Soliton Solutions of the Toda Hierarchy on Quasi-Periodic Backgrounds Revisited

We investigate soliton solutions of the Toda hierarchy on a quasi-periodic finite-gap background by means of the double commutation method and the inverse scattering transform. In particular, we compute the phase shift caused by a soliton on a quasi-periodic finite-gap background. Furthermore, we consider short range perturbations via scattering theory. We give a full description of the effect of the double commutation method on the scattering data and establish the inverse scattering transform in this setting.Comment: 16 page

On the Cauchy Problem for the modified Korteweg-de Vries Equation with Steplike Finite-Gap Initial Data

We solve the Cauchy problem for the modified Korteweg--de Vries equation with steplike quasi-periodic, finite-gap initial conditions under the assumption that the perturbations have a given number of derivatives and moments finite.Comment: 8 page

A Paley-Wiener Theorem for Periodic Scattering with Applications to the Korteweg-de Vries Equation

Consider a one-dimensional Schroedinger operator which is a short range perturbation of a finite-gap operator. We give necessary and sufficient conditions on the left, right reflection coefficient such that the difference of the potentials has finite support to the left, right, respectively. Moreover, we apply these results to show a unique continuation type result for solutions of the Korteweg-de Vries equation in this context. By virtue of the Miura transform an analogous result for the modified Korteweg-de Vries equation is also obtained.Comment: 10 page

Reconstruction of the Transmission Coefficient for Steplike Finite-Gap Backgrounds

We consider scattering theory for one-dimensional Jacobi operators with respect to steplike quasi-periodic finite-gap backgrounds and show how the transmission coefficient can be reconstructed from minimal scattering data. This generalizes the Poisson-Jensen formula for the classical constant background case.Comment: 9 page

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

Long-Time Asymptotics for the Toda Shock Problem: Non-Overlapping Spectra

We derive the long-time asymptotics for the Toda shock problem using the nonlinear steepest descent analysis for oscillatory Riemann--Hilbert factorization problems. We show that the half plane of space/time variables splits into five main regions: The two regions far outside where the solution is close to free backgrounds. The middle region, where the solution can be asymptotically described by a two band solution, and two regions separating them, where the solution is asymptotically given by a slowly modulated two band solution. In particular, the form of this solution in the separating regions verifies a conjecture from Venakides, Deift, and Oba from 1991.Comment: 39 page