assumes integrability, but the previous verification of weak-coupling integrability covered only the su(4) sector of the ABJM gauge theory. Here we derive the complete planar two-loop dilatation generator of N=6 superconformal Chern-Simons theory from osp(6|4) superconformal symmetry. For the osp(4|2) sector, we prove integrability through a Yangian construction. We argue that integrability extends to the full planar two-loop dilatation generator, confirming the applicability of the Bethe equations at weak coupling. Further confirmation follows from an analytic computation of the two-loop twist-one spectrum.Comment: 45 pages, v2: typos in (D.9) fixed, reference added, many small change