Dispersionless equation

Abstract: We derive a new completely integrable dispersive shallow water equation that is bi-Hamiltonian and thus possesses an infinite number of conservation laws in involution. The equation is obtained by using an asymptotic expansion directly in the Hamiltonian for Euler's equations in the shallow water regime. The soliton solution for this equation has a limiting form that has a discontinuity in the first derivative at its peak.

Abstract: A method of analysis of the infinite‐dimensional Hamiltonian equations which avoids the introduction of the Backlund transformation or the use of the Lax equation is suggested. This analysis is based on the possibility of connecting in several ways the conservation laws of special Hamiltonian equations with their symmetries by using symplectic operators. It leads to a simple and sufficiently general model of integrable Hamiltonian equation, of which the Korteweg–de Vries equation, the modified Korteweg–de Vries equation, the nonlinear Schrodinger equation and the so‐called Harry Dym equation turn out to be particular examples.

Abstract: In recent years two nonlinear dispersive partial differential equations have attracted a lot of attention due to their integrable structure. We prove that both equations arise in the modeling of the propagation of shallow water waves over a flat bed. The equations capture stronger nonlinear effects than the classical nonlinear dispersive Benjamin-Bona-Mahoney and Korteweg-de Vries equations. In particular, they accomodate wave breaking phenomena.

Abstract: 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 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.