Compacton Solutions of the Korteweg–de Vries Equation with Constrained Nonlinear Dispersion

Abstract

The numerical solution of initial value problems is used to obtain compacton and kovaton solutions of K(f ^m, g ⁿ) equations generalizing the Korteweg–de Vries K(u², u¹) and Rosenau–Hyman K(u ^m, u ⁿ) equations to more general dependences of the nonlinear and dispersion terms on the solution u. The functions f(u) and g(u) determining their form can be linear or can have the form of a smoothed step. It is shown that peakocompacton and peakosoliton solutions exist depending on the form of the nonlinearity and dispersion. They represent transient forms combining the properties of solitons, compactons, and peakons. It is shown that these solutions can exist against an inhomogeneous and nonstationary background.