Equations of Hydrodynamic Type

TL;DR

This paper explores the universal field equations of hydrodynamic type, revealing their connection to Lagrangian formulations and resurrecting Fourier analysis methods to understand their implicit solutions.

Contribution

It demonstrates that these equations arise from consistency conditions and introduces a Fourier analysis-based method to analyze their solutions.

Findings

Equations admit infinitely many Lagrangian formulations.

Fourier analysis helps conjecture the structure of solutions.

Solutions are implicit and can be studied using algebraic computation.

Abstract

The universal field equations introduced by the author and his collaborators, which admit infinitely many inequivalent Lagrangian formulations are shown to arise as consistency conditions for the existence of non-trivial solutions to the quasi-linear equations, called equations of hydrodynamic type by Novikov , Dubrovin and others. The solutions in closed form are only implicit. A method due to Stokes, which is in essence just Fourier Analysis is resurrected for application to those equations. With the benefit of algebraic computation facilities, this method, allows the general structure of power series solutions to be conjectured.