On a hierarchy of nonlinearly dispersive generalized KdV equations

TL;DR

This paper introduces a hierarchy of nonlinear dispersive equations extending KdV, featuring compact and peaked solutions called peakompactons, with implications for wave propagation in generalized continua.

Contribution

It develops a new hierarchy of equations based on a modified Lagrangian, unifies recent models, and reveals novel peaked, compact solutions with Hamiltonian structure.

Findings

Equations admit peaked, compact traveling wave solutions.

Solutions include peakompactons with explicit implicit forms.

Hierarchy encompasses recent models of wave propagation in generalized media.

Abstract

We propose a hierarchy of nonlinearly dispersive generalized Korteweg--de Vries (KdV) evolution equations based on a modification of the Lagrangian density whose induced action functional the KdV equation extremizes. It is shown that two recent nonlinear evolution equations describing wave propagation in certain generalized continua with an inherent material length scale are members of the proposed hierarchy. Like KdV, the equations from the proposed hierarchy possess Hamiltonian structure. Unlike KdV, however, the solutions to these equations can be compact (i.e., they vanish outside of some open interval) and, in addition, peaked. Implicit solutions for these peaked, compact traveling waves ("peakompactons") are presented.