Quasi-periodic relativistic strings in the Minkowski space $\textbf{R}^{1+n}$

TL;DR

This paper proves the existence of generalized time quasi-periodic solutions for relativistic strings in Minkowski space, modeled by nonlinear wave equations of Born-Infeld type, expanding understanding of their dynamic behaviors.

Contribution

It introduces a Nash-Moser iteration scheme to establish the existence of quasi-periodic solutions for relativistic strings in Minkowski space, a novel approach in this context.

Findings

Existence of time quasi-periodic solutions for relativistic strings.

Construction of a Nash-Moser iteration scheme for nonlinear wave equations.

Solutions are timelike and exhibit generalized quasi-periodic motion.

Abstract

In this article we consider the motion of relativistic strings in the Minkowski space $\textbf{R}^{1+n}$. Those surfaces are known as a timelike minimal surface, and described by a system with $n$ nonlinear wave equations of Born-Infeld type. By constructing a suitable Nash-Moser iteration scheme, we prove that the $n$-dimensional relativistic strings can admit a more generalized time quasi-periodic motion in $\textbf{R}^{1+n}$. Moreover, those time quasi-periodic solutions are also timelike solutions.