Non-Analytic Solutions of Nonlinear Wave Models
Yi A. Li, Peter J. Olver, Philip Rosenau
Abstract
Historically, the mathematical study of physical phenomena has proceeded in three phases. The introductory phase relies on linear models, and is a necessary prerequisite to the deeper understanding of the actual nonlinear regime. Since different wave numbers propagate at different speeds, localized initial data will disperse in time, albeit with conservation of energy. Weakly nonlinear models cannot describe the fully nonlinear, real world phenomena such as wave breaking, shocks, waves of maximal height, large amplitude disturbances, and so on. The classical, analytic traveling wave solutions are given by classical analytic orbits to the dynamical system. The admission of nonanalytic traveling wave solutions indicates that classical dynamical systems analysis must be significantly broadened to explain how a classical analytic orbit can give rise to a nonanalytic solution.