Dispersion estimates for the boundary integral operator associated with the fourth order Schr\"odinger equation posed on the half line

TL;DR

This paper establishes dispersion estimates for the boundary integral operator linked to the fourth order Schrödinger equation on the half line, using a straightforward approach based on the Fokas method, which can extend to other higher order PDEs.

Contribution

It introduces a simple, uniform method for deriving dispersion estimates for boundary integral operators associated with higher order PDEs, expanding the analytical toolkit for such problems.

Findings

Dispersion estimates are proved for the boundary integral operator.

The method is simple, uniform, and adaptable to other higher order PDEs.

The approach avoids complex technical procedures typical in boundary domain estimates.

Abstract

In this paper, we prove dispersion estimates for the boundary integral operator associated with the fourth order Schr\"odinger equation posed on the half line. Proofs of such estimates for domains with boundaries are rare and generally require highly technical approaches, as opposed to our simple treatment which is based on constructing a boundary integral operator of oscillatory nature via the Fokas method. Our method is uniform and can be extended to other higher order partial differential equations where the main equation possibly involves more than one spatial derivatives.