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The stability of obliquely-propagating solitary-wave solutions to Zakharov–Kuznetsov-type equations

In certain circumstances, small amplitude, weakly nonlinear ion-acoustic waves in a magnetized plasma are governed by a Zakharov–Kuznetsov equation or by a reduced form of the equation. Both equations have a plane solitary travelling-wave solution that propagates at an angle αto the magnetic field. The multiple-scale perturbation method developed by Allen and Rowlands is used to calculate the initial growth rate of a small, transverse, long-wavelength perturbation to these solitary-wave solutions. Previous results in the literature are corrected. A numerical determination of the growth rate is given. For k[mid ] secα[mid ][Lt ]1, where k is the wavenumber of the perturbation, there is excellent agreement between our analytical and numerical results.