Symmetry groups of geodesic equations with applications in water waves

2024/08/14 by Bo Gervang, Gervang, Bo, Erwin Luesink +1
Earth and Planetary Sciences · Mathematics · Physics and Astronomy · #Aquatic and Environmental Studies #Differential Equations and Numerical Methods #FOS: Physical sciences #Mathematical Physics (math-ph) #Nonlinear Waves and Solitons

paper · pdf · doi:10.48550/arxiv.2408.07426

openalex publication_date 2024/08/14 · openalex created_date 2024/09/18 · openalex updated_date 2026/07/28

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.

Related

• The Camassa-Holm equation as a geodesic flow for the H¹ right-invariant metric
• Symmetry results for the solutions of a partial differential equation arising in water waves
• Euler equations on homogeneous spaces and Virasoro orbits
• Symmetries and Riemann Invariants
• Symmetric waves are traveling waves