Classification of the travelling waves in the nonlinear dispersive KdV equation

In this paper, we classify the travelling wave solutions to the nonlinear dispersive KdV equation (called K(2, 2) equation). The parameter region is specified and the parameter dependence of its solitary waves is described. Besides the previously known compacton solutions, the equation is shown to admit more new solutions such as cuspons, peakons, loopons, stumpons and fractal-like waves. Furthermore, by the qualitative results, we give some new explicit travelling wave solutions.