Some uniqueness results for stationary solutions to the Maxwell-Born-Infeld field equations and their physical consequences

TL;DR

This paper proves uniqueness of stationary solutions to the Maxwell-Born-Infeld equations under certain conditions, clarifying physical implications such as the non-existence of source-free solitons and uniqueness of electrostatic or magnetostatic fields.

Contribution

It establishes new mathematical uniqueness results for stationary solutions of nonlinear electromagnetic equations, resolving previous conjectures and clarifying physical properties.

Findings

Source-free solitons moving below light speed do not exist.

Purely electrostatic or magnetostatic fields are uniquely determined by their sources.

Uniqueness holds for both Maxwell-Born-Infeld and simplified Maxwell-Born equations.

Abstract

Uniqueness results are established for time-independent finite-energy electromagnetic fields which solve the nonlinear Maxwell--Born--Infeld equations in boundary-free space under the condition that either the charge or current density vanishes. In addition, it is also shown that the simpler Maxwell--Born equations admit at most a unique stationary finite-energy electromagnetic field solution, without the above condition. In these theories of electromagnetism, the following physical consequences emerge: source-free field solitons moving at speeds less than the vacuum speed of light $c$ do not exits; any purely electrostatic (resp. magnetostatic) field is the unique stationary electromagnetic field for the same current-density-free (resp. charge-density-free) sources. Our results put to rest some interesting speculations in the recent physics literature.