Studies of the Zakharov—Kuznetsov equation governing solitons in a strongly magnetized ion-acoustic plasma indicate that a perturbed flat soliton is unstable and evolves into higher-dimensional solitons. The growth rate γ = γ(k) of a small sinusoidal perturbation of wavenumber k to a flat soliton has already been found numerically, and lengthy analytical work has given the value of We introduce a more direct analytical method in the form of an extension to the usual multiple-scale perturbation approach and use it to determine a consistent expansion of γ about k = 0 and the other zero at k2 = 5.By combining these results in the form of a two-point Padé approximant, we obtain an analytical expression for γ valid over the entire range of k for which the solution is unstable. We also present a very efficient numerical method for determining the growth rate curve to great accuracy. The Padé approximant gives excellent agreement with the numerical results.

By determining to first order the growth rate of a small, long-wavelength, perturbation to a Zakharov–Kuznetsov plane soliton moving at an angle α to the magnetic field, it has been found that such solitons are unstable for α α0( ∼ 38 °). To determine the stability for angles greater than α0, one needs the growth rate to higher order. The conventional approach generates a second-order growth rate that is singular at α = α0. We rigorously obtain an expression that is bounded at this point, by developing a method in which exponentially secular terms that arise are regrouped before their subsequent elimination. We then show that these solitons are unstable for all α, although the growth rate is small for α>α and goes to zero as α→½π. The relevant linearized equation is solved numerically, and excellent agreement between analytical and numerical results is obtained.

We present numerical results indicating that the application of a laminar shear flow to an autocatalytic reaction front can lead to an enhanced rate of reaction. This is a result of the formation of a nonplanar wave front propagating at a speed approaching that of the shear amplitude.

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