Nonlinear single degree of freedom system with dynamic responses of smooth, non-smooth, and distorted travelling waves

摘要

In this paper, a Hamiltonian invariant is established for a nonlinear single degree of freedom system, and subsequent criterions are set for various types of travelling wave solutions (i.e. smooth, non-smooth and distorted waves). Using these established criterions, various solutions, such as the periodic wave, soliton, solitary cusp wave, flat-roof soliton, loop soliton and singular-cliff soliton solutions, are obtained. The evolution from a flat-bottom soliton to a flat-plane loop soliton, soliton-like wave with non-smooth bottom, singular-cliff soliton and, finally, flat-roof soliton is also illustrated. Using the proposed criterions, the dynamic behaviour and certain strange phenomena of the new periodic loop solitons can be easily understood and explained, especially for the quasi solitary wave solution which is newly discovered.