Magnetization waves in Landau-Lifshitz Model

The solutions of the Landau-Lifshitz equation with finite-gap behavior at infinity are considered. By means of the inverse scattering method the large-time asymptotics is obtained.Comment: AmSTeX, Ver. 2.1h, 5 pages, amsppt style, 1 figur

Long-Time Asymptotics for Solutions of the NLS Equation with a Delta Potential and Even Initial Data

We consider the one-dimensional focusing nonlinear Schr\"odinger equation (NLS) with a delta potential and even initial data. The problem is equivalent to the solution of the initial/boundary problem for NLS on a half-line with Robin boundary conditions at the origin. We follow the method of Bikbaev and Tarasov which utilizes a B\"acklund transformation to extend the solution on the half-line to a solution of the NLS equation on the whole line. We study the asymptotic stability of the stationary 1-soliton solution of the equation under perturbation by applying the nonlinear steepest-descent method for Riemann-Hilbert problems introduced by Deift and Zhou. Our work strengthens, and extends, earlier work on the problem by Holmer and Zworski

Classical/quantum integrability in AdS/CFT

We discuss the AdS/CFT duality from the perspective of integrable systems and establish a direct relationship between the dimension of single trace local operators composed of two types of scalar fields in N=4 super Yang-Mills and the energy of their dual semiclassical string states in AdS(5) X S(5). The anomalous dimensions can be computed using a set of Bethe equations, which for ``long'' operators reduces to a Riemann-Hilbert problem. We develop a unified approach to the long wavelength Bethe equations, the classical ferromagnet and the classical string solutions in the SU(2) sector and present a general solution, governed by complex curves endowed with meromorphic differentials with integer periods. Using this solution we compute the anomalous dimensions of these long operators up to two loops and demonstrate that they agree with string-theory predictions.Comment: 49 pages, 5 figures, LaTeX; v2: complete proof of the two-loop equivalence between the sigma model and the gauge theory is added. References added; v4,v5,v6: misprints correcte

On the Convexity of the KdV Hamiltonian

We prove that the nonlinear part H 17 of the KdV Hamiltonian Hkdv, when expressed in action variables I=(In)n 651, extends to a real analytic function on the positive quadrant \u21132+(N) of \u21132(N) and is strictly concave near 0. As a consequence, the differential of H 17 defines a local diffeomorphism near 0 of \u21132C(N)