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

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

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

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

Properties of the Scattering Matrix and Dispersion Estimates for Jacobi Operators

We show that for a Jacobi operator with coefficients whose (j+1)'th moments are summable the j'th derivative of the scattering matrix is in the Wiener algebra of functions with summable Fourier coefficients. We use this result to improve the known dispersive estimates with integrable time decay for the time dependent Jacobi equation in the resonant case.Comment: 10 page

Scattering Theory for Jacobi Operators with Quasi-Periodic Background

We develop direct and inverse scattering theory for Jacobi operators which are short range perturbations of quasi-periodic finite-gap operators. We show existence of transformation operators, investigate their properties, derive the corresponding Gel'fand-Levitan-Marchenko equation, and find minimal scattering data which determine the perturbed operator uniquely.Comment: 29 page

Scattering Theory for Jacobi Operators with General Steplike Quasi-Periodic Background

We develop direct and inverse scattering theory for Jacobi operators with steplike coefficients which are asymptotically close to different finite-gap quasi-periodic coefficients on different sides. We give a complete characterization of the scattering data, which allow unique solvability of the inverse scattering problem in the class of perturbations with finite first moment.Comment: 23 page

On the form of dispersive shock waves of the Korteweg-de Vries equation

We show that the long-time behavior of solutions to the Korteweg-de Vries shock problem can be described as a slowly modulated one-gap solution in the dispersive shock region. The modulus of the elliptic function (i.e., the spectrum of the underlying Schr\"odinger operator) depends only on the size of the step of the initial data and on the direction, $\frac{x}{t}=const.$, along which we determine the asymptotic behavior of the solution. In turn, the phase shift (i.e., the Dirichlet spectrum) in this elliptic function depends also on the scattering data, and is computed explicitly via the Jacobi inversion problem.Comment: 11 page

Dispersion Estimates for One-dimensional Discrete Schr\"odinger and Wave Equations

We derive dispersion estimates for solutions of the one-dimensional discrete perturbed Schr\"odinger and wave equations. In particular, we improve upon previous works and weaken the conditions on the potentials. To this end we also provide new results concerning scattering for one-dimensional discrete perturbed Schr\"odinger operators which are of independent interest. Most notably we show that the reflection and transmission coefficients belong to the Wiener algebra.Comment: 23 page