Interface Problems for Dispersive equations

The interface problem for the linear Schr\"odinger equation in one-dimensional piecewise homogeneous domains is examined by providing an explicit solution in each domain. The location of the interfaces is known and the continuity of the wave function and a jump in their derivative at the interface are the only conditions imposed. The problem of two semi-infinite domains and that of two finite-sized domains are examined in detail. The problem and the method considered here extend that of an earlier paper by Deconinck, Pelloni and Sheils (2014). The dispersive nature of the problem presents additional difficulties that are addressed here.Comment: 18 pages, 6 figures. arXiv admin note: text overlap with arXiv:1402.3007, Studies in Applied Mathematics 201

Numerical inverse scattering for the Korteweg–-de Vries and modified Korteweg–-de Vries equations

Recent advances in the numerical solution of Riemann–Hilbert problems allow for the implementation of a Cauchy initial value problem solver for the Korteweg–de Vries equation (KdV) and the defocusing modified Korteweg–de Vries equation (mKdV), without any boundary approximation. Borrowing ideas from the method of nonlinear steepest descent, this method is demonstrated to be asymptotically accurate. The method is straightforward for the case of defocusing mKdV due to the lack of poles in the Riemann–Hilbert problem and the boundedness properties of the reflection coefficient. Solving KdV requires the introduction of poles in the Riemann–Hilbert problem and more complicated deformations. The introduction of a new deformation for KdV allows for the stable asymptotic computation of the solution in the entire (x, t)-plane. KdV and mKdV are dispersive equations and this method can fully capture the dispersion with spectral accuracy. Thus, this method can be used as a benchmarking tool for determining the effectiveness of future numerical methods designed to capture dispersion. This method can easily be adapted to other integrable equations with Riemann–Hilbert formulations, such as the nonlinear Schrödinger equation