Incoming and disappearing solutions for Maxwell's equations

TL;DR

This paper demonstrates the absence of incoming spherical wave solutions for Maxwell's equations, constructs dissipative boundary conditions, and develops solutions that decay or vanish over time, making them invisible in scattering theory.

Contribution

It introduces new dissipative boundary conditions and constructs solutions that decay or disappear, contrasting Maxwell's equations with the free wave equation.

Findings

No incoming spherical wave solutions for Maxwell's equations.

Existence of solutions that decay exponentially over time.

Construction of solutions that are identically zero after a finite time.

Abstract

We prove that in contrast to the free wave equation in $\R^3$ there are no incoming solutions of Maxwell's equations in the form of spherical or modulated spherical waves. We construct solutions which are corrected by lower order incoming waves. With their aid, we construct dissipative boundary conditions and solutions to Maxwell's equations in the exterior of a sphere which decay exponentially as $t \to +\infty$. They are asymptotically disappearing. Disappearing solutions which are identically zero for $t \geq T > 0$ are constructed which satisfy maximal dissipative boundary conditions which depend on time $t$. Both types are invisible in scattering theory.