Spatio-temporal linear instability analysis for arbitrary dispersion relations on the Lefschetz thimble in multidimensional spacetime

TL;DR

This paper introduces a new, mathematically rigorous formalism using the Lefschetz thimble method to analyze linear stability in multidimensional spacetime, overcoming limitations of classical approaches for complex problems.

Contribution

The study develops a novel formalism based on Lefschetz thimbles for explicit asymptotic analysis of linear perturbations in higher-dimensional PDEs, enhancing practical applicability.

Findings

Provides explicit asymptotic expressions for linear perturbations

Improves mathematical rigor over classical stability theory

Applicable to complex, realistic physical problems

Abstract

In linear stability analysis of field quantities described by partial differential equations, the well-established classical theory is all but impossible to apply to concrete problems in its entirety even for uniform backgrounds when the spatial dimension is more than 1. In this study, using the Lefschetz thimble method, we develop a new formalism to give an explicit expression to the asymptotic behavior of linear perturbations. It is not only more mathematically rigorous than the previous theory but also useful practically in its applications to realistic problems, and, as such, has an impact on broad subjects in physics.