Global stability of the plane wave solutions to the relativistic string with non-small perturbations

TL;DR

This paper proves the global stability of plane wave solutions to the relativistic string equation under certain decay conditions, allowing large traveling perturbations but requiring small transversal perturbations, using geometric energy methods.

Contribution

It introduces a gauge and energy framework to establish stability of non-trivial background solutions with large traveling perturbations.

Findings

Global smooth solutions exist under specified decay assumptions.

Energy estimates remain positive despite linear perturbations with undetermined signs.

The approach handles non-trivial backgrounds with robust geometric methods.

Abstract

This paper is concerned with the global stability of the plane wave solutions to the relativistic string equation with non-small perturbations. Under certain decay assumptions on the plane wave, we conclude that the perturbed system admits a globally smooth solution if the perturbation along the transversal direction is sufficiently small, while the travelling direction is allowed to be large. By choosing a gauge adapted to the plane wave solution, we deduce an equivalent Euler-Lagrangian equation for the perturbation whose quasilinear structure is reflected precisely in the induced geometry of the relativistic string. It then helps to proceed a geometrically adapted and weighted energy argument for which robust estimates suffice. Moreover, due to the non-trivial background solutions, the induced metric of the relativistic string involves linear perturbations with undetermined signs, and hence a key observation is needed to guarantee that the energies associated to the multipliers are positive up to lower order terms. %rather than quadratic perturbations as in the case of trivial background solutions.