NPG Nonlinear Processes in Geophysics NPG Nonlin. Processes Geophys. 1607-7946 Copernicus Publications Göttingen, Germany 10.5194/npg-12-643-2005 Hamiltonian formulation of nonlinear travelling Whistler waves Webb G. M. 1 McKenzie J. F. 3 2 1 Dubinin E. M. 4 Sauer K. 4 Institute of Geophysics and Planetary Physics, University of California Riverside, Riverside CA 92521, USA School of Physics and School of Mathematical and Statistical Sci., Univ. of KwaZulu-Natal, Durban, 4041, South Africa University of KwaZulu-Natal (Howard College), Durban, 4041, South Africa Max-Planck-Institute f¨ur Sonnensystemforschung, Katlenburg-Lindau, Germany 22 06 2005 12 5 643 660 Copyright: © 2005 G. M. Webb et al. 2005 This work is licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 2.5 Generic License. To view a copy of this licence, visit https://creativecommons.org/licenses/by-nc-sa/2.5/ This article is available from https://npg.copernicus.org/articles/12/643/2005/npg-12-643-2005.html The full text article is available as a PDF file from https://npg.copernicus.org/articles/12/643/2005/npg-12-643-2005.pdf

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 <IMG WIDTH="12" HEIGHT="13" ALIGN="BOTTOM" BORDER="0" src="npg-12-643-img1.gif" ALT="$u$"> and a phase variable <!-- MATH $\phi{=}\phi_p-\phi_e$ --> <IMG WIDTH="77" HEIGHT="30" ALIGN="MIDDLE" BORDER="0" src="npg-12-643-img2.gif" ALT="$phi{=}phi_p-phi_e$"> 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 <i>H</i>, is used to classify the trajectories in the <IMG WIDTH="44" HEIGHT="32" ALIGN="MIDDLE" BORDER="0" src="npg-12-643-img4.gif" ALT="$(phi,w)$"> phase plane, where <IMG WIDTH="13" HEIGHT="30" ALIGN="MIDDLE" BORDER="0" src="npg-12-643-img5.gif" ALT="$phi$"> and <i>w=u<sup>2</sup></i> are the canonical coordinates. Periodic, oscilliton solitary wave and compacton solutions are obtained, depending on the value of the Hamiltonian integral <i>H</i> and the Alfv&#233;n Mach number <i>M</i> of the travelling wave. The second integral of the equations of motion gives the position <i>x</i> in the travelling wave frame as an elliptic integral. The dependence of the spatial period, <i>L</i>, of the compacton and periodic solutions on the Hamiltonian integral <i>H</i> and the Alfv&#233;n Mach number <i>M</i> 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 <IMG WIDTH="19" HEIGHT="30" ALIGN="MIDDLE" BORDER="0" src="npg-12-643-img10.gif" ALT="$phi_e$"> and <IMG WIDTH="20" HEIGHT="30" ALIGN="MIDDLE" BORDER="0" src="npg-12-643-img11.gif" ALT="$phi_p$"> are determined in terms of <IMG WIDTH="13" HEIGHT="30" ALIGN="MIDDLE" BORDER="0" src="npg-12-643-img5.gif" ALT="$phi$"> and <IMG WIDTH="15" HEIGHT="13" ALIGN="BOTTOM" BORDER="0" src="npg-12-643-img12.gif" ALT="$w$">. An alternative Hamiltonian formulation in which <!-- MATH ${\tilde\phi}{=}\phi_p+\phi_e$ --> <IMG WIDTH="77" HEIGHT="38" ALIGN="MIDDLE" BORDER="0" src="npg-12-643-img13.gif" ALT="${tildephi}{=}phi_p+phi_e$"> is the new independent variable replacing <i>x</i> is used to write the travelling wave solutions parametrically in terms of <IMG WIDTH="13" HEIGHT="38" ALIGN="MIDDLE" BORDER="0" src="npg-12-643-img14.gif">.