Singular Dynamics with Application to Singular Waves in Physical Problems

Abstract

Many physical problems are governed by certain systems of singular ordinary differential equations (ODEs). As a result, singular solutions can arise. In this paper, we provide some theoretical results to deal with these solutions. By applying straightforwardly the notion of weak solutions for partial differential equations (PDEs), a notion of weak solutions for this ODE type is proposed to include these singular solutions. As an application, we consider some traveling-wave solutions of a nonlinear dispersion equation arising in a physical problem. It is shown that compactons can arise in nonlinear elastic rods.