$L^2$-stableness for solution to linearized KdV equation

TL;DR

This paper investigates the $L^2$-stability of the linearized KdV equation by analyzing a transformed nonsymmetric operator, proving it has no eigenvalues, which implies stability properties.

Contribution

It introduces a novel unitary transformation approach to analyze the $L^2$-stability of the linearized KdV operator and proves the absence of eigenvalues for this operator.

Findings

The transformed operator is nonsymmetric but has no eigenvalues.

The analysis provides smoothing estimates for the operator.

The approach advances understanding of stability in linearized KdV equations.

Abstract

The linearized Korteweg-De Vries equation can be written as a Hamilton-like system. However, the Hamilton energy depends on the time, and is a nonsymmetric operator on $L^2({\bf R})$. By performing suitable unitary transforms on the Hamilton energy, we can reduce this operator into one that is not independent on the time but nonsymmetric. In this study, we consider the $L^2$-stability issues and smoothing estimates for this operator, and prove that it has no eigenvalues.