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In this paper, the two-component Degasperis–Procesi system arising in shallow water theory is investigated. By using a special transformation and the characteristic method, a class of perturbational self-similar solutions is constructed. Such solutions are not only more general than those obtained by Yuen in 2011, but also they may have potential applications in the modeling of tsunamis. In addition, the method proposed can be extended to other mathematical physics models like the two-component Camassa–Holm equations.

Perturbational self-similar solutions for the 2-component Degasperis--Procesi system via a characteristic method