Symmetry groups of geodesic equations with applications in water waves

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.