General soliton solutions to the coupled Hirota equation via the Kadomtsev-Petviashvili reduction

Abstract

In this paper, we are concerned with various soliton solutions to the coupled Hirota equation, as well as to the Sasa-Satsuma equation which can be viewed as one reduction case of the coupled Hirota equation. First, we derive bright-bright, dark-dark, and bright-dark soliton solutions of the coupled Hirota equation by using the Kadomtsev-Petviashvili reduction method. Then, we present the bright and dark soliton solutions to the Sasa-Satsuma equation which are expressed by determinants of $N \times N$ instead of $2N \times 2N$ in the literature. The dynamics of first-, second-order solutions are investigated in detail. It is intriguing that, for the SS equation, the bright soliton for \(N=1\) is also the soliton to the complex mKdV equation while the amplitude and velocity of dark soliton for \(N=1\) are determined by the background plane wave. For \(N=2\), the bright soliton can be classified into three types: oscillating, single-hump, and two-hump ones while the dark soliton can be classified into five types: dark (single-hole), anti-dark, Mexican hat, anti-Mexican hat and double-hole. Moreover, the types of bright solitons for the Sasa-Satsuma equation can be changed due to collision.