Quasi-solitons as nonstationary self-similar solutions of the Korteweg-de Vries-Burgers equation

The Korteweg-de Vries-Burgers equation (KdV-B) is studied analytically and numerically for weak dissipation in the nonstationary case (self-similar solutions). Starting the analytical study from the system (S) of two kinetic equations equivalent to KdV-B equation in the exact adiabatic approximation of a symmetrically damped solitary pulse, we renormalize this system by taking into account the asymmetry produced by the damping (tail) and obtain a simple mechanical picture describing qualitatively and quantitatively the properties of the tailed quasi-solitons: the leading pulse is described by an oscillation in a potential well, while the asymmetry is obtained from a transition between two nearby energy levels located at the top of the well. The superposition properties of the quasi-solitons are discussed in the adiabatic approximation, using a Bäcklund transformation directly deduced from system (S). All these analytical results are checked by the numerical study which shows in particular that the tail, once built, becomes completely disconnected from its “mother” soliton and behaves like noise.