A 4 × 4 Matrix Spectral Problem Involving Four Potentials and Its Combined Integrable Hierarchy

This paper introduces a specific matrix spectral problem involving four potentials and derives an associated soliton hierarchy using the zero-curvature formulation. The bi-Hamiltonian formulation is derived via the trace identity, thereby establishing the hierarchy’s Liouville integrability. This is exemplified through two systems: generalized combined NLS-type equations and modified KdV-type equations. Owing to Liouville integrability, each member of the hierarchy admits a bi-Hamiltonian structure and, consequently, possesses infinitely many symmetries and conservation laws.