Interaction of a moving dipole with a soliton in the KdV equation

Abstract

The Korteweg-de Vries equation with the perturbing term εδ'( x − Vt) (a point-like dipole), which models disturbances produced by a small body moving in a liquid layer, is considered. In the case V<0, when the moving dipole emits a quasi-linear monochromatic wave, perturbation of the emission spectrum due to collision of the dipole with a free soliton is investigated. It is demonstrated that prior to the collision (at ft → − ∞) the resultant spectrum's width is exponentially small in ∣ t∣, while after the collision (at t → + ∞) the width is ∾ t −1. Then it is demonstrated that in the case V>0 (a non-emitting dipole) a soliton may be pinned by the moving dipole. In the adiabatic approximation, the pinned state is stable provided ε < 0. In this case a pair of solitons may also be pinned by the dipole, but that pinned state is unstable. Other types of solitary pinned profiles and their stability are discussed. Oscillations of a soliton near the adiabatically stable pinned state are accompanied by emission of quasi-linear waves. The emission intensity is calculated in a general form, and it is demonstrated that the oscillation are subject to radiative instability due to the fact that the energy of the system is not positive definite. The same model is considered with the Bürgers' dissipative term. The dissipation may compensate the radiative instability and render the pinned state of a soliton completely stable. Besides, it is demonstrated that the Bürgers' term gives rise to multisoliton pinned profiles. A maximum possible number of solitons in the profile is found.