On the Analyticity of the Spectral Density for Semiclassical NLS

In a previous work, we have analyzed the semiclassical behavior of solutions to the focusing,completely integrable nonlinear Schroedinger equation, under the assumption of real analytic initial data (among others). We have provided global semiclassical asymptotics under the so-called ”finite gap” assumption. In a subsequent paper, we have justified the ”finite gap” assumption, again under several assumptions, the main assumption being that the limiting spectral density of the eigenvalues of the associated Dirac operator has an analytic extension in the upper half-plane. In the present article, we show that this constraint is unnecessary. In fact, analyticity of the neccessary quantities in the analysis can be recovered via the solution of a scalar Riemann-Hilbert problem

Asymptotics via Steepest Descent for an Operator Riemann-Hilbert Problem

In this paper, we take the first step towards an extension of the nonlinear steepest descent method of Deift, Its and Zhou to the case of operator Riemann-Hilbert problems. In particular, we provide long range asymptotics for a Fredholm determinant arising in the computation of the probability of finding a string of n adjacent parallel spins up in the antiferromagnetic ground state of the spin 1/2 XXX Heisenberg Chain. Such a determinant can be expressed in terms of the solution of an operator Riemann-Hilbert factorization problem

Stability of Periodic Soliton Equations under Short Range Perturbations

We consider the stability of (quasi-)periodic solutions of soliton equations under short range perturbations and give a complete description of the long time asymptotics in this situation. We show that, apart from the phenomenon of the solitons travelling on the quasi-periodic background, the perturbed solution asymptotically approaches a modulated solution. We use the Toda lattice as a model but the same methods and ideas are applicable to all soliton equations in one space dimension. More precisely, let $g$ be the genus of the hyperelliptic Riemann surface associated with the unperturbed solution. We show that the $n/t$-pane contains $g+2$ areas where the perturbed solution is close to a quasi-periodic solution in the same isospectral torus. In between there are $g+1$ regions where the perturbed solution is asymptotically close to a modulated lattice which undergoes a continuous phase transition (in the Jacobian variety) and which interpolates between these isospectral solutions. In the special case of the free solution ($g=0$) the isospectral torus consists of just one point and we recover the classical result. Both the solutions in the isospectral torus and the phase transition are explicitly characterized in terms of Abelian integrals on the underlying hyperelliptic Riemann surface.Comment: 4 pages, 2 figure