• JF McKenzie
  • TB Doyle


Nonlinear stationary waveforms in two completely different systems, namely, electromagnetic-fluid waves in a magnetic plasma and capillary-gravity water waves, are compared and contrasted. These systems display common features and are amenable to a Hamiltonian description. More importantly, however, is the fact that the linear dispersion equations for these systems possess the property that the phase speed has an extremum (a maximum in the case of the former and a minimum for the latter) at a certain wavelength. The wave-number of the associated stationary wave is, as a result, complex for wave speeds in excess (or less than) this extremum value, at which the phase and group velocities are equal. In its asymptotic state this gives rise to nonlinear waveforms with a hump (or depression) resembling a soliton on which oscillations on a finer spatial scale are superimposed. In the case of the former, “whistlers”, these waveforms, called “oscillitons”, have recently been extensively studied and a brief review is given here. In the case of the latter, “generalized  solitons”, a weakly nonlinear theory, based on a 5th order Korteweg-de Vries (KdeV) equation, is used. The resulting 4th order equation for stationary waves, and its first integral, show that “generalized solitons” (i.e. solitons with superimposed oscillations on a finer scale) of depression propagate at subcritical speeds for Bond numbers less than 1/3. The parameter space of the Froude number (F) and the Bond number  (τ) divides into regions exhibiting different waveforms, with lines F = 1, = τ = 1/3 and the curve Fm providing the boundaries of demarcation.

Quaestiones Mathematicae 34(2011), 377–391

Author Biographies

JF McKenzie
Astrophysics and Cosmology Research Unit, School of Mathematical Sciences, University of KwaZulu-Natal, Durban 4041, South Africa
TB Doyle
Materials Research Division, iThemba LABS, Somerset West 7129, South Africa

Journal Identifiers

eISSN: 1727-933X
print ISSN: 1607-3606