Geometry of KdV. I: Addition and the unimodular spectral classes
This is the first of three papers on the geometry of KdV. There is considered a foliation of an extensive function space into which all known invariant manifolds of KdV fit naturally as special leaves. In this paper two main new notions are introduced. The first of them is addition as a generalisation of the Darboux transformation of the Schrödinger operator \(Q=-D\) \(2+q(x)\) with infinitely differentiable real-valued potential vanishing rapidly at \(\pm \infty\) into Q-2D 2 ln \(e_ 1=Q\)-, where \(e_ 1\) is an eigenfunction. The second of them are unimodular spectral classes. The present article investigates the unimodular spectral class with special attention to the invariant \(\sqrt{\det dF(\lambda)}\), where dF(\(\lambda)\) is a (2\(\times 2)\) spectral weight.
scientific article; zbMATH DE number 4061806