Continuous limit, exact rational solutions and combined rational-exponential solutions for a generalized Toda lattice equation with four potentials

Abstract The investigation centers on a generalized Toda lattice equation featuring four potentials. Initially, the continuous counterpart is first established using the continuous limit approach. Subsequently, leveraging its 3 × 3 matrix spectrum problem, the discrete generalized ( m , 3 N − m )-fold Darboux transformation (DT) is formulated for this discrete system. Finally, the derived generalized DT is employed to generate precise rational solutions and hybrid rational-exponential solutions, which are thoroughly analyzed and graphically illustrated. These novel findings may offer insights into the lattice dynamical phenomena described by Toda lattice equation.