The Miura Map on the Line
Spectral Theory
2007-05-23 v1 Analysis of PDEs
Abstract
The Miura map (introduced by Miura) is a nonlinear map between function spaces which transforms smooth solutions of the modified Korteweg - de Vries equation (mKdV) to solutions of the Korteweg - de Vries equation (KdV). In this paper we study relations between the Miura map and Schroedinger operators with real-valued distributional potentials (possibly not decaying at infinity) from various spaces. We also investigate mapping properties of the Miura map in these spaces. As an application we prove existence of solutions of the Korteweg - de Vries equation in the negative Sobolev space H^{-1} for the initial data in the range of the Miura map.
Cite
@article{arxiv.math/0506411,
title = {The Miura Map on the Line},
author = {Thomas Kappeler and Peter Perry and Mikhail Shubin and Peter Topalov},
journal= {arXiv preprint arXiv:math/0506411},
year = {2007}
}
Comments
33 pages