Symmetry groups of geodesic equations with applications in water waves

TL;DR

This paper derives key water wave and liquid crystal equations as geodesic equations on infinite-dimensional groups, analyzes their symmetry groups, and explores improvements for the Hopf equation to aid numerical testing.

Contribution

It introduces a geometric framework for deriving important physical equations and analyzes their symmetry groups to improve their properties and applications.

Findings

Derived equations include Hopf, Camassa-Holm, Hunter-Saxton, and Korteweg-De Vries as geodesic equations.

Identified symmetry groups of these equations to facilitate benchmarking and testing of numerical methods.

Proposed metric and topological corrections to improve the behavior of the Hopf equation.

Abstract

In this work we derive several important equations in water waves and liquid crystals by deriving them as geodesic equations of right-invariant metrics on two infinite-dimensional groups. The equations we obtain this way are the Hopf (inviscid Burgers) equation, the Camassa-Holm equation, the Hunter-Saxton equation and the Korteweg-De Vries equation. We then study the symmetry groups of the equations themselves and show that one can improve the behaviour of the Hopf equation by metric and topological corrections. The symmetry groups of these equations can aid the benchmarking and testing of numerical methods.