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. | Turk J Math (2016) 40: 1237 – 1245 ¨ ITAK ˙ c TUB ⃝ Turkish Journal of Mathematics doi: Research Article Perturbational self-similar solutions for the 2-component Degasperis–Procesi system via a characteristic method Hongli AN1,∗, Ka-Luen CHEUNG2 , Manwai YUEN2 College of Sciences, Nanjing Agricultural University, Nanjing, . China 2 Department of Mathematics and Information Technology, The Hong Kong Institute of Education, Tai Po, New Territories, Hong Kong 1 Received: • Accepted/Published Online: • Final Version: Abstract: 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. Key words: 2-Component Degasperis–Procesi system, special transformation, characteristic method, perturbational solution 1. Introduction In the present work, we consider the following 2-component Degasperis–Procesi shallow water system: { ρt + k2 uρx + (k1 + k2 )ρux = 0, ut − uxxt + 4uux − 3uuxx − uuxxx + k3 ρρx = 0, (1) where k1 , k2 , and k3 are constants. It is noted that when ρ = 0 system (1) reduces to the classical Degasperis– Procesi (DP) equation, which has been extensively studied by many authors (see references [6-8, 10, 13, 19, 20, 22, 31, 41]). For example, Degasperis et al. proved that the DP model was integrable [7]. Lundmark et al. showed that such a system allowed multipeakon solutions [22]. Later, Lin and Liu proved that such peakon solutions were stable under small perturbations [19]. Zhou and Liu et al. .
