Compactons and Riemann Waves of an Extended Modified Korteweg–de Vries Equation with Nonlinear Dispersion

Abstract

The K(f^m, gⁿ) equation is studied, which generalizes the modified Korteweg–de Vries equation K(u³, u¹) and the Rosenau–Hyman equation K(u^m, uⁿ) to other dependences of nonlinearity and dispersion on the solution. The considered functions f(u) and g(u) can be linear or can have the form of a smoothed step. It is found numerically that, depending on the form of nonlinearity and dispersion, the given equation has compacton and kovaton solutions, Riemann-wave solutions, and oscillating wave packets of two types. It is shown that the interaction between solutions of all found types occurs with the preservation of their parameters.