Two types of elliptic dark soliton solutions for the Hirota equation

We primarily study concave-downward and convex-upward types of elliptic dark soliton solutions for the Hirota equation, exhibiting a concave-downward shape on both upper and lower envelope surfaces and showing a convex-upward shape on the lower envelope surface, respectively. By analyzing the supremum and infimum of solutions, we provide the existence conditions for these two types of elliptic dark solitons. Additionally, we study two-elliptic dark soliton solutions combining both types with the same velocity and investigate the elastic collisions between these two types of solutions with different velocities.