Spectral Properties and Time Decay for an Airy Operator with Potential

The linearization of the Korteweg–de Vries equation about a solitary wave is studied with respect to algebraic spatial weights. The decay rate of a solution to the linearized equation is shown to depend on the degree of its initial spatial localization. The operator has the entire imaginary axis as its spectrum, so that special techniques are required to invert the Laplace transform and thereby obtain decay estimates.