A two-component generalization of the reduced Ostrovsky equation and its integrable semi-discrete analogue

TL;DR

This paper introduces a two-component generalization of the reduced Ostrovsky equation, explores its integrability via Lax pairs, derives soliton solutions, and constructs an integrable semi-discrete analogue with explicit solutions.

Contribution

It presents a novel two-component reduced Ostrovsky equation, its bilinear form, soliton solutions, and an integrable semi-discrete version derived through a discrete hodograph transform.

Findings

The two-component reduced Ostrovsky equation is integrable and admits explicit soliton solutions.

An extended BKP hierarchy underpins the derivation of the equation and its solutions.

An integrable semi-discrete analogue of the equation is constructed with explicit N-soliton solutions.

Abstract

In the present paper, we propose a two-component generalization of the reduced Ostrovsky 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 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 B\"acklund transformation of the above extended BKP hierarchy, an integrable semi-discrete analogue of two-component reduced Ostrovsky equation is constructed by defining an appropriate discrete hodograph transform and dependent variable transformations. Especially, 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.