DETERMINATION OF THE GROWTH-RATE FOR THE LINEARIZED ZAKHAROV-KUZNETSOV EQUATION

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 gamma = gamma(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 dgamma/dk at k = 0. 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 gamma about k = 0 and the other zero at k2 = 5. By combining these results in the form of a two-point Pade approximant, we obtain an analytical expression for gamma 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 Pade approximant gives excellent agreement with the numerical results.