Interacting localized solutions for the Zakharov–Kusnetsov equation

Abstract

Analytical investigation of the Zakharov–Kusnetsov equation shows the existence of approximate interacting localized solutions. Using the asymptotic perturbation method, based on Fourier expansion and spatio-temporal rescaling, it is found that the amplitude slow modulation of Fourier modes is described by a C-integrable (solvable via an appropriate change of variables) system of non-linear evolution equations. It is demonstrated the existence of localized solutions (dromions, lumps, ring solitons and breathers) as well as of multiple instanton solutions. The interaction between the localized solutions are completely elastic, because they pass through each other and preserve their shape, the only change being a phase shift.