Perturbative and reversible systems approaches to new families of embedded solitary waves of a generalized fifth-order Korteweg–de Vries equation

Abstract

An earlier approach based on soliton perturbation theory is significantly generalized to obtain an analytical formula for the tail amplitudes of nonlocal solitary waves of a perturbed generalized fifth-order Korteweg–de Vries equation. On isolated curves in the (dispersion, wavespeed) parameter space, these tail amplitudes vanish, thus producing families of localized embedded solitons in large regions of the space. Off these curves, the tail amplitudes of the nonlocal waves are shown to be exponentially small in the small wavespeed limit. These seas of delocalized solitary waves, and the localized solitons embedded on the isolated curves within them, are shown to be entirely distinct from those derived in earlier work for a particular fixed value of the dispersion parameter. These perturbative results are also discussed within the framework of known reversible system results for various families of homoclinic orbits of the corresponding traveling-wave ordinary differential equation of our generalized FKdV equation, which correspond to solitary waves of the original NLPDE. In this setting too, the new families of solitary waves derived here are found to be distinct from those found in a similar earlier treatment for a specific fixed value of the dispersion parameter, and based on a different soliton solution of the unperturbed equation.