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

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 with 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 with the multipliers are positive up to lower-order terms.