RT journal article
T1 A general family of multi-peakon equations and their properties
A1 Anco, Stephen C.
A1 Recio Rodríguez, Elena
A2 Matemáticas
K1 peakon
K1 multi-peakon
K1 nonlinear dispersive wave equation
AB A general family of peakon equations is introduced, involving two arbitraryfunctions of the wave amplitude and the wave gradient. This family containsall of the known breaking wave equations, including the integrable ones:Camassa–Holm equation, Degasperis–Procesi equation, Novikov equation,and FORQ/modified Camassa–Holm equation. One main result is to showthat all of the equations in the general family possess weak solutions givenby multi-peakons which are a linear superposition of peakons with timedependentamplitudes and positions. In particular, neither an integrabilitystructure nor a Hamiltonian structure is needed to derive N-peakon weaksolutions for arbitrary N > 1. As a further result, single peakon travellingwavesolutions are shown to exist under a simple condition on one of the twoarbitrary functions in the general family of equations, and when this conditionfails, generalized single peakon solutions that have a time-dependent amplitudeand a time-dependent speed are shown to exist. An interesting generalizationof the Camassa–Holm and FORQ/modified Camassa–Holm equations isobtained by deriving the most general subfamily of peakon equations thatpossess the Hamiltonian structure shared by the Camassa–Holm and FORQ/modified Camassa–Holm equations. Peakon travelling-wave solutions andtheir features, including a variational formulation (minimizer problem), arederived for these generalized equations. A final main result is that two-peakonweak solutions are investigated and shown to exhibit several novel kinds ofbehaviour, including the formation of a bound pair consisting of a peakon andan anti-peakon that have a maximum finite separation.
SN 1751-8121
YR 2019
FD 2019
LK http://hdl.handle.net/10498/33281
UL http://hdl.handle.net/10498/33281
LA eng
DS Repositorio Institucional de la Universidad de Cádiz
RD 10-may-2026