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

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 double-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.