A New Integrable Shallow Water Equation

Abstract

This chapter discusses about a new integrable shallow water equation. Completely integrable nonlinear partial differential equations arise at various levels of approximation in shallow water theory. Such equations possess soliton solutions-coherent (spatially localized) structures that interact nonlinearly among themselves and then re-emerge, retaining their identity and showing particle-like scattering behavior. This chapter discusses a newly discovered, completely integrable dispersive shallow-water equation found by Camassa and Holm in 1993. This equation is obtained by using a small-wave-amplitude asymptotic expansion directly in the Hamiltonian for the vertically averaged incompressible Euler's equations, after substituting a solution ansatz of columnar fluid motion and restricting to an invariant manifold for unidirectional motion of waves at the free surface under the influence of gravity. Section II of the chapter derives the one-dimensional Green–Naghdi equations. Section III uses Hamiltonian methods to newly discovered equation for unidirectional waves. Section IV analyzes the behavior of the solutions of the equation and shows that certain initial conditions develop a vertical slope in finite time. It is also shown that there exist stable multisoliton solutions. Section V demonstrates the existence of an infinite number of conservation laws for the equation that follow from its bi-Hamiltonian property.