On a hierarchy of nonlinearly dispersive generalized Kortewega€“de Vries evolution equations

摘要

We propose a hierarchy of nonlinearly dispersive generalized Kortewega€“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 (a€?peakompactonsa€?) are presented. Kortewega€“de Vries equation, compact solitary waves, classical field theory, Lagrangian mechanics, Hamiltonian mechanics.