Communications in Mathematical Sciences

Volume 21 (2023)

Number 6

Riemann–Hilbert problem for the focusing Hirota equation with counterpropagating flows

Pages: 1625 – 1654

DOI: https://dx.doi.org/10.4310/CMS.2023.v21.n6.a9

Authors

Xiao-Fan Zhang (School of Mathematics, China University of Mining and Technology, Xuzhou, China)

Shou-Fu Tian (School of Mathematics, China University of Mining and Technology, Xuzhou, China)

Jin-Jie Yang (School of Mathematics, China University of Mining and Technology, Xuzhou, China)

Zhi-Qiang Li (School of Mathematics, China University of Mining and Technology, Xuzhou, China)

Abstract

The focusing Hirota equation is analyzed with a general initial condition via the inverse scattering transform, whose asymptotic behavior at infinity consists of counterpropagating waves. According to some necessary conditions, including jump condition, normalization condition, residue conditions and suitable growth condition near the branch points, the inverse problem is transformed into a matrix Riemann–Hilbert (RH) problem jumping along the branch cuts and real axis, the problem is transformed into a set of linear algebraic integral equations, and the reconstruction formula of potential is successfully obtained. In addition, the zero point of the analytical scattering coefficient on the continuous spectrum is placed on a sufficiently large circle, so a modified piecewise analytical RH problem is further successfully constructed. Finally, the exact expressions of soliton solution and breathing solution of focusing Hirota equation under degenerate initial value conditions are discussed.

Keywords

focusing Hirota equation, Riemann–Hilbert problem, counterpropagating flows

2010 Mathematics Subject Classification

35C08, 35Q15, 35Q51

Full Text (PDF format)

This work was supported by the National Natural Science Foundation of China under Grant No. 11975306, the Natural Science Foundation of Jiangsu Province under Grant No. BK20181351, the Six Talent Peaks Project in Jiangsu Province under Grant No. JY-059, the 333 Project in Jiangsu Province, the Fundamental Research Fund for the Central Universities under the Grant No. 2019ZDPY07, the Graduate Innovation Program of the China University of Mining and Technology under Grant No. 2023WLKXJ118, and the Postgraduate Research & Practice Innovation Program of Jiangsu Province under Grant No. KYCX-232644.

Received 4 July 2022

Received revised 24 November 2022

Accepted 4 December 2022

Published 22 September 2023