### 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 Bcklund 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.

Original language | English |
---|---|

Article number | 055201 |

Journal | Journal of Physics A: Mathematical and Theoretical |

Volume | 50 |

Issue number | 5 |

DOIs | |

Publication status | Published - 2017 Jan 4 |

### Fingerprint

### Keywords

- BKP and modified BKP hierarchy
- hodograph and discrete hodograph transform
- integrable discretization
- pseudo 3-reduction
- short wave model of two-component Degasperis Procesi (DP) equation
- two-component reduced Ostrovsky equation

### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Statistics and Probability
- Modelling and Simulation
- Mathematical Physics
- Physics and Astronomy(all)

### Cite this

**A two-component generalization of the reduced Ostrovsky equation and its integrable semi-discrete analogue.** / Feng, Bao Feng; Maruno, Kenichi; Ohta, Yasuhiro.

Research output: Contribution to journal › Article

*Journal of Physics A: Mathematical and Theoretical*, vol. 50, no. 5, 055201. https://doi.org/10.1088/1751-8121/50/5/055201

}

TY - JOUR

T1 - A two-component generalization of the reduced Ostrovsky equation and its integrable semi-discrete analogue

AU - Feng, Bao Feng

AU - Maruno, Kenichi

AU - Ohta, Yasuhiro

PY - 2017/1/4

Y1 - 2017/1/4

N2 - 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 Bcklund 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.

AB - 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 Bcklund 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.

KW - BKP and modified BKP hierarchy

KW - hodograph and discrete hodograph transform

KW - integrable discretization

KW - pseudo 3-reduction

KW - short wave model of two-component Degasperis Procesi (DP) equation

KW - two-component reduced Ostrovsky equation

UR - http://www.scopus.com/inward/record.url?scp=85010060695&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85010060695&partnerID=8YFLogxK

U2 - 10.1088/1751-8121/50/5/055201

DO - 10.1088/1751-8121/50/5/055201

M3 - Article

AN - SCOPUS:85010060695

VL - 50

JO - Journal of Physics A: Mathematical and Theoretical

JF - Journal of Physics A: Mathematical and Theoretical

SN - 1751-8113

IS - 5

M1 - 055201

ER -