Completeness of Derivatives of Squared Schrödinger Eigenfunctions and Explicit Solutions of the Linearized KdV Equation

Publisher Website
Google Scholar
Abstract
Explicit solutions to the Cauchy problem for the linearized KdV equation are constructed when the initial data is integrable. The method is analogous to the Fourier decomposition for a constant coefficient equation and uses the connection between the one-dimensional Schrödinger equation and the KdV equation, as discovered by Gardner, Greene, Kruskal and Miura [2]. An expansion theorem expressing any integrable function in terms of derivatives of squared Schrödinger (generalized) eigenfunctions is proved. These functions evolve according to the linearized KdV equation, hence the expansion of the initial data leads to a generalized solution of the linearized KdV equation. Under suitable restrictions on the initial data, the solution constructed is classical. The proof of the expansion theorem may be interpreted as the skew-adjoins analogue of the more familiar process of simultaneously diagonalizing two self-adjoins operators.

This publication has 10 references indexed in Scilit: