We investigate the control of solitonic interactions and bound-state formations in the higher-order matrix nonlinear Schrödinger equation with the sign-alternating nonlinearity and third-order dispersion. By employing the binary Darboux transformation, we construct explicit N-soliton solutions from a zero seed background. Adjusting these parameters l_jk(j,k=1,2,3,4) enables systematic generation of various bound-state soliton structures.
For the second-order case (N=2), we obtain several distinct configurations depending on the relations between the matrices C_1 and C_2 (constructed from l_jk) and their determinants. When C_1\neq C_2 with \det(C_2)=i\det(C_1) and \det(C_1C_2)=-i, the components q_1 and q_2 exhibit breathing solitons. Varying the spectral parameters \lambda_j transforms them into the parallel breathing solitons with different amplitudes. For C_1\neq C_2 with \det(C_2)=i\det (C_1) and \det(C_1C_2)=i, the components become linearly independent, yielding a bound state of one ordinary soliton and one breathing soliton. When C_1=C_2 with \det(C_1C_2)=-1, the solitons propagate with the identical amplitude and velocity while displaying periodic attraction and repulsion, which is a typical soliton molecule behavior accompanied by the energy redistribution at the collision points. For C_1=C_2 with \det(C_1C_2)=\frac54-3i, we find the double-peaked bound-state solitons in q_1 and single-peaked intertwined structures in q_2.
In the third-order case (N=3), we explore seven parameter regimes leading to even more complex composite states. These include: one soliton coexisting with two breathing solitons (the elastic interaction with the constant velocity), one soliton with two ordinary solitons exhibiting phase shifts after the collision, one soliton with two double-peaked solitons where the amplitude changes indicate the energy redistribution, and one soliton with two separated double-peaked solitons where both shape and separation are modified by the energy exchange. In all cases, the binary Darboux transformation ensures that the solutions are exact and exhibit either elastic or inelastic interactions characterized by the phase shifts, amplitude modulation, and periodic energy transfer among components.
To verify the robustness of the obtained solutions, we perform the numerical stability tests by adding 10% Gaussian noise to the analytical profile of the breathing soliton at different propagation distances. The shape fidelity remains high, confirming the physical reliability of these structures under small perturbations.
The results provides a theoretical foundation for manipulating multi-component soliton states in conservative nonlinear systems and offers insights into the design of all optical logic devices or soliton-based communication links where controlled interactions are essential.