Dynamics of mixed breathers in the higher-order matrix nonlinear Schrödinger equation

In this work, we study the mixed breather dynamics in the higher-order matrix nonlinear Schrödinger equation. Via the Darboux transformation (DT), we first construct first-order breather solutions and subsequently generalise them to second- and third-order mixed breather solutions incorporating two spectral parameters through a generalised DT. By means of those solutions, we analyse the dynamics of second- and third-order mixed breathers arising from the nonlinear superposition of distinct vector breathers. We demonstrate how double spectral parameters modulate the second-order mixed breathers, highlighting amplitude-dependent behaviours and spectral-parameter-controlled waveform transitions in multi-component nonlinear systems. In addition, we present the third-order mixed breathers, denoting the superposition of a vector breather and a vector degenerate breather.