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The Euler–Poincaré Equations and Semidirect Products with Applications to Continuum Theories
We study Euler–Poincare systems (i.e., the Lagrangian analogue of Lie–Poisson Hamiltonian systems) defined on semidirect product Lie algebras. We first give a derivation of the Euler–Poincare
A New Integrable Equation with Peakon Solutions
We consider a new partial differential equation recently obtained by Degasperis and Procesi using the method of asymptotic integrability; this equation has a form similar to the Camassa–Holm shallow
Nonlinear stability of fluid and plasma equilibria
The Liapunov method for establishing stability has been used in a variety of fluid and plasma problems. For nondissipative systems, this stability method is related to well-known energy principles. A
A New Integrable Shallow Water Equation
Publisher Summary This chapter discusses about a new integrable shallow water equation. Completely integrable nonlinear partial differential equations arise at various levels of approximation in
An integrable shallow water equation with linear and nonlinear dispersion.
We use asymptotic analysis and a near-identity normal form transformation from water wave theory to derive a 1+1 unidirectional nonlinear wave equation that combines the linear dispersion of the
The Three Dimensional Viscous Camassa–Holm Equations, and Their Relation to the Navier–Stokes Equations and Turbulence Theory
We show here the global, in time, regularity of the three dimensional viscous Camassa–Holm (Navier–Stokes-alpha) (NS-α) equations. We also provide estimates, in terms of the physical parameters of
Geometric Mechanics and Symmetry: From Finite to Infinite Dimensions
,by Darryl D. Holm, Tanya Schmah and Cristina Stoica, Oxford University Press,Oxford, 2009, xi + 515 pp., ISBN: 978-0-19-921290-3The purpose of the book is to provide the unifying viewpoint of
Wave Structure and Nonlinear Balances in a Family of Evolutionary PDEs
TLDR
The effects of the balance parameter b and the kernel g(x) on the solitary wave structures are studied and their interactions analytically for $\nu=0$ and numerically for small or zero viscosity are investigated.
Richardson number criterion for the nonlinear stability of three-dimensional stratified flow
With use of a method of Arnol'd, we derive the necessary and sufficient conditions for the formal stability of a parallel shear flow in a three-dimensional stratified fluid. When the local Richardson
S I ] 1 2 M ay 2 00 2 A New Integrable Equation with Peakon Solutions
We consider a new partial differential equation, of a similar form to the Camassa-Holm shallow water wave equation, which was recently obtained by Degasperis and Procesi using the method of
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