A globally smooth solution to the relativistic string equation

TL;DR

This paper proves the global existence of smooth solutions to the relativistic string equation for certain large initial data configurations, extending previous results to a broader, non-small data class.

Contribution

It generalizes prior work by establishing global solutions for non-small data in a quasilinear setting, allowing large wave components in one direction.

Findings

Global smooth solutions exist for large data configurations.

Solutions can be viewed as non-small perturbations of plane waves.

The result extends previous small-data theories to broader initial conditions.

Abstract

We prove the global existence of smooth solution to the relativistic string equation in a class of data that is not small. Our solution admits the feature that the right-travelling wave can be large and the left-travelling wave is sufficiently small, and vice versa. In particular, the large-size solution exists in the whole space, instead of a null strip arising from the short pulse data. This generalizes the result of Liuli-Yang-Yu (Adv. Math. 2018) to the quasilinear setting with non-small data. In addition, in our companion paper, we are able to show the global solution here can also be seen as the non-small perturbations of the plane wave solutions.