Evolution of Benjamin-Ono solitons in the presence of weak Zakharov-Kutznetsov lateral dispersion

Abstract

The effect of weak lateral dispersion of Zakharov-Kutznetsov-type on a Benjamin-Ono solitary wave is studied both asymptotically and numerically. The asymptotic solution is based on an approximate variational solution for the solitary wave, which is then modulated in time through the use of conservation equations. The effect of the dispersive radiation shed as the solitary wave evolves is also included in the modulation equations. It is found that the weak lateral dispersion produces a strongly anisotropic, stable solitary wave which decays algebraically in the direction of propagation, as for the Benjamin-Ono solitary wave, and exponentially in the transverse direction. Moreover, it is found that initial conditions with amplitude above a threshold evolve into solitary waves, while those with amplitude below the threshold evolve as lumps for a short time, then merge into radiation. The modulation equations are found to give a quantitatively accurate description of the evolution of an initial condition into an anisotropic solitary wave. The existence of stable solitary waves is in contrast to previous studies of Benjamin-Ono-type equations subject to the stronger Kadomstev-Petviashvili or Benjamin-Ono-type lateral dispersion, for which the solitary waves either decay or collapse. The present study then completes the catalog of possible behaviors under lateral dispersion.