Differential Galois Groups and Representation of Quivers for Seismic Models with Constant Hessian of Square of Slowness

TL;DR

This paper applies differential Galois theory and quiver representations to analyze the integrability of seismic models with constant Hessian of square of slowness, revealing abelian Galois groups for specific Hamiltonian systems.

Contribution

It introduces a novel application of differential Galois groups and quiver representations to seismic models, linking them to linear Hamiltonian systems with harmonic oscillators.

Findings

Identified abelian differential Galois groups for certain seismic models.

Represented these models using quivers to analyze their algebraic structure.

Showed equivalence to linear Hamiltonian systems with harmonic oscillators.

Abstract

The trajectory of energy is modeled by the solution of the Eikonal equation, which can be solved by solving a Hamiltonian system. This system is amenable of treatment from the point view of the theory of Differential Algebra. In particular, by Morales-Ramis theory it is possible to analyze integrable Hamiltonian systems through the abelian structure of their variational equations. In this paper we obtain the abelian differential Galois groups and the representation of the quiver, that allow us to obtain such abelian differential Galois groups, for some seismic models with constant Hessian of square of slowness, proposed in Cerveny 2001, which are equivalent to linear Hamiltonian systems with three uncoupled harmonic oscillators.