A two-component generalization of the reduced Ostrovsky equation and its integrable semi-discrete analogue. (4th January 2017)

• Record Type: Journal Article
• Authors: Feng, Bao-Feng; Maruno, Ken-ichi; Ohta, Yasuhiro
• Is Part Of: Journal of physics. Volume 50:Number 5(2017)
• Publication Date: 2017-01-04
• Subjects: BKP and modified BKP hierarchy -- pseudo 3-reduction -- hodograph and discrete hodograph transform -- two-component reduced Ostrovsky equation -- short wave model of two-component Degasperis–Procesi (DP) equation -- integrable discretization; Mathematical physics -- Periodicals; Statistical physics -- Periodicals; Quantum theory -- Periodicals; Matter -- Properties -- Periodicals; 530.105
• DOI: 10.1088/1751-8121/50/5/055201
• Languages: English
• ISSNs: 1751-8113

Abstract

Abstract: In the present paper, we propose a two-component generalization of the reduced Ostrovsky (Vakhnenko) equation, whose differential form can be viewed as the short-wave limit of a two-component Degasperis–Procesi (DP) equation. They are integrable due to the existence of Lax pairs. Moreover, we have shown that the two-component reduced Ostrovsky equation can be reduced from an extended BKP hierarchy with negative flow through a pseudo 3-reduction and a hodograph (reciprocal) transform. As a by-product, its bilinear form and N -soliton solution in terms of pfaffians are presented. One- and two-soliton solutions are provided and analyzed. In the second part of the paper, we start with a modified BKP hierarchy, which is a Bäcklund transformation of the above extended BKP hierarchy, an integrable semi-discrete analogue of the two-component reduced Ostrovsky equation is constructed by defining an appropriate discrete hodograph transform and dependent variable transformations. In particular, the backward difference form of above semi-discrete two-component reduced Ostrovsky equation gives rise to the integrable semi-discretization of the short wave limit of a two-component DP equation. Their N -soliton solutions in terms of pffafians are also provided.