Mixed Soliton Solutions of MNLS/DNLS Equations Based on Hirota Method

Using Hirota’s bilinear derivative method to derive single-breather solutions for the modified nonlinear Schrödinger (MNLS) equation and the derivative nonlinear Schrödinger (DNLS) equation under non-vanishing boundary conditions, along with explicit mixed solutions combining breather-type and pure solitons. The collision dynamics between pure and breather-type solitons in a mixed solution has been graphically demonstrated and analyzed. Furthermore, by setting specific parameter to zero, we naturally obtain corresponding single-breather solution and its explicit mixed solutions with pure solitons for the DNLS equation. The mixed soliton solution can asymptotically degenerate into a simple algebraic summation of a simple pure soliton and a breather in the infinite past or the infinite future, which was graphically validated.