Hamiltonian formulation of nonlinear travelling Whistler waves

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 u and a phase variable φ=φp − φe 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 (φ,w) phase plane, where φ and w=u2 are the canonical coordinates. Periodic, oscilliton solitary wave and compacton solutions are obtained, depending on the value of the Hamiltonian integralH 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 φe and φp are determined in terms of φ and w. An alternative Hamiltonian formulation in which φ̃=φp+φe is the new independent variable replacing x is used to write the travelling wave solutions parametrically in terms of φ̃.