On hidden solitons in KdV related systems

Abstract

It is well known that a train of solitons can emerge from the harmonic initial wave in case of the Korteweg–de Vries (KdV) or KdV related evolution equations. Analysis of results of numerical experiments demonstrates that besides clearly visible solitons, which interact with each other, there exist also solitons that either are visible only for a short time due to the fluctuation of the reference level or can be detected only by their influence on other solitons, i.e., by specific changes in the amplitude curves and in the soliton trajectories. Recently I. Christov demonstrated that for integrable PDEs, like the KdV equation, one can apply the periodic inverse scattering transform (PIST) and distinguish “true” soliton modes and modes that fall “in-between” solitons and radiation. However, in nonintegrable cases one cannot apply the PIST and therefore distinction between “true” solitons and nonlinear waves that have solitonic behavior seems to be complicated or even impossible. The existence of hidden solitons in the KdV related systems is discussed in the present paper.