134 research outputs found
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 be the genus of the hyperelliptic Riemann surface
associated with the unperturbed solution. We show that the -pane contains
areas where the perturbed solution is close to a quasi-periodic solution
in the same isospectral torus. In between there are 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 () 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
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