A group theoretical approach to hydrodynamics considers hydrodynamics to be the differential geometry of diffeomorphism groups. The principle of least action implies that the motion of a fluid is described by the geodesics on the group in the right-invariant Riemannian metric given by the kinetic energy. Investigation of the geometry and structure of such groups turns out to be useful for describing the global behavior of fluids for large time intervals.

Topological methods in hydrodynamics

http://authors.library.caltech.edu/19410/1/HoMaRa1998b.pdf

The Euler–Poincaré Equations and Semidirect Products with Applications to Continuum Theories

The peakons are peaked solitary wave solutions of a certain nonlinear dispersive equation that is a model in shallow water theory and the theory of hyperelastic rods We give a very simple proof of the orbital stability of the peakons in the H1 norm © 2000 John Wiley & Sons, Inc

Stability of peakons

On obtient des theoremes d'existence de solutions globales en temps et des resultats sur la formation (le signularites pour une equation qui modelise le phenomene des ondes de surface en eau peu profonde. La solution peut exploser uniquement sous la forme d'un deferlement. En utilisant le fait que cette equation d'ondes decrit le flot geodesique du groupe des diffeomorphismes de la droite verifiant certaines conditions asymptotiques a l'infini, muni d'une structure de variete riemannienne, on donne des conditions suffisantes sur la donnee initiale pour que la solution soit globale en temps ou bien qui impliquent un deferlement au bout d'un temps fini. Ces resultats se traduisent en terme de proprietes des geodesiques du groupe des diffeomorphismes.

/pdf/existence-of-permanent-and-breaking-waves-for-a-shallow-3p105hn7yk.pdf

Existence of permanent and breaking waves for a shallow water equation: a geometric approach

The Camassa–Holm equation has a number of constants of motion arising as eigenvalues of an associated spectral problem. We give a description of the spectral picture and discuss the scattering problem.

Gerard Misio łek

Papers

A shallow water equation as a geodesic flow on the Bott-Virasoro group