Born-Infeld Solitons and Existence & Non-uniqueness of Solutions to the Bj\"orling Problem

TL;DR

This paper explores Born-Infeld soliton surfaces as zero mean curvature surfaces, presents two methods to solve the Björling problem for them, and demonstrates potential non-uniqueness of solutions unlike minimal surfaces.

Contribution

It introduces two approaches to solve the Björling problem for Born-Infeld soliton surfaces and analyzes their solution uniqueness.

Findings

Solutions may not be unique, unlike minimal and maximal surfaces.

Conformal parameters for Born-Infeld soliton surfaces are derived.

Two methods for solving the Björling problem are presented.

Abstract

In this semi-expository article, we study Born-Infeld soliton surfaces as zero mean curvature surfaces and derive conformal parameters for them. Then we present two approaches to solve the Bj\"orling problem for such surfaces, one of them treating them as time-like minimal surfaces and the other one using the Barbashov-Chernikov representation. Finally, we show that the solution to the Bj\"orling problem may not be unique unlike minimal and maximal surfaces.