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Dynamics of waves and multidimensional solitons of the Zakharov–Kuznetsov equation: II. Higher-order calculation; nonlinear development

The Zakharov–Kuznetsov equation describes the propagation of weak ion acoustic waves in a strongly magnetized plasma. Their dynamics have been studied in a series of papers, one of which gives growth rates of instabilities found numerically, as well as pictures of soliton collisions [J. Plasma Phys.64, 397 (2000) – Part I]. In the present paper, we find good approximate formulas for growth rates of the dominant instability, vastly improving those of Part I. This is done by proceeding to higher order in the expansion, combined with an incorporation of exact values for the boundaries of the unstable region in the formulas. The result is better than we had any right to expect. We next depart from linear stability analysis and look at nonlinear dynamics to obtain a pulse in time. The maximum amplitude of this pulse is seen to be proportional to the linear growth rate, a result that was so far suspected from numerics but not derived theoretically. (This paper can be read independently of Part I.)