Dynamics of waves and multidimensional solitons of the Zakharov-Kuznetsov equation

摘要

Nonlinear waves and one-dimensional solitons of the Zakharov-Kuznetsov equation are unstable in two dimensions. Although the wavevector K of a perturbation leading to an instability covers a whole region in (K_x, K_y) parameter space, two classes are of particular interest. One corresponds to the perpendicular, Benjamin-Feir instability (K_x = 0). The second is the wavelength-doubling instability. These two are the only purely growing modes. We concentrate on them. Both analytical and numerical methods for calculating growth rates are employed and results compared. Once a nonlinear wave or soliton breaks up owing to one of these instabilities, an array of cylindrical and/or spherical solitons can emerge. We investigate the interaction of these entities numerically.