Water Waves and Integrability

Abstract

The Euler's equations describe the motion of inviscid fluid. In the case of shallow water, when a perturbative asymtotic expansion of the Euler's equations is taken (to a certain order of smallness of the scale parameters), relations to certain integrable equations emerge. Some recent results concerning the use of integrable equation in modeling the motion of shallow water waves are reviewed in this contribution.

LaTeX, 18 pages, 2 figures, Philos. Trans. Roy. Soc.: Ser. A, Theme Issue 'Water waves' (2007)

References in corpus (10)

• An integrable shallow water equation with peaked solitons
• Inverse Scattering Transform for the Camassa-Holm equation
• Multi-peakon solutions of the Degasperis-Procesi equation
• On asymptotically equivalent shallow water wave equations
• Perturbative Symmetry Approach
• Degasperis-Procesi peakons and the discrete cubic string
• The Camassa-Holm Equation: A Loop Group Approach
• Parametric Representation for the Multisoliton Solution of the Camassa-Holm Equation
• Lie Groups and mechanics: an introduction
• On the Field Equations of Kaluza's Theory

Cited by in corpus (20)

• Two component integrable systems modelling shallow water waves: the constant vorticity case
• Inverse Scattering Transform for the Degasperis-Procesi Equation
• The Degasperis-Procesi equation as a non-metric Euler equation
• Dressing Method for the Degasperis-Procesi Equation
• Two-component equations modelling water waves with constant vorticity
• The periodic b-equation and Euler equations on the circle
• On traveling waves in lattices: The case of Riccati lattices
• Conserved quantities, continuation and compactly supported solutions of some shallow water models
• Stability of smooth solitary waves in the b-Camassa-Holm equation
• Stability of stationary solutions for nonintegrable peakon equations
• One-dimensional weakly nonlinear model equations for Rossby waves
• The periodic $\mu$-$b$-equation and Euler equations on the circle
• A note on multi-dimensional Camassa-Holm type systems on the torus
• Higher-order integrable models for oceanic internal wave-current interactions
• Equations of the Camassa-Holm Hierarchy
• Similarity reductions of peakon equations: integrable cubic equations
• The two-dimensional periodic $b$-equation on the diffeomorphism group of the torus
• Hamiltonian models for the propagation of long gravity waves, higher-order KdV-type equations and integrability
• On the (Non)-Integrability of KdV Hierarchy with Self-consistent Sources
• Solitary Wave Solution of Flat Surface Internal Geophysical Waves with Vorticity