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Hamiltonian formulation of nonlinear travelling Whistler waves
1Institute of Geophysics and Planetary Physics, University of California Riverside, Riverside CA 92521, USA
2School of Physics and School of Mathematical and Statistical Sci., Univ. of KwaZulu-Natal, Durban, 4041, South Africa
3University of KwaZulu-Natal (Howard College), Durban, 4041, South Africa
4Max-Planck-Institute f¨ur Sonnensystemforschung, Katlenburg-Lindau, Germany
Abstract. A Hamiltonian formulation of nonlinear, parallel propagating, travelling whistler waves is developed. The complete system of equations reduces to two coupled differential equations for the transverse electron speed
and a phase variable
representing
the difference in the phases of the transverse complex velocities of the
protons and the electrons. Two integrals of the equations are obtained.
The Hamiltonian integral H, is used to classify the trajectories in the
phase plane, where 
and w=u2 are the canonical
coordinates. Periodic, oscilliton solitary wave and compacton solutions are
obtained, depending on the value of the Hamiltonian integral H and
the Alfvén Mach number M of the travelling wave. The second integral
of the equations of motion gives the position x in the travelling wave
frame as an elliptic integral. The dependence of the spatial period, L,
of the compacton and periodic solutions on the Hamiltonian integral H
and the Alfvén Mach number M is given in terms of complete elliptic
integrals of the first and second kind. A solitary wave solution, with an
embedded rotational discontinuity is obtained in which the transverse
Reynolds stresses of the electrons are balanced by equal and opposite
transverse stresses due to the protons. The individual electron and proton
phase variables 
and 
are determined in terms of
and 
. An alternative Hamiltonian formulation in which
is the new independent variable replacing x is
used to write the travelling wave solutions parametrically in terms of
Full Article (PDF, 791 KB)
Citation: Webb, G. M., McKenzie, J. F., Dubinin, E. M., and Sauer, K.: Hamiltonian formulation of nonlinear travelling Whistler waves, Nonlin. Processes Geophys., 12, 643-660, 2005. Bibtex EndNote Reference Manager
